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2202009
1
PHYS202 ndash SPRING 2009Lecture notes ‐ Electrostatics
What is Electricity
bull Static effects known since ancient times
bull Static charges can be made by rubbing certain materials together
bull Described by Benjamin Franklin as a fluidndash Excess of the fluid was described as positive
Deficit of the fluid was described as negativendash Deficit of the fluid was described as negative
bull Like charges repel unlike charges attract
2202009
2
A Quantitative Approach
bull Charles Augustin de Coulomb first described relationship of force and charges
bull Coulomb invented the torsion balance to measure the force between charges
bull Basic unit of charge in SI system is called the Coulomb
bull One of the basic irreducible measurements
ndash Like mass length and time
Fundamental Charge
bull Due to atomic structure
bull Positively charged heavy nuclei and negative charged electronsPositively charged heavy nuclei and negative charged electrons
bull Charged objects have excess or deficit of electrons
bull Protons and electrons have equal and oppositecharge e = 1602 x 10‐19 C
bull Rubbing some objects causes electrons tomove from one object to another
bull Negatively charged objects have an excessof electrons
bull Positively charged objects have a deficit ofelectrons
bull Net change in charge in a closed system mustbe zero
2202009
3
Fundamental Charge
Problem 186 ‐Water has a mass per molecule of 18 gmol and each molecule has 10 electrons How many electrons are ineach molecule has 10 electrons How many electrons are in one liter of water
h i h h f h l
23 3
26
10electrons 6022times10 molecules 1mol 1g 10 mltimes times1molecule 1mol 18g 1ml 1liter
electrons33 10liter
N
N
= times times
= times
What is the net charge of the electrons
( )( )26 19
7
33 10 electrons 1602 10 C electron
54 10 C
q Ne
q
minus= = times minus times
= minus times
Conservation of Charge
bull Net change in charge in any closed system is zero
bull Or the change of charge in any volume equals the net charge entering andOr the change of charge in any volume equals the net charge entering and leaving the system
bull We denote the amount of charge by the letter Q or q
bull If Charge is created it must be equal and opposite
( ) ( ) ( ) ( )Charge Original Charge Charge in Charge out= + minus
0 in out= + -q q q q
2202009
4
Conservation of Charge
Two identical metal spheres of charge 50 C and ‐16 C are brought together and then separated What is the charge onbrought together and then separated What is the charge on each sphere
Since the charge is distributed equally each must have an equal charge
1 2 = 34Ctotalq q q= +
equal charge
1 2 = 17C2totalqq q= =
Conductors and Insulators
bull Insulatorndash Charge stays where it is depositedCharge stays where it is depositedndash Wood mica Teflon rubber
bull Conductorndash Charge moves ldquofreelyrdquo within the objectndash Metals exhibit high conductivity
bull Due to atomic structurendash Is the outermost electron(s) tightly bound
2202009
5
Charging Objects
bull By frictionndash Rubbing different materialsndash Rubbing different materialsndash Example fur and ebonite
bull By contactndash Bringing a charged object into contact with another (charged or uncharged) object
bull By inductionh d b d llndash Bringing a charged object near a conductor temporally
causes opposite charge to flow towards the charged object
ndash Grounding that conductor ldquodrainsrdquo the excess charge leaving the conductor oppositely charged
Force Between Charges
bull Coulomb found that the force between two charges isbetween two charges is proportional to the product of their charges and inversely proportion to the square of the distance between them
1 22
q qF prop
Where q1 and q2 represent the charges and r12 the distance between them
212r
2202009
6
Coulombrsquos Law
bull By using the torsion balance Coulomb discovered the constant of proportionality k = 899 Nm2C2
bull If both q1 and q2 are both positive the force is positive
1 22
12
q qF kr
=+ +
‐ ‐
‐ +
bull If both q1 and q2 are both negative the force is positive
bull If both q1 is positive and q2 is negative the force is negative
bull Hence the sign of the force is positive for repulsion and negative for attraction
Coulombrsquos LawPrinciple of Superposition
bull Electric force obeys the law of linear superposition ndash the net force on a charge is the vector sum of the forces due to each charge
bull Add the vectors
2ˆi j
i iji j ij
q qk
rne
= sumF rr
1
3
The net force on charge4 is F14+F24+F34=Fnet
Red are positive chargesBlue are negative charges
1
24
F14F24
F34
2202009
7
Coulombrsquos Law Principle of Superposition
bull The forces add for as many charges as you have
2ˆi j
i iji j ij
q qk
rne
= sumF rr
36
Red are positive chargesBlue are negative charges
1
2
3
4
5
7
8
Coulombrsquos Law Principle of Superposition
bull Electric force obeys the law of linear superposition ndash the net force on a charge is the vector sum of the forces due to each charge
2ˆi j
i iji j ij
q qk
rne
= sumF rr
36 The net force on charge 5 is the
vector sum of all the forces on
1
2
3
4
5
7
8charge 5
Red are positive chargesBlue are negative charges
2202009
8
Coulombrsquos Law1814 ndash Two tiny conducting spheres are identical and carry charges of ‐200 μC and
+500 μC They are separated by a distance of 250 cm (a) What is the magnitude of the force that each sphere experiences and is the force attractive or repulsive
( )( )( )( )
9 2 2 6 61 2
22 2
899 10 N m C 200 10 C 500 10 Cq qF kminus minustimes sdot minus times times
= =
The net force is negative so that the force is attractive
(b) The spheres are brought into contact and then separated to a distance of 250 cm Determine the magnitude of the force that each sphere now experiences and state whether the force is attractive or repulsiveSi h h id i l h h h f b i d i
( )22 212
4
250 10 m
144 10 N
r
F
minustimes
= minus times
Since the spheres are identical the charge on each after being separated is one‐half the net charge so q1 + q2 = +15μC
The force is positive so it is repulsive
( )( )( )( )
9 2 2 6 631 2
2 2212
899 10 N m C 150 10 C 150 10 C324 10 N
250 10 m
q qF k
r
minus minus
minus
times sdot times times= = = times
times
Coulombrsquos Law1821 ndash An electrically neutral model airplane is flying in a horizontal circle on a 30 m
guideline which is nearly parallel to the ground The line breaks when the kinetic energy of the plane is 500 J Reconsider the same situation except that now there is a point charge of +q on the plane and a point charge of minusq at the other end of h id li I hi h li b k h h ki i f h l ithe guideline In this case the line breaks when the kinetic energy of the plane is 518 J Find the magnitude of the charges
From the definition of kinetic energy we see that mv2 = 2(KE) so that Equation 53 for the centripetal force becomes
( )2
c2 KEmF
r r= =υ
Sor r
( )( )
( )( )
2charged uncharged
max max2Centripetal
Centripetal forceforce
2 KE 2 KE1 2
k qT T
r rr+ = =
1424314243
2202009
9
Coulombrsquos Law1821 ndash (cont)
( )( )
( )( )
2charged uncharged
max max2Centripetal
Centripetal forcef
2 KE 2 KE1 2
k qT T
r rr+ = =
1424314243
Subtracting Equation (2) from Equation (1) eliminates Tmax and gives
( ) ( )2charged uncharged
2
2 KE ndash KEk qrr
⎡ ⎤⎣ ⎦=
force
Solving for |q| gives
( ) ( ) ( ) ( )charged uncharged ndash59 2 2
2 KE ndash KE 2 30 m 518 J ndash 500 J35 10 C
899 10 N m C
rq
k
⎡ ⎤⎣ ⎦= = = times
times sdot
The Electric Fieldbull Electric charges create and exert and experience the electric force
q q q
bull We define the Electric Field as the force a test charge would feel if placed at that point
2 2ˆ ˆi j j
i ij i i iji j i jij ij
q q qk q k
r rne ne
rArr= =sum sumF r F rr r
bull The electric field exists around all charged objects
2ˆj
i j
qk q
rne
= rArr =sumE r F Er r r
2202009
10
Electric Force Lines
bull Michael Faraday saw the force caused by one charge to becaused by one charge to be from lines emanating from the charges
bull Lines begin on Positive charges and end on Negative charges flowing smoothly through space never crossing
Electric Force Lines
bull Faraday imagined these lines d h fand the force was greater
where the lines are denser weaker where they are spread out
2202009
11
Electric Field Lines
bull Go out from positive charges and into ti hnegative charges
Positive Negative Neutral
The Inverse Square Law
bull Many laws based on the geometry of three‐di i ldimensional space
1 22
2 24 4
g
sound light
Gm mFr
P LI Ir rπ π
= minus
= =
1 22e
q qF kr
=
2202009
12
The Electric Field
1828 ndash Four point charges have the same magnitude of 24 x 10‐12 C and are fixed to the corners of a square that is 40 cm on a side Three of the charges are positive d i ti D t i th it d f th t l t i fi ld th t i tand one is negative Determine the magnitude of the net electric field that exists
at the center of the square
A
B C
D-
++
+
EA
EC
EB
ED
The drawing at the right shows each of the field contributions at the center of the square (see black dot) Each is directed along a diagonal of the square Note that ED and EB point in opposite directions and therefore cancel since they have the same magnitude In contrast EA and EC point in the same direction toward corner A and therefore combine to give a net field thatcorner A and therefore combine to give a net field that is twice the magnitude of EA or EC In other words the net field at the center of the square is given by the following vector equation
2
Σ = + + +
= + + minus
= +
=
A B C D
A B C B
A C
A
E E E E EE E E EE EE
The Electric Field
1828 ndash (cont)
B C++
ED
The magnitude of the net field is A 22 2k q
E Er
Σ = =
A D- +
EA
EC
EB
In this result r is the distance from a corner to the center of the square which is one half of the diagonal distance d Using L for the length of a side of the square and taking advantage of the Pythagorean theorem we have With this substitution for r the magnitude of the net field becomes
r
2r L=
4k k
( )( ) ( )
( )
2 2
9 2 2 12
2
42
2
4 899 10 N m C 24 10 C
0040 m
54 NC
k q k qE
LL
E
E
minus
Σ = =
times sdot timesΣ =
Σ =
2202009
13
Electric Field Lines
1834 ndash Review Conceptual Example 12 before attempting to work this problem The magnitude of each of the charges in Figure 18‐21 is 86 x 10‐12 C The lengths of th id f th t l 3 00 d 5 00 Fi d th it d f ththe sides of the rectangles are 300 cm and 500 cm Find the magnitude of the electric field at the center of the rectangle in Figures 18‐21a and b
1 2
34
1
4
+ q
+ q + q
- qThe figure at the right shows the configuration given in text Figure 1821a The electric field at the center of the rectangle is the resultant of the electric fields at the center due to each of the four charges As discussed in Conceptual Example 11 the magnitudes of the electric field at the center due to each of the four charges are equal However the fields produced by the charges in corners 1 and 3 are in opposite directions Since they have the same magnitudes they combine to give directions Since they have the same magnitudes they combine to givezero resultant The fields produced by the charges in corners 2 and 4 point in the same direction (toward corner 2) Thus EC = EC2 + EC4 where EC is the magnitude of the electric field at the center of the rectangle and EC2 and EC4 are the magnitudes of the electric field at the center due to the charges in corners 2 and 4 respectively Since both EC2 and EC4 have the same magnitude we have EC = 2 EC2
Electric Field Lines1834 ndash (cont)
The distance r from any of the charges to the center of the rectangle can be found using the Pythagorean theorem
1 21
d500 cm
2 2(3 00 ) +(5 00 ) 5 83d
The figure at the right shows the configuration given in text Fi 18 21b All f h t ib t t
344
300 cm
θ
2 2(300 cm) +(500 cm) 583 cmd = =
2Therefore 292 cm 292 10 m2dr minus= = = times
9 2 2 1222
2 2 2 22 2(899 10 N m C )( 860 10 C)2 181 10 NC
(292 10 m)C Ckq
E Er
minus
minustimes sdot minus times
= = = = minus timestimes
- q - q
Figure 1821b All four charges contribute a non‐zero component to the electric field at the center of the rectangle As discussed in Conceptual Example 11 the contribution from the charges in corners 2 and 4 point toward corner 2 and the contribution from the charges in corners 1 and 3 point toward corner 1
1 2
34
1
4+ q + q
C
E 13 E 24
2202009
14
Electric Field Lines1834 ndash (cont)
Notice also the magnitudes of E24 and E13 are equal and from the first part of this problem we know that
E24 = E13 = 181 times 102 NC
1 21- q - q
E 13 E 24
The electric field at the center of the rectangle is the vector sum of E24 and E13 The x components of E24 and E13 are equal in magnitude and opposite in direction hence
(E13)x ndash (E24)x = 0
Therefore
F Fi 2 h th t
344+ q + q
C
C 13 24 13 13( ) ( ) 2( ) 2( )siny y yE E E E E θ= + = =
From Figure 2 we have that
500 cm 500 cmsin 0858583 cmd
θ = = =
( ) ( )( )2 2C 132 sin 2 181 10 NC 0858 311 10 NCE E θ= = times = times
The Electric Field
bull The electric field exists around all charged objects
bull This is shown byelectric fieldlines
bull Constructedid i ll hidentically to theforce lines
2202009
15
Drawing Electric Field Lines
bull Number of lines at a charge is proportional to the chargethe charge
Electric Field Lines
bull The charge with ‐3qmust have three times as many lines as the charge with +q and 15 times the lines for +‐ 2qthe charge with q and 15 times the lines for 2q
2202009
16
Parallel Plates of Charge
bull Adding fields from each point on surface causes field lines parallel to the plates to cancel
PositivelyCharged
NegativelyCharged
bull Away from the edges the field is parallel and perpendicular to the plate(s)
Electric Field amp Conductors
bull In conductors excess electrons will move when an electric field is present
bull For ldquosteady staterdquo (ie no moving charges) the net electric field must be zeror r r r
bull Therefore the charges move to the surface(s) of the conductor until they cancel the electric field inside the conductor
bull Field lines can start and end on a conductor at right angles to the surface of the conductor
0q rArr= = =F E F Er r r r
2202009
17
Electric Field Lines
1844 ndash Two particles are in a uniform electric field whose value is + 2500 NC The mass and charge of particle 1 are m1 = 14 x 10‐5 kg and q1 = ‐70 μC while the
di l f ti l 2 2 6 10 5 k d 18 C I iti llcorresponding values for particle 2 are m2 = 26 x 10‐5 kg and q2 = +18 μC Initially the particles are at rest The particles are both located on the same electric field line but are separated from each other by a distance d When released they accelerate but always remain at this same distance from each other Find d
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The drawing shows the two particles in the electric field Ex They are separated by a distance d If the particles are to stay at the same distance from each other after being released they must have the same acceleration so ax1 = ax2 According to Newtonrsquos second law the acceleration ax of each particle is equal to the net force ΣFx acting on it divided by its mass m or x xa F m= Σ
Electric Field Lines
1844 ndash (cont)
N t th t ti l 1 t b t th l ft f ti l 2 If ti l 1 t th i ht
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
Note that particle 1 must be to the left of particle 2 If particle 1 were to the right of particle 2 the particles would accelerate in opposite directions and the distance between them would not remain constant
1 2 1 1 2
1 21 2 1
11 1
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = minus
minusΣ= =
1 2 2 2 2
1 22 2 2
22 2
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = +
+Σ= =
Note the signs indicated the direction of the vector pointing from charge 2 to charge 1 for the force on charge 1
2202009
18
Electric Field Lines
1844 ndash (cont)
Th l ti f h h t b th if th di t b t th
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The acceleration of each charge must be the same if the distance between the charges remains unchanged
Solving for d
1 2
1 2 1 21 22 2
1 2
x x
x x
a a
q q q qq E k q E k
d dm m
=
minus +=
Solving for d
( )( )1 2 1 2
1 2 2 165 m
x
kq q m md
E q m q m+
= =minus
Electric Fluxbull Flux (from Latin ndash meaning to flow)
bull Electric flux measures amount of electric field passing through a surface
Er
Er
So to calculate find the normal to
φ
E EAΦ =
Flux through surface is Area of surface times the field flowing through that surface
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
( )cosE E AφΦ =
2202009
19
Electric Flux
( )cosE E AφΦ =
Er
So to calculate find the normal to
This can be expressed as the vector dot product
EΦ = sdotE Arr
cosE A φsdot =E Arr
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
cos
x x y y z z
E Aor
E A E A E A
φsdot =
sdot = + +
E A
E Arr
Electric Flux
1848 ndash A rectangular surface (016 m x 038m) is oriented in a uniform electric field of 580 NC What is the maximum possible electric flux through the surface
The maximum possible flux occurs when the electric field is parallel to the normal of the rectangular surface (that is when the angle between the direction of the field and the direction of the normal is zero) Then
( ) ( ) ( )2
( cos ) 580 N C (cos 0 ) 016 m 038 m
35 N m C E
E
E AφΦ = = deg
Φ = sdot
2202009
20
Gaussrsquo Lawbull Electric Flux through a closed surface is proportional to the charge enclosed by that surface
encE
qε
Φ =
bull The fundamental constant ε0 is called the permittivity of free space It is related to Coulombrsquos law by ε0 = 885 x 10‐12 C2(Nm2)
bull Gauss Law is most useful when the electric field on the entire surface is constant
bull Take a point charge Construct a sphere centered on the point charge (all points on the surface equidistant from the charge) ndash the electric field is constant
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so
0ε
0
14
kπε
=
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so φ = 0 cos φ = 1
bull The direction of the field is radially outward from the charge (or inward towards the charge if q is negative)
0 0
cosenc encE
q qEA φε ε
Φ =rArr=
( )22
0 0
44
encq qE r Er
πε πε
= rArr =
Gaussrsquo Law for a plate of chargebull In this case we take our closed surface to be a box oriented such that two faces are
parallel to the plate of charge and the others are perpendicular to the plate
Si th l t i fi ld f l t f h i di l t th f f thbull Since the electric field from a plate of charge is perpendicular to the surface of the plate the sides that are perpendicular to the plate have no flux
Er
bull The charge enclosed by the box is the surface charge density multiplied by the area the box passes through the plate Aσthe plate
bull The area of the box with flux is 2A
Aσ
0 0 0E
q AEA Eσ σε ε ε
rArr rArrΦ = = =
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
2
A Quantitative Approach
bull Charles Augustin de Coulomb first described relationship of force and charges
bull Coulomb invented the torsion balance to measure the force between charges
bull Basic unit of charge in SI system is called the Coulomb
bull One of the basic irreducible measurements
ndash Like mass length and time
Fundamental Charge
bull Due to atomic structure
bull Positively charged heavy nuclei and negative charged electronsPositively charged heavy nuclei and negative charged electrons
bull Charged objects have excess or deficit of electrons
bull Protons and electrons have equal and oppositecharge e = 1602 x 10‐19 C
bull Rubbing some objects causes electrons tomove from one object to another
bull Negatively charged objects have an excessof electrons
bull Positively charged objects have a deficit ofelectrons
bull Net change in charge in a closed system mustbe zero
2202009
3
Fundamental Charge
Problem 186 ‐Water has a mass per molecule of 18 gmol and each molecule has 10 electrons How many electrons are ineach molecule has 10 electrons How many electrons are in one liter of water
h i h h f h l
23 3
26
10electrons 6022times10 molecules 1mol 1g 10 mltimes times1molecule 1mol 18g 1ml 1liter
electrons33 10liter
N
N
= times times
= times
What is the net charge of the electrons
( )( )26 19
7
33 10 electrons 1602 10 C electron
54 10 C
q Ne
q
minus= = times minus times
= minus times
Conservation of Charge
bull Net change in charge in any closed system is zero
bull Or the change of charge in any volume equals the net charge entering andOr the change of charge in any volume equals the net charge entering and leaving the system
bull We denote the amount of charge by the letter Q or q
bull If Charge is created it must be equal and opposite
( ) ( ) ( ) ( )Charge Original Charge Charge in Charge out= + minus
0 in out= + -q q q q
2202009
4
Conservation of Charge
Two identical metal spheres of charge 50 C and ‐16 C are brought together and then separated What is the charge onbrought together and then separated What is the charge on each sphere
Since the charge is distributed equally each must have an equal charge
1 2 = 34Ctotalq q q= +
equal charge
1 2 = 17C2totalqq q= =
Conductors and Insulators
bull Insulatorndash Charge stays where it is depositedCharge stays where it is depositedndash Wood mica Teflon rubber
bull Conductorndash Charge moves ldquofreelyrdquo within the objectndash Metals exhibit high conductivity
bull Due to atomic structurendash Is the outermost electron(s) tightly bound
2202009
5
Charging Objects
bull By frictionndash Rubbing different materialsndash Rubbing different materialsndash Example fur and ebonite
bull By contactndash Bringing a charged object into contact with another (charged or uncharged) object
bull By inductionh d b d llndash Bringing a charged object near a conductor temporally
causes opposite charge to flow towards the charged object
ndash Grounding that conductor ldquodrainsrdquo the excess charge leaving the conductor oppositely charged
Force Between Charges
bull Coulomb found that the force between two charges isbetween two charges is proportional to the product of their charges and inversely proportion to the square of the distance between them
1 22
q qF prop
Where q1 and q2 represent the charges and r12 the distance between them
212r
2202009
6
Coulombrsquos Law
bull By using the torsion balance Coulomb discovered the constant of proportionality k = 899 Nm2C2
bull If both q1 and q2 are both positive the force is positive
1 22
12
q qF kr
=+ +
‐ ‐
‐ +
bull If both q1 and q2 are both negative the force is positive
bull If both q1 is positive and q2 is negative the force is negative
bull Hence the sign of the force is positive for repulsion and negative for attraction
Coulombrsquos LawPrinciple of Superposition
bull Electric force obeys the law of linear superposition ndash the net force on a charge is the vector sum of the forces due to each charge
bull Add the vectors
2ˆi j
i iji j ij
q qk
rne
= sumF rr
1
3
The net force on charge4 is F14+F24+F34=Fnet
Red are positive chargesBlue are negative charges
1
24
F14F24
F34
2202009
7
Coulombrsquos Law Principle of Superposition
bull The forces add for as many charges as you have
2ˆi j
i iji j ij
q qk
rne
= sumF rr
36
Red are positive chargesBlue are negative charges
1
2
3
4
5
7
8
Coulombrsquos Law Principle of Superposition
bull Electric force obeys the law of linear superposition ndash the net force on a charge is the vector sum of the forces due to each charge
2ˆi j
i iji j ij
q qk
rne
= sumF rr
36 The net force on charge 5 is the
vector sum of all the forces on
1
2
3
4
5
7
8charge 5
Red are positive chargesBlue are negative charges
2202009
8
Coulombrsquos Law1814 ndash Two tiny conducting spheres are identical and carry charges of ‐200 μC and
+500 μC They are separated by a distance of 250 cm (a) What is the magnitude of the force that each sphere experiences and is the force attractive or repulsive
( )( )( )( )
9 2 2 6 61 2
22 2
899 10 N m C 200 10 C 500 10 Cq qF kminus minustimes sdot minus times times
= =
The net force is negative so that the force is attractive
(b) The spheres are brought into contact and then separated to a distance of 250 cm Determine the magnitude of the force that each sphere now experiences and state whether the force is attractive or repulsiveSi h h id i l h h h f b i d i
( )22 212
4
250 10 m
144 10 N
r
F
minustimes
= minus times
Since the spheres are identical the charge on each after being separated is one‐half the net charge so q1 + q2 = +15μC
The force is positive so it is repulsive
( )( )( )( )
9 2 2 6 631 2
2 2212
899 10 N m C 150 10 C 150 10 C324 10 N
250 10 m
q qF k
r
minus minus
minus
times sdot times times= = = times
times
Coulombrsquos Law1821 ndash An electrically neutral model airplane is flying in a horizontal circle on a 30 m
guideline which is nearly parallel to the ground The line breaks when the kinetic energy of the plane is 500 J Reconsider the same situation except that now there is a point charge of +q on the plane and a point charge of minusq at the other end of h id li I hi h li b k h h ki i f h l ithe guideline In this case the line breaks when the kinetic energy of the plane is 518 J Find the magnitude of the charges
From the definition of kinetic energy we see that mv2 = 2(KE) so that Equation 53 for the centripetal force becomes
( )2
c2 KEmF
r r= =υ
Sor r
( )( )
( )( )
2charged uncharged
max max2Centripetal
Centripetal forceforce
2 KE 2 KE1 2
k qT T
r rr+ = =
1424314243
2202009
9
Coulombrsquos Law1821 ndash (cont)
( )( )
( )( )
2charged uncharged
max max2Centripetal
Centripetal forcef
2 KE 2 KE1 2
k qT T
r rr+ = =
1424314243
Subtracting Equation (2) from Equation (1) eliminates Tmax and gives
( ) ( )2charged uncharged
2
2 KE ndash KEk qrr
⎡ ⎤⎣ ⎦=
force
Solving for |q| gives
( ) ( ) ( ) ( )charged uncharged ndash59 2 2
2 KE ndash KE 2 30 m 518 J ndash 500 J35 10 C
899 10 N m C
rq
k
⎡ ⎤⎣ ⎦= = = times
times sdot
The Electric Fieldbull Electric charges create and exert and experience the electric force
q q q
bull We define the Electric Field as the force a test charge would feel if placed at that point
2 2ˆ ˆi j j
i ij i i iji j i jij ij
q q qk q k
r rne ne
rArr= =sum sumF r F rr r
bull The electric field exists around all charged objects
2ˆj
i j
qk q
rne
= rArr =sumE r F Er r r
2202009
10
Electric Force Lines
bull Michael Faraday saw the force caused by one charge to becaused by one charge to be from lines emanating from the charges
bull Lines begin on Positive charges and end on Negative charges flowing smoothly through space never crossing
Electric Force Lines
bull Faraday imagined these lines d h fand the force was greater
where the lines are denser weaker where they are spread out
2202009
11
Electric Field Lines
bull Go out from positive charges and into ti hnegative charges
Positive Negative Neutral
The Inverse Square Law
bull Many laws based on the geometry of three‐di i ldimensional space
1 22
2 24 4
g
sound light
Gm mFr
P LI Ir rπ π
= minus
= =
1 22e
q qF kr
=
2202009
12
The Electric Field
1828 ndash Four point charges have the same magnitude of 24 x 10‐12 C and are fixed to the corners of a square that is 40 cm on a side Three of the charges are positive d i ti D t i th it d f th t l t i fi ld th t i tand one is negative Determine the magnitude of the net electric field that exists
at the center of the square
A
B C
D-
++
+
EA
EC
EB
ED
The drawing at the right shows each of the field contributions at the center of the square (see black dot) Each is directed along a diagonal of the square Note that ED and EB point in opposite directions and therefore cancel since they have the same magnitude In contrast EA and EC point in the same direction toward corner A and therefore combine to give a net field thatcorner A and therefore combine to give a net field that is twice the magnitude of EA or EC In other words the net field at the center of the square is given by the following vector equation
2
Σ = + + +
= + + minus
= +
=
A B C D
A B C B
A C
A
E E E E EE E E EE EE
The Electric Field
1828 ndash (cont)
B C++
ED
The magnitude of the net field is A 22 2k q
E Er
Σ = =
A D- +
EA
EC
EB
In this result r is the distance from a corner to the center of the square which is one half of the diagonal distance d Using L for the length of a side of the square and taking advantage of the Pythagorean theorem we have With this substitution for r the magnitude of the net field becomes
r
2r L=
4k k
( )( ) ( )
( )
2 2
9 2 2 12
2
42
2
4 899 10 N m C 24 10 C
0040 m
54 NC
k q k qE
LL
E
E
minus
Σ = =
times sdot timesΣ =
Σ =
2202009
13
Electric Field Lines
1834 ndash Review Conceptual Example 12 before attempting to work this problem The magnitude of each of the charges in Figure 18‐21 is 86 x 10‐12 C The lengths of th id f th t l 3 00 d 5 00 Fi d th it d f ththe sides of the rectangles are 300 cm and 500 cm Find the magnitude of the electric field at the center of the rectangle in Figures 18‐21a and b
1 2
34
1
4
+ q
+ q + q
- qThe figure at the right shows the configuration given in text Figure 1821a The electric field at the center of the rectangle is the resultant of the electric fields at the center due to each of the four charges As discussed in Conceptual Example 11 the magnitudes of the electric field at the center due to each of the four charges are equal However the fields produced by the charges in corners 1 and 3 are in opposite directions Since they have the same magnitudes they combine to give directions Since they have the same magnitudes they combine to givezero resultant The fields produced by the charges in corners 2 and 4 point in the same direction (toward corner 2) Thus EC = EC2 + EC4 where EC is the magnitude of the electric field at the center of the rectangle and EC2 and EC4 are the magnitudes of the electric field at the center due to the charges in corners 2 and 4 respectively Since both EC2 and EC4 have the same magnitude we have EC = 2 EC2
Electric Field Lines1834 ndash (cont)
The distance r from any of the charges to the center of the rectangle can be found using the Pythagorean theorem
1 21
d500 cm
2 2(3 00 ) +(5 00 ) 5 83d
The figure at the right shows the configuration given in text Fi 18 21b All f h t ib t t
344
300 cm
θ
2 2(300 cm) +(500 cm) 583 cmd = =
2Therefore 292 cm 292 10 m2dr minus= = = times
9 2 2 1222
2 2 2 22 2(899 10 N m C )( 860 10 C)2 181 10 NC
(292 10 m)C Ckq
E Er
minus
minustimes sdot minus times
= = = = minus timestimes
- q - q
Figure 1821b All four charges contribute a non‐zero component to the electric field at the center of the rectangle As discussed in Conceptual Example 11 the contribution from the charges in corners 2 and 4 point toward corner 2 and the contribution from the charges in corners 1 and 3 point toward corner 1
1 2
34
1
4+ q + q
C
E 13 E 24
2202009
14
Electric Field Lines1834 ndash (cont)
Notice also the magnitudes of E24 and E13 are equal and from the first part of this problem we know that
E24 = E13 = 181 times 102 NC
1 21- q - q
E 13 E 24
The electric field at the center of the rectangle is the vector sum of E24 and E13 The x components of E24 and E13 are equal in magnitude and opposite in direction hence
(E13)x ndash (E24)x = 0
Therefore
F Fi 2 h th t
344+ q + q
C
C 13 24 13 13( ) ( ) 2( ) 2( )siny y yE E E E E θ= + = =
From Figure 2 we have that
500 cm 500 cmsin 0858583 cmd
θ = = =
( ) ( )( )2 2C 132 sin 2 181 10 NC 0858 311 10 NCE E θ= = times = times
The Electric Field
bull The electric field exists around all charged objects
bull This is shown byelectric fieldlines
bull Constructedid i ll hidentically to theforce lines
2202009
15
Drawing Electric Field Lines
bull Number of lines at a charge is proportional to the chargethe charge
Electric Field Lines
bull The charge with ‐3qmust have three times as many lines as the charge with +q and 15 times the lines for +‐ 2qthe charge with q and 15 times the lines for 2q
2202009
16
Parallel Plates of Charge
bull Adding fields from each point on surface causes field lines parallel to the plates to cancel
PositivelyCharged
NegativelyCharged
bull Away from the edges the field is parallel and perpendicular to the plate(s)
Electric Field amp Conductors
bull In conductors excess electrons will move when an electric field is present
bull For ldquosteady staterdquo (ie no moving charges) the net electric field must be zeror r r r
bull Therefore the charges move to the surface(s) of the conductor until they cancel the electric field inside the conductor
bull Field lines can start and end on a conductor at right angles to the surface of the conductor
0q rArr= = =F E F Er r r r
2202009
17
Electric Field Lines
1844 ndash Two particles are in a uniform electric field whose value is + 2500 NC The mass and charge of particle 1 are m1 = 14 x 10‐5 kg and q1 = ‐70 μC while the
di l f ti l 2 2 6 10 5 k d 18 C I iti llcorresponding values for particle 2 are m2 = 26 x 10‐5 kg and q2 = +18 μC Initially the particles are at rest The particles are both located on the same electric field line but are separated from each other by a distance d When released they accelerate but always remain at this same distance from each other Find d
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The drawing shows the two particles in the electric field Ex They are separated by a distance d If the particles are to stay at the same distance from each other after being released they must have the same acceleration so ax1 = ax2 According to Newtonrsquos second law the acceleration ax of each particle is equal to the net force ΣFx acting on it divided by its mass m or x xa F m= Σ
Electric Field Lines
1844 ndash (cont)
N t th t ti l 1 t b t th l ft f ti l 2 If ti l 1 t th i ht
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
Note that particle 1 must be to the left of particle 2 If particle 1 were to the right of particle 2 the particles would accelerate in opposite directions and the distance between them would not remain constant
1 2 1 1 2
1 21 2 1
11 1
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = minus
minusΣ= =
1 2 2 2 2
1 22 2 2
22 2
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = +
+Σ= =
Note the signs indicated the direction of the vector pointing from charge 2 to charge 1 for the force on charge 1
2202009
18
Electric Field Lines
1844 ndash (cont)
Th l ti f h h t b th if th di t b t th
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The acceleration of each charge must be the same if the distance between the charges remains unchanged
Solving for d
1 2
1 2 1 21 22 2
1 2
x x
x x
a a
q q q qq E k q E k
d dm m
=
minus +=
Solving for d
( )( )1 2 1 2
1 2 2 165 m
x
kq q m md
E q m q m+
= =minus
Electric Fluxbull Flux (from Latin ndash meaning to flow)
bull Electric flux measures amount of electric field passing through a surface
Er
Er
So to calculate find the normal to
φ
E EAΦ =
Flux through surface is Area of surface times the field flowing through that surface
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
( )cosE E AφΦ =
2202009
19
Electric Flux
( )cosE E AφΦ =
Er
So to calculate find the normal to
This can be expressed as the vector dot product
EΦ = sdotE Arr
cosE A φsdot =E Arr
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
cos
x x y y z z
E Aor
E A E A E A
φsdot =
sdot = + +
E A
E Arr
Electric Flux
1848 ndash A rectangular surface (016 m x 038m) is oriented in a uniform electric field of 580 NC What is the maximum possible electric flux through the surface
The maximum possible flux occurs when the electric field is parallel to the normal of the rectangular surface (that is when the angle between the direction of the field and the direction of the normal is zero) Then
( ) ( ) ( )2
( cos ) 580 N C (cos 0 ) 016 m 038 m
35 N m C E
E
E AφΦ = = deg
Φ = sdot
2202009
20
Gaussrsquo Lawbull Electric Flux through a closed surface is proportional to the charge enclosed by that surface
encE
qε
Φ =
bull The fundamental constant ε0 is called the permittivity of free space It is related to Coulombrsquos law by ε0 = 885 x 10‐12 C2(Nm2)
bull Gauss Law is most useful when the electric field on the entire surface is constant
bull Take a point charge Construct a sphere centered on the point charge (all points on the surface equidistant from the charge) ndash the electric field is constant
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so
0ε
0
14
kπε
=
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so φ = 0 cos φ = 1
bull The direction of the field is radially outward from the charge (or inward towards the charge if q is negative)
0 0
cosenc encE
q qEA φε ε
Φ =rArr=
( )22
0 0
44
encq qE r Er
πε πε
= rArr =
Gaussrsquo Law for a plate of chargebull In this case we take our closed surface to be a box oriented such that two faces are
parallel to the plate of charge and the others are perpendicular to the plate
Si th l t i fi ld f l t f h i di l t th f f thbull Since the electric field from a plate of charge is perpendicular to the surface of the plate the sides that are perpendicular to the plate have no flux
Er
bull The charge enclosed by the box is the surface charge density multiplied by the area the box passes through the plate Aσthe plate
bull The area of the box with flux is 2A
Aσ
0 0 0E
q AEA Eσ σε ε ε
rArr rArrΦ = = =
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
3
Fundamental Charge
Problem 186 ‐Water has a mass per molecule of 18 gmol and each molecule has 10 electrons How many electrons are ineach molecule has 10 electrons How many electrons are in one liter of water
h i h h f h l
23 3
26
10electrons 6022times10 molecules 1mol 1g 10 mltimes times1molecule 1mol 18g 1ml 1liter
electrons33 10liter
N
N
= times times
= times
What is the net charge of the electrons
( )( )26 19
7
33 10 electrons 1602 10 C electron
54 10 C
q Ne
q
minus= = times minus times
= minus times
Conservation of Charge
bull Net change in charge in any closed system is zero
bull Or the change of charge in any volume equals the net charge entering andOr the change of charge in any volume equals the net charge entering and leaving the system
bull We denote the amount of charge by the letter Q or q
bull If Charge is created it must be equal and opposite
( ) ( ) ( ) ( )Charge Original Charge Charge in Charge out= + minus
0 in out= + -q q q q
2202009
4
Conservation of Charge
Two identical metal spheres of charge 50 C and ‐16 C are brought together and then separated What is the charge onbrought together and then separated What is the charge on each sphere
Since the charge is distributed equally each must have an equal charge
1 2 = 34Ctotalq q q= +
equal charge
1 2 = 17C2totalqq q= =
Conductors and Insulators
bull Insulatorndash Charge stays where it is depositedCharge stays where it is depositedndash Wood mica Teflon rubber
bull Conductorndash Charge moves ldquofreelyrdquo within the objectndash Metals exhibit high conductivity
bull Due to atomic structurendash Is the outermost electron(s) tightly bound
2202009
5
Charging Objects
bull By frictionndash Rubbing different materialsndash Rubbing different materialsndash Example fur and ebonite
bull By contactndash Bringing a charged object into contact with another (charged or uncharged) object
bull By inductionh d b d llndash Bringing a charged object near a conductor temporally
causes opposite charge to flow towards the charged object
ndash Grounding that conductor ldquodrainsrdquo the excess charge leaving the conductor oppositely charged
Force Between Charges
bull Coulomb found that the force between two charges isbetween two charges is proportional to the product of their charges and inversely proportion to the square of the distance between them
1 22
q qF prop
Where q1 and q2 represent the charges and r12 the distance between them
212r
2202009
6
Coulombrsquos Law
bull By using the torsion balance Coulomb discovered the constant of proportionality k = 899 Nm2C2
bull If both q1 and q2 are both positive the force is positive
1 22
12
q qF kr
=+ +
‐ ‐
‐ +
bull If both q1 and q2 are both negative the force is positive
bull If both q1 is positive and q2 is negative the force is negative
bull Hence the sign of the force is positive for repulsion and negative for attraction
Coulombrsquos LawPrinciple of Superposition
bull Electric force obeys the law of linear superposition ndash the net force on a charge is the vector sum of the forces due to each charge
bull Add the vectors
2ˆi j
i iji j ij
q qk
rne
= sumF rr
1
3
The net force on charge4 is F14+F24+F34=Fnet
Red are positive chargesBlue are negative charges
1
24
F14F24
F34
2202009
7
Coulombrsquos Law Principle of Superposition
bull The forces add for as many charges as you have
2ˆi j
i iji j ij
q qk
rne
= sumF rr
36
Red are positive chargesBlue are negative charges
1
2
3
4
5
7
8
Coulombrsquos Law Principle of Superposition
bull Electric force obeys the law of linear superposition ndash the net force on a charge is the vector sum of the forces due to each charge
2ˆi j
i iji j ij
q qk
rne
= sumF rr
36 The net force on charge 5 is the
vector sum of all the forces on
1
2
3
4
5
7
8charge 5
Red are positive chargesBlue are negative charges
2202009
8
Coulombrsquos Law1814 ndash Two tiny conducting spheres are identical and carry charges of ‐200 μC and
+500 μC They are separated by a distance of 250 cm (a) What is the magnitude of the force that each sphere experiences and is the force attractive or repulsive
( )( )( )( )
9 2 2 6 61 2
22 2
899 10 N m C 200 10 C 500 10 Cq qF kminus minustimes sdot minus times times
= =
The net force is negative so that the force is attractive
(b) The spheres are brought into contact and then separated to a distance of 250 cm Determine the magnitude of the force that each sphere now experiences and state whether the force is attractive or repulsiveSi h h id i l h h h f b i d i
( )22 212
4
250 10 m
144 10 N
r
F
minustimes
= minus times
Since the spheres are identical the charge on each after being separated is one‐half the net charge so q1 + q2 = +15μC
The force is positive so it is repulsive
( )( )( )( )
9 2 2 6 631 2
2 2212
899 10 N m C 150 10 C 150 10 C324 10 N
250 10 m
q qF k
r
minus minus
minus
times sdot times times= = = times
times
Coulombrsquos Law1821 ndash An electrically neutral model airplane is flying in a horizontal circle on a 30 m
guideline which is nearly parallel to the ground The line breaks when the kinetic energy of the plane is 500 J Reconsider the same situation except that now there is a point charge of +q on the plane and a point charge of minusq at the other end of h id li I hi h li b k h h ki i f h l ithe guideline In this case the line breaks when the kinetic energy of the plane is 518 J Find the magnitude of the charges
From the definition of kinetic energy we see that mv2 = 2(KE) so that Equation 53 for the centripetal force becomes
( )2
c2 KEmF
r r= =υ
Sor r
( )( )
( )( )
2charged uncharged
max max2Centripetal
Centripetal forceforce
2 KE 2 KE1 2
k qT T
r rr+ = =
1424314243
2202009
9
Coulombrsquos Law1821 ndash (cont)
( )( )
( )( )
2charged uncharged
max max2Centripetal
Centripetal forcef
2 KE 2 KE1 2
k qT T
r rr+ = =
1424314243
Subtracting Equation (2) from Equation (1) eliminates Tmax and gives
( ) ( )2charged uncharged
2
2 KE ndash KEk qrr
⎡ ⎤⎣ ⎦=
force
Solving for |q| gives
( ) ( ) ( ) ( )charged uncharged ndash59 2 2
2 KE ndash KE 2 30 m 518 J ndash 500 J35 10 C
899 10 N m C
rq
k
⎡ ⎤⎣ ⎦= = = times
times sdot
The Electric Fieldbull Electric charges create and exert and experience the electric force
q q q
bull We define the Electric Field as the force a test charge would feel if placed at that point
2 2ˆ ˆi j j
i ij i i iji j i jij ij
q q qk q k
r rne ne
rArr= =sum sumF r F rr r
bull The electric field exists around all charged objects
2ˆj
i j
qk q
rne
= rArr =sumE r F Er r r
2202009
10
Electric Force Lines
bull Michael Faraday saw the force caused by one charge to becaused by one charge to be from lines emanating from the charges
bull Lines begin on Positive charges and end on Negative charges flowing smoothly through space never crossing
Electric Force Lines
bull Faraday imagined these lines d h fand the force was greater
where the lines are denser weaker where they are spread out
2202009
11
Electric Field Lines
bull Go out from positive charges and into ti hnegative charges
Positive Negative Neutral
The Inverse Square Law
bull Many laws based on the geometry of three‐di i ldimensional space
1 22
2 24 4
g
sound light
Gm mFr
P LI Ir rπ π
= minus
= =
1 22e
q qF kr
=
2202009
12
The Electric Field
1828 ndash Four point charges have the same magnitude of 24 x 10‐12 C and are fixed to the corners of a square that is 40 cm on a side Three of the charges are positive d i ti D t i th it d f th t l t i fi ld th t i tand one is negative Determine the magnitude of the net electric field that exists
at the center of the square
A
B C
D-
++
+
EA
EC
EB
ED
The drawing at the right shows each of the field contributions at the center of the square (see black dot) Each is directed along a diagonal of the square Note that ED and EB point in opposite directions and therefore cancel since they have the same magnitude In contrast EA and EC point in the same direction toward corner A and therefore combine to give a net field thatcorner A and therefore combine to give a net field that is twice the magnitude of EA or EC In other words the net field at the center of the square is given by the following vector equation
2
Σ = + + +
= + + minus
= +
=
A B C D
A B C B
A C
A
E E E E EE E E EE EE
The Electric Field
1828 ndash (cont)
B C++
ED
The magnitude of the net field is A 22 2k q
E Er
Σ = =
A D- +
EA
EC
EB
In this result r is the distance from a corner to the center of the square which is one half of the diagonal distance d Using L for the length of a side of the square and taking advantage of the Pythagorean theorem we have With this substitution for r the magnitude of the net field becomes
r
2r L=
4k k
( )( ) ( )
( )
2 2
9 2 2 12
2
42
2
4 899 10 N m C 24 10 C
0040 m
54 NC
k q k qE
LL
E
E
minus
Σ = =
times sdot timesΣ =
Σ =
2202009
13
Electric Field Lines
1834 ndash Review Conceptual Example 12 before attempting to work this problem The magnitude of each of the charges in Figure 18‐21 is 86 x 10‐12 C The lengths of th id f th t l 3 00 d 5 00 Fi d th it d f ththe sides of the rectangles are 300 cm and 500 cm Find the magnitude of the electric field at the center of the rectangle in Figures 18‐21a and b
1 2
34
1
4
+ q
+ q + q
- qThe figure at the right shows the configuration given in text Figure 1821a The electric field at the center of the rectangle is the resultant of the electric fields at the center due to each of the four charges As discussed in Conceptual Example 11 the magnitudes of the electric field at the center due to each of the four charges are equal However the fields produced by the charges in corners 1 and 3 are in opposite directions Since they have the same magnitudes they combine to give directions Since they have the same magnitudes they combine to givezero resultant The fields produced by the charges in corners 2 and 4 point in the same direction (toward corner 2) Thus EC = EC2 + EC4 where EC is the magnitude of the electric field at the center of the rectangle and EC2 and EC4 are the magnitudes of the electric field at the center due to the charges in corners 2 and 4 respectively Since both EC2 and EC4 have the same magnitude we have EC = 2 EC2
Electric Field Lines1834 ndash (cont)
The distance r from any of the charges to the center of the rectangle can be found using the Pythagorean theorem
1 21
d500 cm
2 2(3 00 ) +(5 00 ) 5 83d
The figure at the right shows the configuration given in text Fi 18 21b All f h t ib t t
344
300 cm
θ
2 2(300 cm) +(500 cm) 583 cmd = =
2Therefore 292 cm 292 10 m2dr minus= = = times
9 2 2 1222
2 2 2 22 2(899 10 N m C )( 860 10 C)2 181 10 NC
(292 10 m)C Ckq
E Er
minus
minustimes sdot minus times
= = = = minus timestimes
- q - q
Figure 1821b All four charges contribute a non‐zero component to the electric field at the center of the rectangle As discussed in Conceptual Example 11 the contribution from the charges in corners 2 and 4 point toward corner 2 and the contribution from the charges in corners 1 and 3 point toward corner 1
1 2
34
1
4+ q + q
C
E 13 E 24
2202009
14
Electric Field Lines1834 ndash (cont)
Notice also the magnitudes of E24 and E13 are equal and from the first part of this problem we know that
E24 = E13 = 181 times 102 NC
1 21- q - q
E 13 E 24
The electric field at the center of the rectangle is the vector sum of E24 and E13 The x components of E24 and E13 are equal in magnitude and opposite in direction hence
(E13)x ndash (E24)x = 0
Therefore
F Fi 2 h th t
344+ q + q
C
C 13 24 13 13( ) ( ) 2( ) 2( )siny y yE E E E E θ= + = =
From Figure 2 we have that
500 cm 500 cmsin 0858583 cmd
θ = = =
( ) ( )( )2 2C 132 sin 2 181 10 NC 0858 311 10 NCE E θ= = times = times
The Electric Field
bull The electric field exists around all charged objects
bull This is shown byelectric fieldlines
bull Constructedid i ll hidentically to theforce lines
2202009
15
Drawing Electric Field Lines
bull Number of lines at a charge is proportional to the chargethe charge
Electric Field Lines
bull The charge with ‐3qmust have three times as many lines as the charge with +q and 15 times the lines for +‐ 2qthe charge with q and 15 times the lines for 2q
2202009
16
Parallel Plates of Charge
bull Adding fields from each point on surface causes field lines parallel to the plates to cancel
PositivelyCharged
NegativelyCharged
bull Away from the edges the field is parallel and perpendicular to the plate(s)
Electric Field amp Conductors
bull In conductors excess electrons will move when an electric field is present
bull For ldquosteady staterdquo (ie no moving charges) the net electric field must be zeror r r r
bull Therefore the charges move to the surface(s) of the conductor until they cancel the electric field inside the conductor
bull Field lines can start and end on a conductor at right angles to the surface of the conductor
0q rArr= = =F E F Er r r r
2202009
17
Electric Field Lines
1844 ndash Two particles are in a uniform electric field whose value is + 2500 NC The mass and charge of particle 1 are m1 = 14 x 10‐5 kg and q1 = ‐70 μC while the
di l f ti l 2 2 6 10 5 k d 18 C I iti llcorresponding values for particle 2 are m2 = 26 x 10‐5 kg and q2 = +18 μC Initially the particles are at rest The particles are both located on the same electric field line but are separated from each other by a distance d When released they accelerate but always remain at this same distance from each other Find d
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The drawing shows the two particles in the electric field Ex They are separated by a distance d If the particles are to stay at the same distance from each other after being released they must have the same acceleration so ax1 = ax2 According to Newtonrsquos second law the acceleration ax of each particle is equal to the net force ΣFx acting on it divided by its mass m or x xa F m= Σ
Electric Field Lines
1844 ndash (cont)
N t th t ti l 1 t b t th l ft f ti l 2 If ti l 1 t th i ht
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
Note that particle 1 must be to the left of particle 2 If particle 1 were to the right of particle 2 the particles would accelerate in opposite directions and the distance between them would not remain constant
1 2 1 1 2
1 21 2 1
11 1
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = minus
minusΣ= =
1 2 2 2 2
1 22 2 2
22 2
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = +
+Σ= =
Note the signs indicated the direction of the vector pointing from charge 2 to charge 1 for the force on charge 1
2202009
18
Electric Field Lines
1844 ndash (cont)
Th l ti f h h t b th if th di t b t th
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The acceleration of each charge must be the same if the distance between the charges remains unchanged
Solving for d
1 2
1 2 1 21 22 2
1 2
x x
x x
a a
q q q qq E k q E k
d dm m
=
minus +=
Solving for d
( )( )1 2 1 2
1 2 2 165 m
x
kq q m md
E q m q m+
= =minus
Electric Fluxbull Flux (from Latin ndash meaning to flow)
bull Electric flux measures amount of electric field passing through a surface
Er
Er
So to calculate find the normal to
φ
E EAΦ =
Flux through surface is Area of surface times the field flowing through that surface
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
( )cosE E AφΦ =
2202009
19
Electric Flux
( )cosE E AφΦ =
Er
So to calculate find the normal to
This can be expressed as the vector dot product
EΦ = sdotE Arr
cosE A φsdot =E Arr
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
cos
x x y y z z
E Aor
E A E A E A
φsdot =
sdot = + +
E A
E Arr
Electric Flux
1848 ndash A rectangular surface (016 m x 038m) is oriented in a uniform electric field of 580 NC What is the maximum possible electric flux through the surface
The maximum possible flux occurs when the electric field is parallel to the normal of the rectangular surface (that is when the angle between the direction of the field and the direction of the normal is zero) Then
( ) ( ) ( )2
( cos ) 580 N C (cos 0 ) 016 m 038 m
35 N m C E
E
E AφΦ = = deg
Φ = sdot
2202009
20
Gaussrsquo Lawbull Electric Flux through a closed surface is proportional to the charge enclosed by that surface
encE
qε
Φ =
bull The fundamental constant ε0 is called the permittivity of free space It is related to Coulombrsquos law by ε0 = 885 x 10‐12 C2(Nm2)
bull Gauss Law is most useful when the electric field on the entire surface is constant
bull Take a point charge Construct a sphere centered on the point charge (all points on the surface equidistant from the charge) ndash the electric field is constant
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so
0ε
0
14
kπε
=
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so φ = 0 cos φ = 1
bull The direction of the field is radially outward from the charge (or inward towards the charge if q is negative)
0 0
cosenc encE
q qEA φε ε
Φ =rArr=
( )22
0 0
44
encq qE r Er
πε πε
= rArr =
Gaussrsquo Law for a plate of chargebull In this case we take our closed surface to be a box oriented such that two faces are
parallel to the plate of charge and the others are perpendicular to the plate
Si th l t i fi ld f l t f h i di l t th f f thbull Since the electric field from a plate of charge is perpendicular to the surface of the plate the sides that are perpendicular to the plate have no flux
Er
bull The charge enclosed by the box is the surface charge density multiplied by the area the box passes through the plate Aσthe plate
bull The area of the box with flux is 2A
Aσ
0 0 0E
q AEA Eσ σε ε ε
rArr rArrΦ = = =
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
4
Conservation of Charge
Two identical metal spheres of charge 50 C and ‐16 C are brought together and then separated What is the charge onbrought together and then separated What is the charge on each sphere
Since the charge is distributed equally each must have an equal charge
1 2 = 34Ctotalq q q= +
equal charge
1 2 = 17C2totalqq q= =
Conductors and Insulators
bull Insulatorndash Charge stays where it is depositedCharge stays where it is depositedndash Wood mica Teflon rubber
bull Conductorndash Charge moves ldquofreelyrdquo within the objectndash Metals exhibit high conductivity
bull Due to atomic structurendash Is the outermost electron(s) tightly bound
2202009
5
Charging Objects
bull By frictionndash Rubbing different materialsndash Rubbing different materialsndash Example fur and ebonite
bull By contactndash Bringing a charged object into contact with another (charged or uncharged) object
bull By inductionh d b d llndash Bringing a charged object near a conductor temporally
causes opposite charge to flow towards the charged object
ndash Grounding that conductor ldquodrainsrdquo the excess charge leaving the conductor oppositely charged
Force Between Charges
bull Coulomb found that the force between two charges isbetween two charges is proportional to the product of their charges and inversely proportion to the square of the distance between them
1 22
q qF prop
Where q1 and q2 represent the charges and r12 the distance between them
212r
2202009
6
Coulombrsquos Law
bull By using the torsion balance Coulomb discovered the constant of proportionality k = 899 Nm2C2
bull If both q1 and q2 are both positive the force is positive
1 22
12
q qF kr
=+ +
‐ ‐
‐ +
bull If both q1 and q2 are both negative the force is positive
bull If both q1 is positive and q2 is negative the force is negative
bull Hence the sign of the force is positive for repulsion and negative for attraction
Coulombrsquos LawPrinciple of Superposition
bull Electric force obeys the law of linear superposition ndash the net force on a charge is the vector sum of the forces due to each charge
bull Add the vectors
2ˆi j
i iji j ij
q qk
rne
= sumF rr
1
3
The net force on charge4 is F14+F24+F34=Fnet
Red are positive chargesBlue are negative charges
1
24
F14F24
F34
2202009
7
Coulombrsquos Law Principle of Superposition
bull The forces add for as many charges as you have
2ˆi j
i iji j ij
q qk
rne
= sumF rr
36
Red are positive chargesBlue are negative charges
1
2
3
4
5
7
8
Coulombrsquos Law Principle of Superposition
bull Electric force obeys the law of linear superposition ndash the net force on a charge is the vector sum of the forces due to each charge
2ˆi j
i iji j ij
q qk
rne
= sumF rr
36 The net force on charge 5 is the
vector sum of all the forces on
1
2
3
4
5
7
8charge 5
Red are positive chargesBlue are negative charges
2202009
8
Coulombrsquos Law1814 ndash Two tiny conducting spheres are identical and carry charges of ‐200 μC and
+500 μC They are separated by a distance of 250 cm (a) What is the magnitude of the force that each sphere experiences and is the force attractive or repulsive
( )( )( )( )
9 2 2 6 61 2
22 2
899 10 N m C 200 10 C 500 10 Cq qF kminus minustimes sdot minus times times
= =
The net force is negative so that the force is attractive
(b) The spheres are brought into contact and then separated to a distance of 250 cm Determine the magnitude of the force that each sphere now experiences and state whether the force is attractive or repulsiveSi h h id i l h h h f b i d i
( )22 212
4
250 10 m
144 10 N
r
F
minustimes
= minus times
Since the spheres are identical the charge on each after being separated is one‐half the net charge so q1 + q2 = +15μC
The force is positive so it is repulsive
( )( )( )( )
9 2 2 6 631 2
2 2212
899 10 N m C 150 10 C 150 10 C324 10 N
250 10 m
q qF k
r
minus minus
minus
times sdot times times= = = times
times
Coulombrsquos Law1821 ndash An electrically neutral model airplane is flying in a horizontal circle on a 30 m
guideline which is nearly parallel to the ground The line breaks when the kinetic energy of the plane is 500 J Reconsider the same situation except that now there is a point charge of +q on the plane and a point charge of minusq at the other end of h id li I hi h li b k h h ki i f h l ithe guideline In this case the line breaks when the kinetic energy of the plane is 518 J Find the magnitude of the charges
From the definition of kinetic energy we see that mv2 = 2(KE) so that Equation 53 for the centripetal force becomes
( )2
c2 KEmF
r r= =υ
Sor r
( )( )
( )( )
2charged uncharged
max max2Centripetal
Centripetal forceforce
2 KE 2 KE1 2
k qT T
r rr+ = =
1424314243
2202009
9
Coulombrsquos Law1821 ndash (cont)
( )( )
( )( )
2charged uncharged
max max2Centripetal
Centripetal forcef
2 KE 2 KE1 2
k qT T
r rr+ = =
1424314243
Subtracting Equation (2) from Equation (1) eliminates Tmax and gives
( ) ( )2charged uncharged
2
2 KE ndash KEk qrr
⎡ ⎤⎣ ⎦=
force
Solving for |q| gives
( ) ( ) ( ) ( )charged uncharged ndash59 2 2
2 KE ndash KE 2 30 m 518 J ndash 500 J35 10 C
899 10 N m C
rq
k
⎡ ⎤⎣ ⎦= = = times
times sdot
The Electric Fieldbull Electric charges create and exert and experience the electric force
q q q
bull We define the Electric Field as the force a test charge would feel if placed at that point
2 2ˆ ˆi j j
i ij i i iji j i jij ij
q q qk q k
r rne ne
rArr= =sum sumF r F rr r
bull The electric field exists around all charged objects
2ˆj
i j
qk q
rne
= rArr =sumE r F Er r r
2202009
10
Electric Force Lines
bull Michael Faraday saw the force caused by one charge to becaused by one charge to be from lines emanating from the charges
bull Lines begin on Positive charges and end on Negative charges flowing smoothly through space never crossing
Electric Force Lines
bull Faraday imagined these lines d h fand the force was greater
where the lines are denser weaker where they are spread out
2202009
11
Electric Field Lines
bull Go out from positive charges and into ti hnegative charges
Positive Negative Neutral
The Inverse Square Law
bull Many laws based on the geometry of three‐di i ldimensional space
1 22
2 24 4
g
sound light
Gm mFr
P LI Ir rπ π
= minus
= =
1 22e
q qF kr
=
2202009
12
The Electric Field
1828 ndash Four point charges have the same magnitude of 24 x 10‐12 C and are fixed to the corners of a square that is 40 cm on a side Three of the charges are positive d i ti D t i th it d f th t l t i fi ld th t i tand one is negative Determine the magnitude of the net electric field that exists
at the center of the square
A
B C
D-
++
+
EA
EC
EB
ED
The drawing at the right shows each of the field contributions at the center of the square (see black dot) Each is directed along a diagonal of the square Note that ED and EB point in opposite directions and therefore cancel since they have the same magnitude In contrast EA and EC point in the same direction toward corner A and therefore combine to give a net field thatcorner A and therefore combine to give a net field that is twice the magnitude of EA or EC In other words the net field at the center of the square is given by the following vector equation
2
Σ = + + +
= + + minus
= +
=
A B C D
A B C B
A C
A
E E E E EE E E EE EE
The Electric Field
1828 ndash (cont)
B C++
ED
The magnitude of the net field is A 22 2k q
E Er
Σ = =
A D- +
EA
EC
EB
In this result r is the distance from a corner to the center of the square which is one half of the diagonal distance d Using L for the length of a side of the square and taking advantage of the Pythagorean theorem we have With this substitution for r the magnitude of the net field becomes
r
2r L=
4k k
( )( ) ( )
( )
2 2
9 2 2 12
2
42
2
4 899 10 N m C 24 10 C
0040 m
54 NC
k q k qE
LL
E
E
minus
Σ = =
times sdot timesΣ =
Σ =
2202009
13
Electric Field Lines
1834 ndash Review Conceptual Example 12 before attempting to work this problem The magnitude of each of the charges in Figure 18‐21 is 86 x 10‐12 C The lengths of th id f th t l 3 00 d 5 00 Fi d th it d f ththe sides of the rectangles are 300 cm and 500 cm Find the magnitude of the electric field at the center of the rectangle in Figures 18‐21a and b
1 2
34
1
4
+ q
+ q + q
- qThe figure at the right shows the configuration given in text Figure 1821a The electric field at the center of the rectangle is the resultant of the electric fields at the center due to each of the four charges As discussed in Conceptual Example 11 the magnitudes of the electric field at the center due to each of the four charges are equal However the fields produced by the charges in corners 1 and 3 are in opposite directions Since they have the same magnitudes they combine to give directions Since they have the same magnitudes they combine to givezero resultant The fields produced by the charges in corners 2 and 4 point in the same direction (toward corner 2) Thus EC = EC2 + EC4 where EC is the magnitude of the electric field at the center of the rectangle and EC2 and EC4 are the magnitudes of the electric field at the center due to the charges in corners 2 and 4 respectively Since both EC2 and EC4 have the same magnitude we have EC = 2 EC2
Electric Field Lines1834 ndash (cont)
The distance r from any of the charges to the center of the rectangle can be found using the Pythagorean theorem
1 21
d500 cm
2 2(3 00 ) +(5 00 ) 5 83d
The figure at the right shows the configuration given in text Fi 18 21b All f h t ib t t
344
300 cm
θ
2 2(300 cm) +(500 cm) 583 cmd = =
2Therefore 292 cm 292 10 m2dr minus= = = times
9 2 2 1222
2 2 2 22 2(899 10 N m C )( 860 10 C)2 181 10 NC
(292 10 m)C Ckq
E Er
minus
minustimes sdot minus times
= = = = minus timestimes
- q - q
Figure 1821b All four charges contribute a non‐zero component to the electric field at the center of the rectangle As discussed in Conceptual Example 11 the contribution from the charges in corners 2 and 4 point toward corner 2 and the contribution from the charges in corners 1 and 3 point toward corner 1
1 2
34
1
4+ q + q
C
E 13 E 24
2202009
14
Electric Field Lines1834 ndash (cont)
Notice also the magnitudes of E24 and E13 are equal and from the first part of this problem we know that
E24 = E13 = 181 times 102 NC
1 21- q - q
E 13 E 24
The electric field at the center of the rectangle is the vector sum of E24 and E13 The x components of E24 and E13 are equal in magnitude and opposite in direction hence
(E13)x ndash (E24)x = 0
Therefore
F Fi 2 h th t
344+ q + q
C
C 13 24 13 13( ) ( ) 2( ) 2( )siny y yE E E E E θ= + = =
From Figure 2 we have that
500 cm 500 cmsin 0858583 cmd
θ = = =
( ) ( )( )2 2C 132 sin 2 181 10 NC 0858 311 10 NCE E θ= = times = times
The Electric Field
bull The electric field exists around all charged objects
bull This is shown byelectric fieldlines
bull Constructedid i ll hidentically to theforce lines
2202009
15
Drawing Electric Field Lines
bull Number of lines at a charge is proportional to the chargethe charge
Electric Field Lines
bull The charge with ‐3qmust have three times as many lines as the charge with +q and 15 times the lines for +‐ 2qthe charge with q and 15 times the lines for 2q
2202009
16
Parallel Plates of Charge
bull Adding fields from each point on surface causes field lines parallel to the plates to cancel
PositivelyCharged
NegativelyCharged
bull Away from the edges the field is parallel and perpendicular to the plate(s)
Electric Field amp Conductors
bull In conductors excess electrons will move when an electric field is present
bull For ldquosteady staterdquo (ie no moving charges) the net electric field must be zeror r r r
bull Therefore the charges move to the surface(s) of the conductor until they cancel the electric field inside the conductor
bull Field lines can start and end on a conductor at right angles to the surface of the conductor
0q rArr= = =F E F Er r r r
2202009
17
Electric Field Lines
1844 ndash Two particles are in a uniform electric field whose value is + 2500 NC The mass and charge of particle 1 are m1 = 14 x 10‐5 kg and q1 = ‐70 μC while the
di l f ti l 2 2 6 10 5 k d 18 C I iti llcorresponding values for particle 2 are m2 = 26 x 10‐5 kg and q2 = +18 μC Initially the particles are at rest The particles are both located on the same electric field line but are separated from each other by a distance d When released they accelerate but always remain at this same distance from each other Find d
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The drawing shows the two particles in the electric field Ex They are separated by a distance d If the particles are to stay at the same distance from each other after being released they must have the same acceleration so ax1 = ax2 According to Newtonrsquos second law the acceleration ax of each particle is equal to the net force ΣFx acting on it divided by its mass m or x xa F m= Σ
Electric Field Lines
1844 ndash (cont)
N t th t ti l 1 t b t th l ft f ti l 2 If ti l 1 t th i ht
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
Note that particle 1 must be to the left of particle 2 If particle 1 were to the right of particle 2 the particles would accelerate in opposite directions and the distance between them would not remain constant
1 2 1 1 2
1 21 2 1
11 1
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = minus
minusΣ= =
1 2 2 2 2
1 22 2 2
22 2
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = +
+Σ= =
Note the signs indicated the direction of the vector pointing from charge 2 to charge 1 for the force on charge 1
2202009
18
Electric Field Lines
1844 ndash (cont)
Th l ti f h h t b th if th di t b t th
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The acceleration of each charge must be the same if the distance between the charges remains unchanged
Solving for d
1 2
1 2 1 21 22 2
1 2
x x
x x
a a
q q q qq E k q E k
d dm m
=
minus +=
Solving for d
( )( )1 2 1 2
1 2 2 165 m
x
kq q m md
E q m q m+
= =minus
Electric Fluxbull Flux (from Latin ndash meaning to flow)
bull Electric flux measures amount of electric field passing through a surface
Er
Er
So to calculate find the normal to
φ
E EAΦ =
Flux through surface is Area of surface times the field flowing through that surface
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
( )cosE E AφΦ =
2202009
19
Electric Flux
( )cosE E AφΦ =
Er
So to calculate find the normal to
This can be expressed as the vector dot product
EΦ = sdotE Arr
cosE A φsdot =E Arr
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
cos
x x y y z z
E Aor
E A E A E A
φsdot =
sdot = + +
E A
E Arr
Electric Flux
1848 ndash A rectangular surface (016 m x 038m) is oriented in a uniform electric field of 580 NC What is the maximum possible electric flux through the surface
The maximum possible flux occurs when the electric field is parallel to the normal of the rectangular surface (that is when the angle between the direction of the field and the direction of the normal is zero) Then
( ) ( ) ( )2
( cos ) 580 N C (cos 0 ) 016 m 038 m
35 N m C E
E
E AφΦ = = deg
Φ = sdot
2202009
20
Gaussrsquo Lawbull Electric Flux through a closed surface is proportional to the charge enclosed by that surface
encE
qε
Φ =
bull The fundamental constant ε0 is called the permittivity of free space It is related to Coulombrsquos law by ε0 = 885 x 10‐12 C2(Nm2)
bull Gauss Law is most useful when the electric field on the entire surface is constant
bull Take a point charge Construct a sphere centered on the point charge (all points on the surface equidistant from the charge) ndash the electric field is constant
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so
0ε
0
14
kπε
=
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so φ = 0 cos φ = 1
bull The direction of the field is radially outward from the charge (or inward towards the charge if q is negative)
0 0
cosenc encE
q qEA φε ε
Φ =rArr=
( )22
0 0
44
encq qE r Er
πε πε
= rArr =
Gaussrsquo Law for a plate of chargebull In this case we take our closed surface to be a box oriented such that two faces are
parallel to the plate of charge and the others are perpendicular to the plate
Si th l t i fi ld f l t f h i di l t th f f thbull Since the electric field from a plate of charge is perpendicular to the surface of the plate the sides that are perpendicular to the plate have no flux
Er
bull The charge enclosed by the box is the surface charge density multiplied by the area the box passes through the plate Aσthe plate
bull The area of the box with flux is 2A
Aσ
0 0 0E
q AEA Eσ σε ε ε
rArr rArrΦ = = =
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
5
Charging Objects
bull By frictionndash Rubbing different materialsndash Rubbing different materialsndash Example fur and ebonite
bull By contactndash Bringing a charged object into contact with another (charged or uncharged) object
bull By inductionh d b d llndash Bringing a charged object near a conductor temporally
causes opposite charge to flow towards the charged object
ndash Grounding that conductor ldquodrainsrdquo the excess charge leaving the conductor oppositely charged
Force Between Charges
bull Coulomb found that the force between two charges isbetween two charges is proportional to the product of their charges and inversely proportion to the square of the distance between them
1 22
q qF prop
Where q1 and q2 represent the charges and r12 the distance between them
212r
2202009
6
Coulombrsquos Law
bull By using the torsion balance Coulomb discovered the constant of proportionality k = 899 Nm2C2
bull If both q1 and q2 are both positive the force is positive
1 22
12
q qF kr
=+ +
‐ ‐
‐ +
bull If both q1 and q2 are both negative the force is positive
bull If both q1 is positive and q2 is negative the force is negative
bull Hence the sign of the force is positive for repulsion and negative for attraction
Coulombrsquos LawPrinciple of Superposition
bull Electric force obeys the law of linear superposition ndash the net force on a charge is the vector sum of the forces due to each charge
bull Add the vectors
2ˆi j
i iji j ij
q qk
rne
= sumF rr
1
3
The net force on charge4 is F14+F24+F34=Fnet
Red are positive chargesBlue are negative charges
1
24
F14F24
F34
2202009
7
Coulombrsquos Law Principle of Superposition
bull The forces add for as many charges as you have
2ˆi j
i iji j ij
q qk
rne
= sumF rr
36
Red are positive chargesBlue are negative charges
1
2
3
4
5
7
8
Coulombrsquos Law Principle of Superposition
bull Electric force obeys the law of linear superposition ndash the net force on a charge is the vector sum of the forces due to each charge
2ˆi j
i iji j ij
q qk
rne
= sumF rr
36 The net force on charge 5 is the
vector sum of all the forces on
1
2
3
4
5
7
8charge 5
Red are positive chargesBlue are negative charges
2202009
8
Coulombrsquos Law1814 ndash Two tiny conducting spheres are identical and carry charges of ‐200 μC and
+500 μC They are separated by a distance of 250 cm (a) What is the magnitude of the force that each sphere experiences and is the force attractive or repulsive
( )( )( )( )
9 2 2 6 61 2
22 2
899 10 N m C 200 10 C 500 10 Cq qF kminus minustimes sdot minus times times
= =
The net force is negative so that the force is attractive
(b) The spheres are brought into contact and then separated to a distance of 250 cm Determine the magnitude of the force that each sphere now experiences and state whether the force is attractive or repulsiveSi h h id i l h h h f b i d i
( )22 212
4
250 10 m
144 10 N
r
F
minustimes
= minus times
Since the spheres are identical the charge on each after being separated is one‐half the net charge so q1 + q2 = +15μC
The force is positive so it is repulsive
( )( )( )( )
9 2 2 6 631 2
2 2212
899 10 N m C 150 10 C 150 10 C324 10 N
250 10 m
q qF k
r
minus minus
minus
times sdot times times= = = times
times
Coulombrsquos Law1821 ndash An electrically neutral model airplane is flying in a horizontal circle on a 30 m
guideline which is nearly parallel to the ground The line breaks when the kinetic energy of the plane is 500 J Reconsider the same situation except that now there is a point charge of +q on the plane and a point charge of minusq at the other end of h id li I hi h li b k h h ki i f h l ithe guideline In this case the line breaks when the kinetic energy of the plane is 518 J Find the magnitude of the charges
From the definition of kinetic energy we see that mv2 = 2(KE) so that Equation 53 for the centripetal force becomes
( )2
c2 KEmF
r r= =υ
Sor r
( )( )
( )( )
2charged uncharged
max max2Centripetal
Centripetal forceforce
2 KE 2 KE1 2
k qT T
r rr+ = =
1424314243
2202009
9
Coulombrsquos Law1821 ndash (cont)
( )( )
( )( )
2charged uncharged
max max2Centripetal
Centripetal forcef
2 KE 2 KE1 2
k qT T
r rr+ = =
1424314243
Subtracting Equation (2) from Equation (1) eliminates Tmax and gives
( ) ( )2charged uncharged
2
2 KE ndash KEk qrr
⎡ ⎤⎣ ⎦=
force
Solving for |q| gives
( ) ( ) ( ) ( )charged uncharged ndash59 2 2
2 KE ndash KE 2 30 m 518 J ndash 500 J35 10 C
899 10 N m C
rq
k
⎡ ⎤⎣ ⎦= = = times
times sdot
The Electric Fieldbull Electric charges create and exert and experience the electric force
q q q
bull We define the Electric Field as the force a test charge would feel if placed at that point
2 2ˆ ˆi j j
i ij i i iji j i jij ij
q q qk q k
r rne ne
rArr= =sum sumF r F rr r
bull The electric field exists around all charged objects
2ˆj
i j
qk q
rne
= rArr =sumE r F Er r r
2202009
10
Electric Force Lines
bull Michael Faraday saw the force caused by one charge to becaused by one charge to be from lines emanating from the charges
bull Lines begin on Positive charges and end on Negative charges flowing smoothly through space never crossing
Electric Force Lines
bull Faraday imagined these lines d h fand the force was greater
where the lines are denser weaker where they are spread out
2202009
11
Electric Field Lines
bull Go out from positive charges and into ti hnegative charges
Positive Negative Neutral
The Inverse Square Law
bull Many laws based on the geometry of three‐di i ldimensional space
1 22
2 24 4
g
sound light
Gm mFr
P LI Ir rπ π
= minus
= =
1 22e
q qF kr
=
2202009
12
The Electric Field
1828 ndash Four point charges have the same magnitude of 24 x 10‐12 C and are fixed to the corners of a square that is 40 cm on a side Three of the charges are positive d i ti D t i th it d f th t l t i fi ld th t i tand one is negative Determine the magnitude of the net electric field that exists
at the center of the square
A
B C
D-
++
+
EA
EC
EB
ED
The drawing at the right shows each of the field contributions at the center of the square (see black dot) Each is directed along a diagonal of the square Note that ED and EB point in opposite directions and therefore cancel since they have the same magnitude In contrast EA and EC point in the same direction toward corner A and therefore combine to give a net field thatcorner A and therefore combine to give a net field that is twice the magnitude of EA or EC In other words the net field at the center of the square is given by the following vector equation
2
Σ = + + +
= + + minus
= +
=
A B C D
A B C B
A C
A
E E E E EE E E EE EE
The Electric Field
1828 ndash (cont)
B C++
ED
The magnitude of the net field is A 22 2k q
E Er
Σ = =
A D- +
EA
EC
EB
In this result r is the distance from a corner to the center of the square which is one half of the diagonal distance d Using L for the length of a side of the square and taking advantage of the Pythagorean theorem we have With this substitution for r the magnitude of the net field becomes
r
2r L=
4k k
( )( ) ( )
( )
2 2
9 2 2 12
2
42
2
4 899 10 N m C 24 10 C
0040 m
54 NC
k q k qE
LL
E
E
minus
Σ = =
times sdot timesΣ =
Σ =
2202009
13
Electric Field Lines
1834 ndash Review Conceptual Example 12 before attempting to work this problem The magnitude of each of the charges in Figure 18‐21 is 86 x 10‐12 C The lengths of th id f th t l 3 00 d 5 00 Fi d th it d f ththe sides of the rectangles are 300 cm and 500 cm Find the magnitude of the electric field at the center of the rectangle in Figures 18‐21a and b
1 2
34
1
4
+ q
+ q + q
- qThe figure at the right shows the configuration given in text Figure 1821a The electric field at the center of the rectangle is the resultant of the electric fields at the center due to each of the four charges As discussed in Conceptual Example 11 the magnitudes of the electric field at the center due to each of the four charges are equal However the fields produced by the charges in corners 1 and 3 are in opposite directions Since they have the same magnitudes they combine to give directions Since they have the same magnitudes they combine to givezero resultant The fields produced by the charges in corners 2 and 4 point in the same direction (toward corner 2) Thus EC = EC2 + EC4 where EC is the magnitude of the electric field at the center of the rectangle and EC2 and EC4 are the magnitudes of the electric field at the center due to the charges in corners 2 and 4 respectively Since both EC2 and EC4 have the same magnitude we have EC = 2 EC2
Electric Field Lines1834 ndash (cont)
The distance r from any of the charges to the center of the rectangle can be found using the Pythagorean theorem
1 21
d500 cm
2 2(3 00 ) +(5 00 ) 5 83d
The figure at the right shows the configuration given in text Fi 18 21b All f h t ib t t
344
300 cm
θ
2 2(300 cm) +(500 cm) 583 cmd = =
2Therefore 292 cm 292 10 m2dr minus= = = times
9 2 2 1222
2 2 2 22 2(899 10 N m C )( 860 10 C)2 181 10 NC
(292 10 m)C Ckq
E Er
minus
minustimes sdot minus times
= = = = minus timestimes
- q - q
Figure 1821b All four charges contribute a non‐zero component to the electric field at the center of the rectangle As discussed in Conceptual Example 11 the contribution from the charges in corners 2 and 4 point toward corner 2 and the contribution from the charges in corners 1 and 3 point toward corner 1
1 2
34
1
4+ q + q
C
E 13 E 24
2202009
14
Electric Field Lines1834 ndash (cont)
Notice also the magnitudes of E24 and E13 are equal and from the first part of this problem we know that
E24 = E13 = 181 times 102 NC
1 21- q - q
E 13 E 24
The electric field at the center of the rectangle is the vector sum of E24 and E13 The x components of E24 and E13 are equal in magnitude and opposite in direction hence
(E13)x ndash (E24)x = 0
Therefore
F Fi 2 h th t
344+ q + q
C
C 13 24 13 13( ) ( ) 2( ) 2( )siny y yE E E E E θ= + = =
From Figure 2 we have that
500 cm 500 cmsin 0858583 cmd
θ = = =
( ) ( )( )2 2C 132 sin 2 181 10 NC 0858 311 10 NCE E θ= = times = times
The Electric Field
bull The electric field exists around all charged objects
bull This is shown byelectric fieldlines
bull Constructedid i ll hidentically to theforce lines
2202009
15
Drawing Electric Field Lines
bull Number of lines at a charge is proportional to the chargethe charge
Electric Field Lines
bull The charge with ‐3qmust have three times as many lines as the charge with +q and 15 times the lines for +‐ 2qthe charge with q and 15 times the lines for 2q
2202009
16
Parallel Plates of Charge
bull Adding fields from each point on surface causes field lines parallel to the plates to cancel
PositivelyCharged
NegativelyCharged
bull Away from the edges the field is parallel and perpendicular to the plate(s)
Electric Field amp Conductors
bull In conductors excess electrons will move when an electric field is present
bull For ldquosteady staterdquo (ie no moving charges) the net electric field must be zeror r r r
bull Therefore the charges move to the surface(s) of the conductor until they cancel the electric field inside the conductor
bull Field lines can start and end on a conductor at right angles to the surface of the conductor
0q rArr= = =F E F Er r r r
2202009
17
Electric Field Lines
1844 ndash Two particles are in a uniform electric field whose value is + 2500 NC The mass and charge of particle 1 are m1 = 14 x 10‐5 kg and q1 = ‐70 μC while the
di l f ti l 2 2 6 10 5 k d 18 C I iti llcorresponding values for particle 2 are m2 = 26 x 10‐5 kg and q2 = +18 μC Initially the particles are at rest The particles are both located on the same electric field line but are separated from each other by a distance d When released they accelerate but always remain at this same distance from each other Find d
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The drawing shows the two particles in the electric field Ex They are separated by a distance d If the particles are to stay at the same distance from each other after being released they must have the same acceleration so ax1 = ax2 According to Newtonrsquos second law the acceleration ax of each particle is equal to the net force ΣFx acting on it divided by its mass m or x xa F m= Σ
Electric Field Lines
1844 ndash (cont)
N t th t ti l 1 t b t th l ft f ti l 2 If ti l 1 t th i ht
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
Note that particle 1 must be to the left of particle 2 If particle 1 were to the right of particle 2 the particles would accelerate in opposite directions and the distance between them would not remain constant
1 2 1 1 2
1 21 2 1
11 1
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = minus
minusΣ= =
1 2 2 2 2
1 22 2 2
22 2
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = +
+Σ= =
Note the signs indicated the direction of the vector pointing from charge 2 to charge 1 for the force on charge 1
2202009
18
Electric Field Lines
1844 ndash (cont)
Th l ti f h h t b th if th di t b t th
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The acceleration of each charge must be the same if the distance between the charges remains unchanged
Solving for d
1 2
1 2 1 21 22 2
1 2
x x
x x
a a
q q q qq E k q E k
d dm m
=
minus +=
Solving for d
( )( )1 2 1 2
1 2 2 165 m
x
kq q m md
E q m q m+
= =minus
Electric Fluxbull Flux (from Latin ndash meaning to flow)
bull Electric flux measures amount of electric field passing through a surface
Er
Er
So to calculate find the normal to
φ
E EAΦ =
Flux through surface is Area of surface times the field flowing through that surface
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
( )cosE E AφΦ =
2202009
19
Electric Flux
( )cosE E AφΦ =
Er
So to calculate find the normal to
This can be expressed as the vector dot product
EΦ = sdotE Arr
cosE A φsdot =E Arr
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
cos
x x y y z z
E Aor
E A E A E A
φsdot =
sdot = + +
E A
E Arr
Electric Flux
1848 ndash A rectangular surface (016 m x 038m) is oriented in a uniform electric field of 580 NC What is the maximum possible electric flux through the surface
The maximum possible flux occurs when the electric field is parallel to the normal of the rectangular surface (that is when the angle between the direction of the field and the direction of the normal is zero) Then
( ) ( ) ( )2
( cos ) 580 N C (cos 0 ) 016 m 038 m
35 N m C E
E
E AφΦ = = deg
Φ = sdot
2202009
20
Gaussrsquo Lawbull Electric Flux through a closed surface is proportional to the charge enclosed by that surface
encE
qε
Φ =
bull The fundamental constant ε0 is called the permittivity of free space It is related to Coulombrsquos law by ε0 = 885 x 10‐12 C2(Nm2)
bull Gauss Law is most useful when the electric field on the entire surface is constant
bull Take a point charge Construct a sphere centered on the point charge (all points on the surface equidistant from the charge) ndash the electric field is constant
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so
0ε
0
14
kπε
=
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so φ = 0 cos φ = 1
bull The direction of the field is radially outward from the charge (or inward towards the charge if q is negative)
0 0
cosenc encE
q qEA φε ε
Φ =rArr=
( )22
0 0
44
encq qE r Er
πε πε
= rArr =
Gaussrsquo Law for a plate of chargebull In this case we take our closed surface to be a box oriented such that two faces are
parallel to the plate of charge and the others are perpendicular to the plate
Si th l t i fi ld f l t f h i di l t th f f thbull Since the electric field from a plate of charge is perpendicular to the surface of the plate the sides that are perpendicular to the plate have no flux
Er
bull The charge enclosed by the box is the surface charge density multiplied by the area the box passes through the plate Aσthe plate
bull The area of the box with flux is 2A
Aσ
0 0 0E
q AEA Eσ σε ε ε
rArr rArrΦ = = =
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
6
Coulombrsquos Law
bull By using the torsion balance Coulomb discovered the constant of proportionality k = 899 Nm2C2
bull If both q1 and q2 are both positive the force is positive
1 22
12
q qF kr
=+ +
‐ ‐
‐ +
bull If both q1 and q2 are both negative the force is positive
bull If both q1 is positive and q2 is negative the force is negative
bull Hence the sign of the force is positive for repulsion and negative for attraction
Coulombrsquos LawPrinciple of Superposition
bull Electric force obeys the law of linear superposition ndash the net force on a charge is the vector sum of the forces due to each charge
bull Add the vectors
2ˆi j
i iji j ij
q qk
rne
= sumF rr
1
3
The net force on charge4 is F14+F24+F34=Fnet
Red are positive chargesBlue are negative charges
1
24
F14F24
F34
2202009
7
Coulombrsquos Law Principle of Superposition
bull The forces add for as many charges as you have
2ˆi j
i iji j ij
q qk
rne
= sumF rr
36
Red are positive chargesBlue are negative charges
1
2
3
4
5
7
8
Coulombrsquos Law Principle of Superposition
bull Electric force obeys the law of linear superposition ndash the net force on a charge is the vector sum of the forces due to each charge
2ˆi j
i iji j ij
q qk
rne
= sumF rr
36 The net force on charge 5 is the
vector sum of all the forces on
1
2
3
4
5
7
8charge 5
Red are positive chargesBlue are negative charges
2202009
8
Coulombrsquos Law1814 ndash Two tiny conducting spheres are identical and carry charges of ‐200 μC and
+500 μC They are separated by a distance of 250 cm (a) What is the magnitude of the force that each sphere experiences and is the force attractive or repulsive
( )( )( )( )
9 2 2 6 61 2
22 2
899 10 N m C 200 10 C 500 10 Cq qF kminus minustimes sdot minus times times
= =
The net force is negative so that the force is attractive
(b) The spheres are brought into contact and then separated to a distance of 250 cm Determine the magnitude of the force that each sphere now experiences and state whether the force is attractive or repulsiveSi h h id i l h h h f b i d i
( )22 212
4
250 10 m
144 10 N
r
F
minustimes
= minus times
Since the spheres are identical the charge on each after being separated is one‐half the net charge so q1 + q2 = +15μC
The force is positive so it is repulsive
( )( )( )( )
9 2 2 6 631 2
2 2212
899 10 N m C 150 10 C 150 10 C324 10 N
250 10 m
q qF k
r
minus minus
minus
times sdot times times= = = times
times
Coulombrsquos Law1821 ndash An electrically neutral model airplane is flying in a horizontal circle on a 30 m
guideline which is nearly parallel to the ground The line breaks when the kinetic energy of the plane is 500 J Reconsider the same situation except that now there is a point charge of +q on the plane and a point charge of minusq at the other end of h id li I hi h li b k h h ki i f h l ithe guideline In this case the line breaks when the kinetic energy of the plane is 518 J Find the magnitude of the charges
From the definition of kinetic energy we see that mv2 = 2(KE) so that Equation 53 for the centripetal force becomes
( )2
c2 KEmF
r r= =υ
Sor r
( )( )
( )( )
2charged uncharged
max max2Centripetal
Centripetal forceforce
2 KE 2 KE1 2
k qT T
r rr+ = =
1424314243
2202009
9
Coulombrsquos Law1821 ndash (cont)
( )( )
( )( )
2charged uncharged
max max2Centripetal
Centripetal forcef
2 KE 2 KE1 2
k qT T
r rr+ = =
1424314243
Subtracting Equation (2) from Equation (1) eliminates Tmax and gives
( ) ( )2charged uncharged
2
2 KE ndash KEk qrr
⎡ ⎤⎣ ⎦=
force
Solving for |q| gives
( ) ( ) ( ) ( )charged uncharged ndash59 2 2
2 KE ndash KE 2 30 m 518 J ndash 500 J35 10 C
899 10 N m C
rq
k
⎡ ⎤⎣ ⎦= = = times
times sdot
The Electric Fieldbull Electric charges create and exert and experience the electric force
q q q
bull We define the Electric Field as the force a test charge would feel if placed at that point
2 2ˆ ˆi j j
i ij i i iji j i jij ij
q q qk q k
r rne ne
rArr= =sum sumF r F rr r
bull The electric field exists around all charged objects
2ˆj
i j
qk q
rne
= rArr =sumE r F Er r r
2202009
10
Electric Force Lines
bull Michael Faraday saw the force caused by one charge to becaused by one charge to be from lines emanating from the charges
bull Lines begin on Positive charges and end on Negative charges flowing smoothly through space never crossing
Electric Force Lines
bull Faraday imagined these lines d h fand the force was greater
where the lines are denser weaker where they are spread out
2202009
11
Electric Field Lines
bull Go out from positive charges and into ti hnegative charges
Positive Negative Neutral
The Inverse Square Law
bull Many laws based on the geometry of three‐di i ldimensional space
1 22
2 24 4
g
sound light
Gm mFr
P LI Ir rπ π
= minus
= =
1 22e
q qF kr
=
2202009
12
The Electric Field
1828 ndash Four point charges have the same magnitude of 24 x 10‐12 C and are fixed to the corners of a square that is 40 cm on a side Three of the charges are positive d i ti D t i th it d f th t l t i fi ld th t i tand one is negative Determine the magnitude of the net electric field that exists
at the center of the square
A
B C
D-
++
+
EA
EC
EB
ED
The drawing at the right shows each of the field contributions at the center of the square (see black dot) Each is directed along a diagonal of the square Note that ED and EB point in opposite directions and therefore cancel since they have the same magnitude In contrast EA and EC point in the same direction toward corner A and therefore combine to give a net field thatcorner A and therefore combine to give a net field that is twice the magnitude of EA or EC In other words the net field at the center of the square is given by the following vector equation
2
Σ = + + +
= + + minus
= +
=
A B C D
A B C B
A C
A
E E E E EE E E EE EE
The Electric Field
1828 ndash (cont)
B C++
ED
The magnitude of the net field is A 22 2k q
E Er
Σ = =
A D- +
EA
EC
EB
In this result r is the distance from a corner to the center of the square which is one half of the diagonal distance d Using L for the length of a side of the square and taking advantage of the Pythagorean theorem we have With this substitution for r the magnitude of the net field becomes
r
2r L=
4k k
( )( ) ( )
( )
2 2
9 2 2 12
2
42
2
4 899 10 N m C 24 10 C
0040 m
54 NC
k q k qE
LL
E
E
minus
Σ = =
times sdot timesΣ =
Σ =
2202009
13
Electric Field Lines
1834 ndash Review Conceptual Example 12 before attempting to work this problem The magnitude of each of the charges in Figure 18‐21 is 86 x 10‐12 C The lengths of th id f th t l 3 00 d 5 00 Fi d th it d f ththe sides of the rectangles are 300 cm and 500 cm Find the magnitude of the electric field at the center of the rectangle in Figures 18‐21a and b
1 2
34
1
4
+ q
+ q + q
- qThe figure at the right shows the configuration given in text Figure 1821a The electric field at the center of the rectangle is the resultant of the electric fields at the center due to each of the four charges As discussed in Conceptual Example 11 the magnitudes of the electric field at the center due to each of the four charges are equal However the fields produced by the charges in corners 1 and 3 are in opposite directions Since they have the same magnitudes they combine to give directions Since they have the same magnitudes they combine to givezero resultant The fields produced by the charges in corners 2 and 4 point in the same direction (toward corner 2) Thus EC = EC2 + EC4 where EC is the magnitude of the electric field at the center of the rectangle and EC2 and EC4 are the magnitudes of the electric field at the center due to the charges in corners 2 and 4 respectively Since both EC2 and EC4 have the same magnitude we have EC = 2 EC2
Electric Field Lines1834 ndash (cont)
The distance r from any of the charges to the center of the rectangle can be found using the Pythagorean theorem
1 21
d500 cm
2 2(3 00 ) +(5 00 ) 5 83d
The figure at the right shows the configuration given in text Fi 18 21b All f h t ib t t
344
300 cm
θ
2 2(300 cm) +(500 cm) 583 cmd = =
2Therefore 292 cm 292 10 m2dr minus= = = times
9 2 2 1222
2 2 2 22 2(899 10 N m C )( 860 10 C)2 181 10 NC
(292 10 m)C Ckq
E Er
minus
minustimes sdot minus times
= = = = minus timestimes
- q - q
Figure 1821b All four charges contribute a non‐zero component to the electric field at the center of the rectangle As discussed in Conceptual Example 11 the contribution from the charges in corners 2 and 4 point toward corner 2 and the contribution from the charges in corners 1 and 3 point toward corner 1
1 2
34
1
4+ q + q
C
E 13 E 24
2202009
14
Electric Field Lines1834 ndash (cont)
Notice also the magnitudes of E24 and E13 are equal and from the first part of this problem we know that
E24 = E13 = 181 times 102 NC
1 21- q - q
E 13 E 24
The electric field at the center of the rectangle is the vector sum of E24 and E13 The x components of E24 and E13 are equal in magnitude and opposite in direction hence
(E13)x ndash (E24)x = 0
Therefore
F Fi 2 h th t
344+ q + q
C
C 13 24 13 13( ) ( ) 2( ) 2( )siny y yE E E E E θ= + = =
From Figure 2 we have that
500 cm 500 cmsin 0858583 cmd
θ = = =
( ) ( )( )2 2C 132 sin 2 181 10 NC 0858 311 10 NCE E θ= = times = times
The Electric Field
bull The electric field exists around all charged objects
bull This is shown byelectric fieldlines
bull Constructedid i ll hidentically to theforce lines
2202009
15
Drawing Electric Field Lines
bull Number of lines at a charge is proportional to the chargethe charge
Electric Field Lines
bull The charge with ‐3qmust have three times as many lines as the charge with +q and 15 times the lines for +‐ 2qthe charge with q and 15 times the lines for 2q
2202009
16
Parallel Plates of Charge
bull Adding fields from each point on surface causes field lines parallel to the plates to cancel
PositivelyCharged
NegativelyCharged
bull Away from the edges the field is parallel and perpendicular to the plate(s)
Electric Field amp Conductors
bull In conductors excess electrons will move when an electric field is present
bull For ldquosteady staterdquo (ie no moving charges) the net electric field must be zeror r r r
bull Therefore the charges move to the surface(s) of the conductor until they cancel the electric field inside the conductor
bull Field lines can start and end on a conductor at right angles to the surface of the conductor
0q rArr= = =F E F Er r r r
2202009
17
Electric Field Lines
1844 ndash Two particles are in a uniform electric field whose value is + 2500 NC The mass and charge of particle 1 are m1 = 14 x 10‐5 kg and q1 = ‐70 μC while the
di l f ti l 2 2 6 10 5 k d 18 C I iti llcorresponding values for particle 2 are m2 = 26 x 10‐5 kg and q2 = +18 μC Initially the particles are at rest The particles are both located on the same electric field line but are separated from each other by a distance d When released they accelerate but always remain at this same distance from each other Find d
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The drawing shows the two particles in the electric field Ex They are separated by a distance d If the particles are to stay at the same distance from each other after being released they must have the same acceleration so ax1 = ax2 According to Newtonrsquos second law the acceleration ax of each particle is equal to the net force ΣFx acting on it divided by its mass m or x xa F m= Σ
Electric Field Lines
1844 ndash (cont)
N t th t ti l 1 t b t th l ft f ti l 2 If ti l 1 t th i ht
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
Note that particle 1 must be to the left of particle 2 If particle 1 were to the right of particle 2 the particles would accelerate in opposite directions and the distance between them would not remain constant
1 2 1 1 2
1 21 2 1
11 1
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = minus
minusΣ= =
1 2 2 2 2
1 22 2 2
22 2
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = +
+Σ= =
Note the signs indicated the direction of the vector pointing from charge 2 to charge 1 for the force on charge 1
2202009
18
Electric Field Lines
1844 ndash (cont)
Th l ti f h h t b th if th di t b t th
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The acceleration of each charge must be the same if the distance between the charges remains unchanged
Solving for d
1 2
1 2 1 21 22 2
1 2
x x
x x
a a
q q q qq E k q E k
d dm m
=
minus +=
Solving for d
( )( )1 2 1 2
1 2 2 165 m
x
kq q m md
E q m q m+
= =minus
Electric Fluxbull Flux (from Latin ndash meaning to flow)
bull Electric flux measures amount of electric field passing through a surface
Er
Er
So to calculate find the normal to
φ
E EAΦ =
Flux through surface is Area of surface times the field flowing through that surface
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
( )cosE E AφΦ =
2202009
19
Electric Flux
( )cosE E AφΦ =
Er
So to calculate find the normal to
This can be expressed as the vector dot product
EΦ = sdotE Arr
cosE A φsdot =E Arr
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
cos
x x y y z z
E Aor
E A E A E A
φsdot =
sdot = + +
E A
E Arr
Electric Flux
1848 ndash A rectangular surface (016 m x 038m) is oriented in a uniform electric field of 580 NC What is the maximum possible electric flux through the surface
The maximum possible flux occurs when the electric field is parallel to the normal of the rectangular surface (that is when the angle between the direction of the field and the direction of the normal is zero) Then
( ) ( ) ( )2
( cos ) 580 N C (cos 0 ) 016 m 038 m
35 N m C E
E
E AφΦ = = deg
Φ = sdot
2202009
20
Gaussrsquo Lawbull Electric Flux through a closed surface is proportional to the charge enclosed by that surface
encE
qε
Φ =
bull The fundamental constant ε0 is called the permittivity of free space It is related to Coulombrsquos law by ε0 = 885 x 10‐12 C2(Nm2)
bull Gauss Law is most useful when the electric field on the entire surface is constant
bull Take a point charge Construct a sphere centered on the point charge (all points on the surface equidistant from the charge) ndash the electric field is constant
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so
0ε
0
14
kπε
=
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so φ = 0 cos φ = 1
bull The direction of the field is radially outward from the charge (or inward towards the charge if q is negative)
0 0
cosenc encE
q qEA φε ε
Φ =rArr=
( )22
0 0
44
encq qE r Er
πε πε
= rArr =
Gaussrsquo Law for a plate of chargebull In this case we take our closed surface to be a box oriented such that two faces are
parallel to the plate of charge and the others are perpendicular to the plate
Si th l t i fi ld f l t f h i di l t th f f thbull Since the electric field from a plate of charge is perpendicular to the surface of the plate the sides that are perpendicular to the plate have no flux
Er
bull The charge enclosed by the box is the surface charge density multiplied by the area the box passes through the plate Aσthe plate
bull The area of the box with flux is 2A
Aσ
0 0 0E
q AEA Eσ σε ε ε
rArr rArrΦ = = =
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
7
Coulombrsquos Law Principle of Superposition
bull The forces add for as many charges as you have
2ˆi j
i iji j ij
q qk
rne
= sumF rr
36
Red are positive chargesBlue are negative charges
1
2
3
4
5
7
8
Coulombrsquos Law Principle of Superposition
bull Electric force obeys the law of linear superposition ndash the net force on a charge is the vector sum of the forces due to each charge
2ˆi j
i iji j ij
q qk
rne
= sumF rr
36 The net force on charge 5 is the
vector sum of all the forces on
1
2
3
4
5
7
8charge 5
Red are positive chargesBlue are negative charges
2202009
8
Coulombrsquos Law1814 ndash Two tiny conducting spheres are identical and carry charges of ‐200 μC and
+500 μC They are separated by a distance of 250 cm (a) What is the magnitude of the force that each sphere experiences and is the force attractive or repulsive
( )( )( )( )
9 2 2 6 61 2
22 2
899 10 N m C 200 10 C 500 10 Cq qF kminus minustimes sdot minus times times
= =
The net force is negative so that the force is attractive
(b) The spheres are brought into contact and then separated to a distance of 250 cm Determine the magnitude of the force that each sphere now experiences and state whether the force is attractive or repulsiveSi h h id i l h h h f b i d i
( )22 212
4
250 10 m
144 10 N
r
F
minustimes
= minus times
Since the spheres are identical the charge on each after being separated is one‐half the net charge so q1 + q2 = +15μC
The force is positive so it is repulsive
( )( )( )( )
9 2 2 6 631 2
2 2212
899 10 N m C 150 10 C 150 10 C324 10 N
250 10 m
q qF k
r
minus minus
minus
times sdot times times= = = times
times
Coulombrsquos Law1821 ndash An electrically neutral model airplane is flying in a horizontal circle on a 30 m
guideline which is nearly parallel to the ground The line breaks when the kinetic energy of the plane is 500 J Reconsider the same situation except that now there is a point charge of +q on the plane and a point charge of minusq at the other end of h id li I hi h li b k h h ki i f h l ithe guideline In this case the line breaks when the kinetic energy of the plane is 518 J Find the magnitude of the charges
From the definition of kinetic energy we see that mv2 = 2(KE) so that Equation 53 for the centripetal force becomes
( )2
c2 KEmF
r r= =υ
Sor r
( )( )
( )( )
2charged uncharged
max max2Centripetal
Centripetal forceforce
2 KE 2 KE1 2
k qT T
r rr+ = =
1424314243
2202009
9
Coulombrsquos Law1821 ndash (cont)
( )( )
( )( )
2charged uncharged
max max2Centripetal
Centripetal forcef
2 KE 2 KE1 2
k qT T
r rr+ = =
1424314243
Subtracting Equation (2) from Equation (1) eliminates Tmax and gives
( ) ( )2charged uncharged
2
2 KE ndash KEk qrr
⎡ ⎤⎣ ⎦=
force
Solving for |q| gives
( ) ( ) ( ) ( )charged uncharged ndash59 2 2
2 KE ndash KE 2 30 m 518 J ndash 500 J35 10 C
899 10 N m C
rq
k
⎡ ⎤⎣ ⎦= = = times
times sdot
The Electric Fieldbull Electric charges create and exert and experience the electric force
q q q
bull We define the Electric Field as the force a test charge would feel if placed at that point
2 2ˆ ˆi j j
i ij i i iji j i jij ij
q q qk q k
r rne ne
rArr= =sum sumF r F rr r
bull The electric field exists around all charged objects
2ˆj
i j
qk q
rne
= rArr =sumE r F Er r r
2202009
10
Electric Force Lines
bull Michael Faraday saw the force caused by one charge to becaused by one charge to be from lines emanating from the charges
bull Lines begin on Positive charges and end on Negative charges flowing smoothly through space never crossing
Electric Force Lines
bull Faraday imagined these lines d h fand the force was greater
where the lines are denser weaker where they are spread out
2202009
11
Electric Field Lines
bull Go out from positive charges and into ti hnegative charges
Positive Negative Neutral
The Inverse Square Law
bull Many laws based on the geometry of three‐di i ldimensional space
1 22
2 24 4
g
sound light
Gm mFr
P LI Ir rπ π
= minus
= =
1 22e
q qF kr
=
2202009
12
The Electric Field
1828 ndash Four point charges have the same magnitude of 24 x 10‐12 C and are fixed to the corners of a square that is 40 cm on a side Three of the charges are positive d i ti D t i th it d f th t l t i fi ld th t i tand one is negative Determine the magnitude of the net electric field that exists
at the center of the square
A
B C
D-
++
+
EA
EC
EB
ED
The drawing at the right shows each of the field contributions at the center of the square (see black dot) Each is directed along a diagonal of the square Note that ED and EB point in opposite directions and therefore cancel since they have the same magnitude In contrast EA and EC point in the same direction toward corner A and therefore combine to give a net field thatcorner A and therefore combine to give a net field that is twice the magnitude of EA or EC In other words the net field at the center of the square is given by the following vector equation
2
Σ = + + +
= + + minus
= +
=
A B C D
A B C B
A C
A
E E E E EE E E EE EE
The Electric Field
1828 ndash (cont)
B C++
ED
The magnitude of the net field is A 22 2k q
E Er
Σ = =
A D- +
EA
EC
EB
In this result r is the distance from a corner to the center of the square which is one half of the diagonal distance d Using L for the length of a side of the square and taking advantage of the Pythagorean theorem we have With this substitution for r the magnitude of the net field becomes
r
2r L=
4k k
( )( ) ( )
( )
2 2
9 2 2 12
2
42
2
4 899 10 N m C 24 10 C
0040 m
54 NC
k q k qE
LL
E
E
minus
Σ = =
times sdot timesΣ =
Σ =
2202009
13
Electric Field Lines
1834 ndash Review Conceptual Example 12 before attempting to work this problem The magnitude of each of the charges in Figure 18‐21 is 86 x 10‐12 C The lengths of th id f th t l 3 00 d 5 00 Fi d th it d f ththe sides of the rectangles are 300 cm and 500 cm Find the magnitude of the electric field at the center of the rectangle in Figures 18‐21a and b
1 2
34
1
4
+ q
+ q + q
- qThe figure at the right shows the configuration given in text Figure 1821a The electric field at the center of the rectangle is the resultant of the electric fields at the center due to each of the four charges As discussed in Conceptual Example 11 the magnitudes of the electric field at the center due to each of the four charges are equal However the fields produced by the charges in corners 1 and 3 are in opposite directions Since they have the same magnitudes they combine to give directions Since they have the same magnitudes they combine to givezero resultant The fields produced by the charges in corners 2 and 4 point in the same direction (toward corner 2) Thus EC = EC2 + EC4 where EC is the magnitude of the electric field at the center of the rectangle and EC2 and EC4 are the magnitudes of the electric field at the center due to the charges in corners 2 and 4 respectively Since both EC2 and EC4 have the same magnitude we have EC = 2 EC2
Electric Field Lines1834 ndash (cont)
The distance r from any of the charges to the center of the rectangle can be found using the Pythagorean theorem
1 21
d500 cm
2 2(3 00 ) +(5 00 ) 5 83d
The figure at the right shows the configuration given in text Fi 18 21b All f h t ib t t
344
300 cm
θ
2 2(300 cm) +(500 cm) 583 cmd = =
2Therefore 292 cm 292 10 m2dr minus= = = times
9 2 2 1222
2 2 2 22 2(899 10 N m C )( 860 10 C)2 181 10 NC
(292 10 m)C Ckq
E Er
minus
minustimes sdot minus times
= = = = minus timestimes
- q - q
Figure 1821b All four charges contribute a non‐zero component to the electric field at the center of the rectangle As discussed in Conceptual Example 11 the contribution from the charges in corners 2 and 4 point toward corner 2 and the contribution from the charges in corners 1 and 3 point toward corner 1
1 2
34
1
4+ q + q
C
E 13 E 24
2202009
14
Electric Field Lines1834 ndash (cont)
Notice also the magnitudes of E24 and E13 are equal and from the first part of this problem we know that
E24 = E13 = 181 times 102 NC
1 21- q - q
E 13 E 24
The electric field at the center of the rectangle is the vector sum of E24 and E13 The x components of E24 and E13 are equal in magnitude and opposite in direction hence
(E13)x ndash (E24)x = 0
Therefore
F Fi 2 h th t
344+ q + q
C
C 13 24 13 13( ) ( ) 2( ) 2( )siny y yE E E E E θ= + = =
From Figure 2 we have that
500 cm 500 cmsin 0858583 cmd
θ = = =
( ) ( )( )2 2C 132 sin 2 181 10 NC 0858 311 10 NCE E θ= = times = times
The Electric Field
bull The electric field exists around all charged objects
bull This is shown byelectric fieldlines
bull Constructedid i ll hidentically to theforce lines
2202009
15
Drawing Electric Field Lines
bull Number of lines at a charge is proportional to the chargethe charge
Electric Field Lines
bull The charge with ‐3qmust have three times as many lines as the charge with +q and 15 times the lines for +‐ 2qthe charge with q and 15 times the lines for 2q
2202009
16
Parallel Plates of Charge
bull Adding fields from each point on surface causes field lines parallel to the plates to cancel
PositivelyCharged
NegativelyCharged
bull Away from the edges the field is parallel and perpendicular to the plate(s)
Electric Field amp Conductors
bull In conductors excess electrons will move when an electric field is present
bull For ldquosteady staterdquo (ie no moving charges) the net electric field must be zeror r r r
bull Therefore the charges move to the surface(s) of the conductor until they cancel the electric field inside the conductor
bull Field lines can start and end on a conductor at right angles to the surface of the conductor
0q rArr= = =F E F Er r r r
2202009
17
Electric Field Lines
1844 ndash Two particles are in a uniform electric field whose value is + 2500 NC The mass and charge of particle 1 are m1 = 14 x 10‐5 kg and q1 = ‐70 μC while the
di l f ti l 2 2 6 10 5 k d 18 C I iti llcorresponding values for particle 2 are m2 = 26 x 10‐5 kg and q2 = +18 μC Initially the particles are at rest The particles are both located on the same electric field line but are separated from each other by a distance d When released they accelerate but always remain at this same distance from each other Find d
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The drawing shows the two particles in the electric field Ex They are separated by a distance d If the particles are to stay at the same distance from each other after being released they must have the same acceleration so ax1 = ax2 According to Newtonrsquos second law the acceleration ax of each particle is equal to the net force ΣFx acting on it divided by its mass m or x xa F m= Σ
Electric Field Lines
1844 ndash (cont)
N t th t ti l 1 t b t th l ft f ti l 2 If ti l 1 t th i ht
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
Note that particle 1 must be to the left of particle 2 If particle 1 were to the right of particle 2 the particles would accelerate in opposite directions and the distance between them would not remain constant
1 2 1 1 2
1 21 2 1
11 1
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = minus
minusΣ= =
1 2 2 2 2
1 22 2 2
22 2
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = +
+Σ= =
Note the signs indicated the direction of the vector pointing from charge 2 to charge 1 for the force on charge 1
2202009
18
Electric Field Lines
1844 ndash (cont)
Th l ti f h h t b th if th di t b t th
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The acceleration of each charge must be the same if the distance between the charges remains unchanged
Solving for d
1 2
1 2 1 21 22 2
1 2
x x
x x
a a
q q q qq E k q E k
d dm m
=
minus +=
Solving for d
( )( )1 2 1 2
1 2 2 165 m
x
kq q m md
E q m q m+
= =minus
Electric Fluxbull Flux (from Latin ndash meaning to flow)
bull Electric flux measures amount of electric field passing through a surface
Er
Er
So to calculate find the normal to
φ
E EAΦ =
Flux through surface is Area of surface times the field flowing through that surface
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
( )cosE E AφΦ =
2202009
19
Electric Flux
( )cosE E AφΦ =
Er
So to calculate find the normal to
This can be expressed as the vector dot product
EΦ = sdotE Arr
cosE A φsdot =E Arr
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
cos
x x y y z z
E Aor
E A E A E A
φsdot =
sdot = + +
E A
E Arr
Electric Flux
1848 ndash A rectangular surface (016 m x 038m) is oriented in a uniform electric field of 580 NC What is the maximum possible electric flux through the surface
The maximum possible flux occurs when the electric field is parallel to the normal of the rectangular surface (that is when the angle between the direction of the field and the direction of the normal is zero) Then
( ) ( ) ( )2
( cos ) 580 N C (cos 0 ) 016 m 038 m
35 N m C E
E
E AφΦ = = deg
Φ = sdot
2202009
20
Gaussrsquo Lawbull Electric Flux through a closed surface is proportional to the charge enclosed by that surface
encE
qε
Φ =
bull The fundamental constant ε0 is called the permittivity of free space It is related to Coulombrsquos law by ε0 = 885 x 10‐12 C2(Nm2)
bull Gauss Law is most useful when the electric field on the entire surface is constant
bull Take a point charge Construct a sphere centered on the point charge (all points on the surface equidistant from the charge) ndash the electric field is constant
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so
0ε
0
14
kπε
=
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so φ = 0 cos φ = 1
bull The direction of the field is radially outward from the charge (or inward towards the charge if q is negative)
0 0
cosenc encE
q qEA φε ε
Φ =rArr=
( )22
0 0
44
encq qE r Er
πε πε
= rArr =
Gaussrsquo Law for a plate of chargebull In this case we take our closed surface to be a box oriented such that two faces are
parallel to the plate of charge and the others are perpendicular to the plate
Si th l t i fi ld f l t f h i di l t th f f thbull Since the electric field from a plate of charge is perpendicular to the surface of the plate the sides that are perpendicular to the plate have no flux
Er
bull The charge enclosed by the box is the surface charge density multiplied by the area the box passes through the plate Aσthe plate
bull The area of the box with flux is 2A
Aσ
0 0 0E
q AEA Eσ σε ε ε
rArr rArrΦ = = =
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
8
Coulombrsquos Law1814 ndash Two tiny conducting spheres are identical and carry charges of ‐200 μC and
+500 μC They are separated by a distance of 250 cm (a) What is the magnitude of the force that each sphere experiences and is the force attractive or repulsive
( )( )( )( )
9 2 2 6 61 2
22 2
899 10 N m C 200 10 C 500 10 Cq qF kminus minustimes sdot minus times times
= =
The net force is negative so that the force is attractive
(b) The spheres are brought into contact and then separated to a distance of 250 cm Determine the magnitude of the force that each sphere now experiences and state whether the force is attractive or repulsiveSi h h id i l h h h f b i d i
( )22 212
4
250 10 m
144 10 N
r
F
minustimes
= minus times
Since the spheres are identical the charge on each after being separated is one‐half the net charge so q1 + q2 = +15μC
The force is positive so it is repulsive
( )( )( )( )
9 2 2 6 631 2
2 2212
899 10 N m C 150 10 C 150 10 C324 10 N
250 10 m
q qF k
r
minus minus
minus
times sdot times times= = = times
times
Coulombrsquos Law1821 ndash An electrically neutral model airplane is flying in a horizontal circle on a 30 m
guideline which is nearly parallel to the ground The line breaks when the kinetic energy of the plane is 500 J Reconsider the same situation except that now there is a point charge of +q on the plane and a point charge of minusq at the other end of h id li I hi h li b k h h ki i f h l ithe guideline In this case the line breaks when the kinetic energy of the plane is 518 J Find the magnitude of the charges
From the definition of kinetic energy we see that mv2 = 2(KE) so that Equation 53 for the centripetal force becomes
( )2
c2 KEmF
r r= =υ
Sor r
( )( )
( )( )
2charged uncharged
max max2Centripetal
Centripetal forceforce
2 KE 2 KE1 2
k qT T
r rr+ = =
1424314243
2202009
9
Coulombrsquos Law1821 ndash (cont)
( )( )
( )( )
2charged uncharged
max max2Centripetal
Centripetal forcef
2 KE 2 KE1 2
k qT T
r rr+ = =
1424314243
Subtracting Equation (2) from Equation (1) eliminates Tmax and gives
( ) ( )2charged uncharged
2
2 KE ndash KEk qrr
⎡ ⎤⎣ ⎦=
force
Solving for |q| gives
( ) ( ) ( ) ( )charged uncharged ndash59 2 2
2 KE ndash KE 2 30 m 518 J ndash 500 J35 10 C
899 10 N m C
rq
k
⎡ ⎤⎣ ⎦= = = times
times sdot
The Electric Fieldbull Electric charges create and exert and experience the electric force
q q q
bull We define the Electric Field as the force a test charge would feel if placed at that point
2 2ˆ ˆi j j
i ij i i iji j i jij ij
q q qk q k
r rne ne
rArr= =sum sumF r F rr r
bull The electric field exists around all charged objects
2ˆj
i j
qk q
rne
= rArr =sumE r F Er r r
2202009
10
Electric Force Lines
bull Michael Faraday saw the force caused by one charge to becaused by one charge to be from lines emanating from the charges
bull Lines begin on Positive charges and end on Negative charges flowing smoothly through space never crossing
Electric Force Lines
bull Faraday imagined these lines d h fand the force was greater
where the lines are denser weaker where they are spread out
2202009
11
Electric Field Lines
bull Go out from positive charges and into ti hnegative charges
Positive Negative Neutral
The Inverse Square Law
bull Many laws based on the geometry of three‐di i ldimensional space
1 22
2 24 4
g
sound light
Gm mFr
P LI Ir rπ π
= minus
= =
1 22e
q qF kr
=
2202009
12
The Electric Field
1828 ndash Four point charges have the same magnitude of 24 x 10‐12 C and are fixed to the corners of a square that is 40 cm on a side Three of the charges are positive d i ti D t i th it d f th t l t i fi ld th t i tand one is negative Determine the magnitude of the net electric field that exists
at the center of the square
A
B C
D-
++
+
EA
EC
EB
ED
The drawing at the right shows each of the field contributions at the center of the square (see black dot) Each is directed along a diagonal of the square Note that ED and EB point in opposite directions and therefore cancel since they have the same magnitude In contrast EA and EC point in the same direction toward corner A and therefore combine to give a net field thatcorner A and therefore combine to give a net field that is twice the magnitude of EA or EC In other words the net field at the center of the square is given by the following vector equation
2
Σ = + + +
= + + minus
= +
=
A B C D
A B C B
A C
A
E E E E EE E E EE EE
The Electric Field
1828 ndash (cont)
B C++
ED
The magnitude of the net field is A 22 2k q
E Er
Σ = =
A D- +
EA
EC
EB
In this result r is the distance from a corner to the center of the square which is one half of the diagonal distance d Using L for the length of a side of the square and taking advantage of the Pythagorean theorem we have With this substitution for r the magnitude of the net field becomes
r
2r L=
4k k
( )( ) ( )
( )
2 2
9 2 2 12
2
42
2
4 899 10 N m C 24 10 C
0040 m
54 NC
k q k qE
LL
E
E
minus
Σ = =
times sdot timesΣ =
Σ =
2202009
13
Electric Field Lines
1834 ndash Review Conceptual Example 12 before attempting to work this problem The magnitude of each of the charges in Figure 18‐21 is 86 x 10‐12 C The lengths of th id f th t l 3 00 d 5 00 Fi d th it d f ththe sides of the rectangles are 300 cm and 500 cm Find the magnitude of the electric field at the center of the rectangle in Figures 18‐21a and b
1 2
34
1
4
+ q
+ q + q
- qThe figure at the right shows the configuration given in text Figure 1821a The electric field at the center of the rectangle is the resultant of the electric fields at the center due to each of the four charges As discussed in Conceptual Example 11 the magnitudes of the electric field at the center due to each of the four charges are equal However the fields produced by the charges in corners 1 and 3 are in opposite directions Since they have the same magnitudes they combine to give directions Since they have the same magnitudes they combine to givezero resultant The fields produced by the charges in corners 2 and 4 point in the same direction (toward corner 2) Thus EC = EC2 + EC4 where EC is the magnitude of the electric field at the center of the rectangle and EC2 and EC4 are the magnitudes of the electric field at the center due to the charges in corners 2 and 4 respectively Since both EC2 and EC4 have the same magnitude we have EC = 2 EC2
Electric Field Lines1834 ndash (cont)
The distance r from any of the charges to the center of the rectangle can be found using the Pythagorean theorem
1 21
d500 cm
2 2(3 00 ) +(5 00 ) 5 83d
The figure at the right shows the configuration given in text Fi 18 21b All f h t ib t t
344
300 cm
θ
2 2(300 cm) +(500 cm) 583 cmd = =
2Therefore 292 cm 292 10 m2dr minus= = = times
9 2 2 1222
2 2 2 22 2(899 10 N m C )( 860 10 C)2 181 10 NC
(292 10 m)C Ckq
E Er
minus
minustimes sdot minus times
= = = = minus timestimes
- q - q
Figure 1821b All four charges contribute a non‐zero component to the electric field at the center of the rectangle As discussed in Conceptual Example 11 the contribution from the charges in corners 2 and 4 point toward corner 2 and the contribution from the charges in corners 1 and 3 point toward corner 1
1 2
34
1
4+ q + q
C
E 13 E 24
2202009
14
Electric Field Lines1834 ndash (cont)
Notice also the magnitudes of E24 and E13 are equal and from the first part of this problem we know that
E24 = E13 = 181 times 102 NC
1 21- q - q
E 13 E 24
The electric field at the center of the rectangle is the vector sum of E24 and E13 The x components of E24 and E13 are equal in magnitude and opposite in direction hence
(E13)x ndash (E24)x = 0
Therefore
F Fi 2 h th t
344+ q + q
C
C 13 24 13 13( ) ( ) 2( ) 2( )siny y yE E E E E θ= + = =
From Figure 2 we have that
500 cm 500 cmsin 0858583 cmd
θ = = =
( ) ( )( )2 2C 132 sin 2 181 10 NC 0858 311 10 NCE E θ= = times = times
The Electric Field
bull The electric field exists around all charged objects
bull This is shown byelectric fieldlines
bull Constructedid i ll hidentically to theforce lines
2202009
15
Drawing Electric Field Lines
bull Number of lines at a charge is proportional to the chargethe charge
Electric Field Lines
bull The charge with ‐3qmust have three times as many lines as the charge with +q and 15 times the lines for +‐ 2qthe charge with q and 15 times the lines for 2q
2202009
16
Parallel Plates of Charge
bull Adding fields from each point on surface causes field lines parallel to the plates to cancel
PositivelyCharged
NegativelyCharged
bull Away from the edges the field is parallel and perpendicular to the plate(s)
Electric Field amp Conductors
bull In conductors excess electrons will move when an electric field is present
bull For ldquosteady staterdquo (ie no moving charges) the net electric field must be zeror r r r
bull Therefore the charges move to the surface(s) of the conductor until they cancel the electric field inside the conductor
bull Field lines can start and end on a conductor at right angles to the surface of the conductor
0q rArr= = =F E F Er r r r
2202009
17
Electric Field Lines
1844 ndash Two particles are in a uniform electric field whose value is + 2500 NC The mass and charge of particle 1 are m1 = 14 x 10‐5 kg and q1 = ‐70 μC while the
di l f ti l 2 2 6 10 5 k d 18 C I iti llcorresponding values for particle 2 are m2 = 26 x 10‐5 kg and q2 = +18 μC Initially the particles are at rest The particles are both located on the same electric field line but are separated from each other by a distance d When released they accelerate but always remain at this same distance from each other Find d
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The drawing shows the two particles in the electric field Ex They are separated by a distance d If the particles are to stay at the same distance from each other after being released they must have the same acceleration so ax1 = ax2 According to Newtonrsquos second law the acceleration ax of each particle is equal to the net force ΣFx acting on it divided by its mass m or x xa F m= Σ
Electric Field Lines
1844 ndash (cont)
N t th t ti l 1 t b t th l ft f ti l 2 If ti l 1 t th i ht
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
Note that particle 1 must be to the left of particle 2 If particle 1 were to the right of particle 2 the particles would accelerate in opposite directions and the distance between them would not remain constant
1 2 1 1 2
1 21 2 1
11 1
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = minus
minusΣ= =
1 2 2 2 2
1 22 2 2
22 2
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = +
+Σ= =
Note the signs indicated the direction of the vector pointing from charge 2 to charge 1 for the force on charge 1
2202009
18
Electric Field Lines
1844 ndash (cont)
Th l ti f h h t b th if th di t b t th
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The acceleration of each charge must be the same if the distance between the charges remains unchanged
Solving for d
1 2
1 2 1 21 22 2
1 2
x x
x x
a a
q q q qq E k q E k
d dm m
=
minus +=
Solving for d
( )( )1 2 1 2
1 2 2 165 m
x
kq q m md
E q m q m+
= =minus
Electric Fluxbull Flux (from Latin ndash meaning to flow)
bull Electric flux measures amount of electric field passing through a surface
Er
Er
So to calculate find the normal to
φ
E EAΦ =
Flux through surface is Area of surface times the field flowing through that surface
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
( )cosE E AφΦ =
2202009
19
Electric Flux
( )cosE E AφΦ =
Er
So to calculate find the normal to
This can be expressed as the vector dot product
EΦ = sdotE Arr
cosE A φsdot =E Arr
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
cos
x x y y z z
E Aor
E A E A E A
φsdot =
sdot = + +
E A
E Arr
Electric Flux
1848 ndash A rectangular surface (016 m x 038m) is oriented in a uniform electric field of 580 NC What is the maximum possible electric flux through the surface
The maximum possible flux occurs when the electric field is parallel to the normal of the rectangular surface (that is when the angle between the direction of the field and the direction of the normal is zero) Then
( ) ( ) ( )2
( cos ) 580 N C (cos 0 ) 016 m 038 m
35 N m C E
E
E AφΦ = = deg
Φ = sdot
2202009
20
Gaussrsquo Lawbull Electric Flux through a closed surface is proportional to the charge enclosed by that surface
encE
qε
Φ =
bull The fundamental constant ε0 is called the permittivity of free space It is related to Coulombrsquos law by ε0 = 885 x 10‐12 C2(Nm2)
bull Gauss Law is most useful when the electric field on the entire surface is constant
bull Take a point charge Construct a sphere centered on the point charge (all points on the surface equidistant from the charge) ndash the electric field is constant
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so
0ε
0
14
kπε
=
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so φ = 0 cos φ = 1
bull The direction of the field is radially outward from the charge (or inward towards the charge if q is negative)
0 0
cosenc encE
q qEA φε ε
Φ =rArr=
( )22
0 0
44
encq qE r Er
πε πε
= rArr =
Gaussrsquo Law for a plate of chargebull In this case we take our closed surface to be a box oriented such that two faces are
parallel to the plate of charge and the others are perpendicular to the plate
Si th l t i fi ld f l t f h i di l t th f f thbull Since the electric field from a plate of charge is perpendicular to the surface of the plate the sides that are perpendicular to the plate have no flux
Er
bull The charge enclosed by the box is the surface charge density multiplied by the area the box passes through the plate Aσthe plate
bull The area of the box with flux is 2A
Aσ
0 0 0E
q AEA Eσ σε ε ε
rArr rArrΦ = = =
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
9
Coulombrsquos Law1821 ndash (cont)
( )( )
( )( )
2charged uncharged
max max2Centripetal
Centripetal forcef
2 KE 2 KE1 2
k qT T
r rr+ = =
1424314243
Subtracting Equation (2) from Equation (1) eliminates Tmax and gives
( ) ( )2charged uncharged
2
2 KE ndash KEk qrr
⎡ ⎤⎣ ⎦=
force
Solving for |q| gives
( ) ( ) ( ) ( )charged uncharged ndash59 2 2
2 KE ndash KE 2 30 m 518 J ndash 500 J35 10 C
899 10 N m C
rq
k
⎡ ⎤⎣ ⎦= = = times
times sdot
The Electric Fieldbull Electric charges create and exert and experience the electric force
q q q
bull We define the Electric Field as the force a test charge would feel if placed at that point
2 2ˆ ˆi j j
i ij i i iji j i jij ij
q q qk q k
r rne ne
rArr= =sum sumF r F rr r
bull The electric field exists around all charged objects
2ˆj
i j
qk q
rne
= rArr =sumE r F Er r r
2202009
10
Electric Force Lines
bull Michael Faraday saw the force caused by one charge to becaused by one charge to be from lines emanating from the charges
bull Lines begin on Positive charges and end on Negative charges flowing smoothly through space never crossing
Electric Force Lines
bull Faraday imagined these lines d h fand the force was greater
where the lines are denser weaker where they are spread out
2202009
11
Electric Field Lines
bull Go out from positive charges and into ti hnegative charges
Positive Negative Neutral
The Inverse Square Law
bull Many laws based on the geometry of three‐di i ldimensional space
1 22
2 24 4
g
sound light
Gm mFr
P LI Ir rπ π
= minus
= =
1 22e
q qF kr
=
2202009
12
The Electric Field
1828 ndash Four point charges have the same magnitude of 24 x 10‐12 C and are fixed to the corners of a square that is 40 cm on a side Three of the charges are positive d i ti D t i th it d f th t l t i fi ld th t i tand one is negative Determine the magnitude of the net electric field that exists
at the center of the square
A
B C
D-
++
+
EA
EC
EB
ED
The drawing at the right shows each of the field contributions at the center of the square (see black dot) Each is directed along a diagonal of the square Note that ED and EB point in opposite directions and therefore cancel since they have the same magnitude In contrast EA and EC point in the same direction toward corner A and therefore combine to give a net field thatcorner A and therefore combine to give a net field that is twice the magnitude of EA or EC In other words the net field at the center of the square is given by the following vector equation
2
Σ = + + +
= + + minus
= +
=
A B C D
A B C B
A C
A
E E E E EE E E EE EE
The Electric Field
1828 ndash (cont)
B C++
ED
The magnitude of the net field is A 22 2k q
E Er
Σ = =
A D- +
EA
EC
EB
In this result r is the distance from a corner to the center of the square which is one half of the diagonal distance d Using L for the length of a side of the square and taking advantage of the Pythagorean theorem we have With this substitution for r the magnitude of the net field becomes
r
2r L=
4k k
( )( ) ( )
( )
2 2
9 2 2 12
2
42
2
4 899 10 N m C 24 10 C
0040 m
54 NC
k q k qE
LL
E
E
minus
Σ = =
times sdot timesΣ =
Σ =
2202009
13
Electric Field Lines
1834 ndash Review Conceptual Example 12 before attempting to work this problem The magnitude of each of the charges in Figure 18‐21 is 86 x 10‐12 C The lengths of th id f th t l 3 00 d 5 00 Fi d th it d f ththe sides of the rectangles are 300 cm and 500 cm Find the magnitude of the electric field at the center of the rectangle in Figures 18‐21a and b
1 2
34
1
4
+ q
+ q + q
- qThe figure at the right shows the configuration given in text Figure 1821a The electric field at the center of the rectangle is the resultant of the electric fields at the center due to each of the four charges As discussed in Conceptual Example 11 the magnitudes of the electric field at the center due to each of the four charges are equal However the fields produced by the charges in corners 1 and 3 are in opposite directions Since they have the same magnitudes they combine to give directions Since they have the same magnitudes they combine to givezero resultant The fields produced by the charges in corners 2 and 4 point in the same direction (toward corner 2) Thus EC = EC2 + EC4 where EC is the magnitude of the electric field at the center of the rectangle and EC2 and EC4 are the magnitudes of the electric field at the center due to the charges in corners 2 and 4 respectively Since both EC2 and EC4 have the same magnitude we have EC = 2 EC2
Electric Field Lines1834 ndash (cont)
The distance r from any of the charges to the center of the rectangle can be found using the Pythagorean theorem
1 21
d500 cm
2 2(3 00 ) +(5 00 ) 5 83d
The figure at the right shows the configuration given in text Fi 18 21b All f h t ib t t
344
300 cm
θ
2 2(300 cm) +(500 cm) 583 cmd = =
2Therefore 292 cm 292 10 m2dr minus= = = times
9 2 2 1222
2 2 2 22 2(899 10 N m C )( 860 10 C)2 181 10 NC
(292 10 m)C Ckq
E Er
minus
minustimes sdot minus times
= = = = minus timestimes
- q - q
Figure 1821b All four charges contribute a non‐zero component to the electric field at the center of the rectangle As discussed in Conceptual Example 11 the contribution from the charges in corners 2 and 4 point toward corner 2 and the contribution from the charges in corners 1 and 3 point toward corner 1
1 2
34
1
4+ q + q
C
E 13 E 24
2202009
14
Electric Field Lines1834 ndash (cont)
Notice also the magnitudes of E24 and E13 are equal and from the first part of this problem we know that
E24 = E13 = 181 times 102 NC
1 21- q - q
E 13 E 24
The electric field at the center of the rectangle is the vector sum of E24 and E13 The x components of E24 and E13 are equal in magnitude and opposite in direction hence
(E13)x ndash (E24)x = 0
Therefore
F Fi 2 h th t
344+ q + q
C
C 13 24 13 13( ) ( ) 2( ) 2( )siny y yE E E E E θ= + = =
From Figure 2 we have that
500 cm 500 cmsin 0858583 cmd
θ = = =
( ) ( )( )2 2C 132 sin 2 181 10 NC 0858 311 10 NCE E θ= = times = times
The Electric Field
bull The electric field exists around all charged objects
bull This is shown byelectric fieldlines
bull Constructedid i ll hidentically to theforce lines
2202009
15
Drawing Electric Field Lines
bull Number of lines at a charge is proportional to the chargethe charge
Electric Field Lines
bull The charge with ‐3qmust have three times as many lines as the charge with +q and 15 times the lines for +‐ 2qthe charge with q and 15 times the lines for 2q
2202009
16
Parallel Plates of Charge
bull Adding fields from each point on surface causes field lines parallel to the plates to cancel
PositivelyCharged
NegativelyCharged
bull Away from the edges the field is parallel and perpendicular to the plate(s)
Electric Field amp Conductors
bull In conductors excess electrons will move when an electric field is present
bull For ldquosteady staterdquo (ie no moving charges) the net electric field must be zeror r r r
bull Therefore the charges move to the surface(s) of the conductor until they cancel the electric field inside the conductor
bull Field lines can start and end on a conductor at right angles to the surface of the conductor
0q rArr= = =F E F Er r r r
2202009
17
Electric Field Lines
1844 ndash Two particles are in a uniform electric field whose value is + 2500 NC The mass and charge of particle 1 are m1 = 14 x 10‐5 kg and q1 = ‐70 μC while the
di l f ti l 2 2 6 10 5 k d 18 C I iti llcorresponding values for particle 2 are m2 = 26 x 10‐5 kg and q2 = +18 μC Initially the particles are at rest The particles are both located on the same electric field line but are separated from each other by a distance d When released they accelerate but always remain at this same distance from each other Find d
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The drawing shows the two particles in the electric field Ex They are separated by a distance d If the particles are to stay at the same distance from each other after being released they must have the same acceleration so ax1 = ax2 According to Newtonrsquos second law the acceleration ax of each particle is equal to the net force ΣFx acting on it divided by its mass m or x xa F m= Σ
Electric Field Lines
1844 ndash (cont)
N t th t ti l 1 t b t th l ft f ti l 2 If ti l 1 t th i ht
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
Note that particle 1 must be to the left of particle 2 If particle 1 were to the right of particle 2 the particles would accelerate in opposite directions and the distance between them would not remain constant
1 2 1 1 2
1 21 2 1
11 1
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = minus
minusΣ= =
1 2 2 2 2
1 22 2 2
22 2
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = +
+Σ= =
Note the signs indicated the direction of the vector pointing from charge 2 to charge 1 for the force on charge 1
2202009
18
Electric Field Lines
1844 ndash (cont)
Th l ti f h h t b th if th di t b t th
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The acceleration of each charge must be the same if the distance between the charges remains unchanged
Solving for d
1 2
1 2 1 21 22 2
1 2
x x
x x
a a
q q q qq E k q E k
d dm m
=
minus +=
Solving for d
( )( )1 2 1 2
1 2 2 165 m
x
kq q m md
E q m q m+
= =minus
Electric Fluxbull Flux (from Latin ndash meaning to flow)
bull Electric flux measures amount of electric field passing through a surface
Er
Er
So to calculate find the normal to
φ
E EAΦ =
Flux through surface is Area of surface times the field flowing through that surface
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
( )cosE E AφΦ =
2202009
19
Electric Flux
( )cosE E AφΦ =
Er
So to calculate find the normal to
This can be expressed as the vector dot product
EΦ = sdotE Arr
cosE A φsdot =E Arr
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
cos
x x y y z z
E Aor
E A E A E A
φsdot =
sdot = + +
E A
E Arr
Electric Flux
1848 ndash A rectangular surface (016 m x 038m) is oriented in a uniform electric field of 580 NC What is the maximum possible electric flux through the surface
The maximum possible flux occurs when the electric field is parallel to the normal of the rectangular surface (that is when the angle between the direction of the field and the direction of the normal is zero) Then
( ) ( ) ( )2
( cos ) 580 N C (cos 0 ) 016 m 038 m
35 N m C E
E
E AφΦ = = deg
Φ = sdot
2202009
20
Gaussrsquo Lawbull Electric Flux through a closed surface is proportional to the charge enclosed by that surface
encE
qε
Φ =
bull The fundamental constant ε0 is called the permittivity of free space It is related to Coulombrsquos law by ε0 = 885 x 10‐12 C2(Nm2)
bull Gauss Law is most useful when the electric field on the entire surface is constant
bull Take a point charge Construct a sphere centered on the point charge (all points on the surface equidistant from the charge) ndash the electric field is constant
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so
0ε
0
14
kπε
=
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so φ = 0 cos φ = 1
bull The direction of the field is radially outward from the charge (or inward towards the charge if q is negative)
0 0
cosenc encE
q qEA φε ε
Φ =rArr=
( )22
0 0
44
encq qE r Er
πε πε
= rArr =
Gaussrsquo Law for a plate of chargebull In this case we take our closed surface to be a box oriented such that two faces are
parallel to the plate of charge and the others are perpendicular to the plate
Si th l t i fi ld f l t f h i di l t th f f thbull Since the electric field from a plate of charge is perpendicular to the surface of the plate the sides that are perpendicular to the plate have no flux
Er
bull The charge enclosed by the box is the surface charge density multiplied by the area the box passes through the plate Aσthe plate
bull The area of the box with flux is 2A
Aσ
0 0 0E
q AEA Eσ σε ε ε
rArr rArrΦ = = =
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
10
Electric Force Lines
bull Michael Faraday saw the force caused by one charge to becaused by one charge to be from lines emanating from the charges
bull Lines begin on Positive charges and end on Negative charges flowing smoothly through space never crossing
Electric Force Lines
bull Faraday imagined these lines d h fand the force was greater
where the lines are denser weaker where they are spread out
2202009
11
Electric Field Lines
bull Go out from positive charges and into ti hnegative charges
Positive Negative Neutral
The Inverse Square Law
bull Many laws based on the geometry of three‐di i ldimensional space
1 22
2 24 4
g
sound light
Gm mFr
P LI Ir rπ π
= minus
= =
1 22e
q qF kr
=
2202009
12
The Electric Field
1828 ndash Four point charges have the same magnitude of 24 x 10‐12 C and are fixed to the corners of a square that is 40 cm on a side Three of the charges are positive d i ti D t i th it d f th t l t i fi ld th t i tand one is negative Determine the magnitude of the net electric field that exists
at the center of the square
A
B C
D-
++
+
EA
EC
EB
ED
The drawing at the right shows each of the field contributions at the center of the square (see black dot) Each is directed along a diagonal of the square Note that ED and EB point in opposite directions and therefore cancel since they have the same magnitude In contrast EA and EC point in the same direction toward corner A and therefore combine to give a net field thatcorner A and therefore combine to give a net field that is twice the magnitude of EA or EC In other words the net field at the center of the square is given by the following vector equation
2
Σ = + + +
= + + minus
= +
=
A B C D
A B C B
A C
A
E E E E EE E E EE EE
The Electric Field
1828 ndash (cont)
B C++
ED
The magnitude of the net field is A 22 2k q
E Er
Σ = =
A D- +
EA
EC
EB
In this result r is the distance from a corner to the center of the square which is one half of the diagonal distance d Using L for the length of a side of the square and taking advantage of the Pythagorean theorem we have With this substitution for r the magnitude of the net field becomes
r
2r L=
4k k
( )( ) ( )
( )
2 2
9 2 2 12
2
42
2
4 899 10 N m C 24 10 C
0040 m
54 NC
k q k qE
LL
E
E
minus
Σ = =
times sdot timesΣ =
Σ =
2202009
13
Electric Field Lines
1834 ndash Review Conceptual Example 12 before attempting to work this problem The magnitude of each of the charges in Figure 18‐21 is 86 x 10‐12 C The lengths of th id f th t l 3 00 d 5 00 Fi d th it d f ththe sides of the rectangles are 300 cm and 500 cm Find the magnitude of the electric field at the center of the rectangle in Figures 18‐21a and b
1 2
34
1
4
+ q
+ q + q
- qThe figure at the right shows the configuration given in text Figure 1821a The electric field at the center of the rectangle is the resultant of the electric fields at the center due to each of the four charges As discussed in Conceptual Example 11 the magnitudes of the electric field at the center due to each of the four charges are equal However the fields produced by the charges in corners 1 and 3 are in opposite directions Since they have the same magnitudes they combine to give directions Since they have the same magnitudes they combine to givezero resultant The fields produced by the charges in corners 2 and 4 point in the same direction (toward corner 2) Thus EC = EC2 + EC4 where EC is the magnitude of the electric field at the center of the rectangle and EC2 and EC4 are the magnitudes of the electric field at the center due to the charges in corners 2 and 4 respectively Since both EC2 and EC4 have the same magnitude we have EC = 2 EC2
Electric Field Lines1834 ndash (cont)
The distance r from any of the charges to the center of the rectangle can be found using the Pythagorean theorem
1 21
d500 cm
2 2(3 00 ) +(5 00 ) 5 83d
The figure at the right shows the configuration given in text Fi 18 21b All f h t ib t t
344
300 cm
θ
2 2(300 cm) +(500 cm) 583 cmd = =
2Therefore 292 cm 292 10 m2dr minus= = = times
9 2 2 1222
2 2 2 22 2(899 10 N m C )( 860 10 C)2 181 10 NC
(292 10 m)C Ckq
E Er
minus
minustimes sdot minus times
= = = = minus timestimes
- q - q
Figure 1821b All four charges contribute a non‐zero component to the electric field at the center of the rectangle As discussed in Conceptual Example 11 the contribution from the charges in corners 2 and 4 point toward corner 2 and the contribution from the charges in corners 1 and 3 point toward corner 1
1 2
34
1
4+ q + q
C
E 13 E 24
2202009
14
Electric Field Lines1834 ndash (cont)
Notice also the magnitudes of E24 and E13 are equal and from the first part of this problem we know that
E24 = E13 = 181 times 102 NC
1 21- q - q
E 13 E 24
The electric field at the center of the rectangle is the vector sum of E24 and E13 The x components of E24 and E13 are equal in magnitude and opposite in direction hence
(E13)x ndash (E24)x = 0
Therefore
F Fi 2 h th t
344+ q + q
C
C 13 24 13 13( ) ( ) 2( ) 2( )siny y yE E E E E θ= + = =
From Figure 2 we have that
500 cm 500 cmsin 0858583 cmd
θ = = =
( ) ( )( )2 2C 132 sin 2 181 10 NC 0858 311 10 NCE E θ= = times = times
The Electric Field
bull The electric field exists around all charged objects
bull This is shown byelectric fieldlines
bull Constructedid i ll hidentically to theforce lines
2202009
15
Drawing Electric Field Lines
bull Number of lines at a charge is proportional to the chargethe charge
Electric Field Lines
bull The charge with ‐3qmust have three times as many lines as the charge with +q and 15 times the lines for +‐ 2qthe charge with q and 15 times the lines for 2q
2202009
16
Parallel Plates of Charge
bull Adding fields from each point on surface causes field lines parallel to the plates to cancel
PositivelyCharged
NegativelyCharged
bull Away from the edges the field is parallel and perpendicular to the plate(s)
Electric Field amp Conductors
bull In conductors excess electrons will move when an electric field is present
bull For ldquosteady staterdquo (ie no moving charges) the net electric field must be zeror r r r
bull Therefore the charges move to the surface(s) of the conductor until they cancel the electric field inside the conductor
bull Field lines can start and end on a conductor at right angles to the surface of the conductor
0q rArr= = =F E F Er r r r
2202009
17
Electric Field Lines
1844 ndash Two particles are in a uniform electric field whose value is + 2500 NC The mass and charge of particle 1 are m1 = 14 x 10‐5 kg and q1 = ‐70 μC while the
di l f ti l 2 2 6 10 5 k d 18 C I iti llcorresponding values for particle 2 are m2 = 26 x 10‐5 kg and q2 = +18 μC Initially the particles are at rest The particles are both located on the same electric field line but are separated from each other by a distance d When released they accelerate but always remain at this same distance from each other Find d
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The drawing shows the two particles in the electric field Ex They are separated by a distance d If the particles are to stay at the same distance from each other after being released they must have the same acceleration so ax1 = ax2 According to Newtonrsquos second law the acceleration ax of each particle is equal to the net force ΣFx acting on it divided by its mass m or x xa F m= Σ
Electric Field Lines
1844 ndash (cont)
N t th t ti l 1 t b t th l ft f ti l 2 If ti l 1 t th i ht
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
Note that particle 1 must be to the left of particle 2 If particle 1 were to the right of particle 2 the particles would accelerate in opposite directions and the distance between them would not remain constant
1 2 1 1 2
1 21 2 1
11 1
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = minus
minusΣ= =
1 2 2 2 2
1 22 2 2
22 2
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = +
+Σ= =
Note the signs indicated the direction of the vector pointing from charge 2 to charge 1 for the force on charge 1
2202009
18
Electric Field Lines
1844 ndash (cont)
Th l ti f h h t b th if th di t b t th
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The acceleration of each charge must be the same if the distance between the charges remains unchanged
Solving for d
1 2
1 2 1 21 22 2
1 2
x x
x x
a a
q q q qq E k q E k
d dm m
=
minus +=
Solving for d
( )( )1 2 1 2
1 2 2 165 m
x
kq q m md
E q m q m+
= =minus
Electric Fluxbull Flux (from Latin ndash meaning to flow)
bull Electric flux measures amount of electric field passing through a surface
Er
Er
So to calculate find the normal to
φ
E EAΦ =
Flux through surface is Area of surface times the field flowing through that surface
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
( )cosE E AφΦ =
2202009
19
Electric Flux
( )cosE E AφΦ =
Er
So to calculate find the normal to
This can be expressed as the vector dot product
EΦ = sdotE Arr
cosE A φsdot =E Arr
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
cos
x x y y z z
E Aor
E A E A E A
φsdot =
sdot = + +
E A
E Arr
Electric Flux
1848 ndash A rectangular surface (016 m x 038m) is oriented in a uniform electric field of 580 NC What is the maximum possible electric flux through the surface
The maximum possible flux occurs when the electric field is parallel to the normal of the rectangular surface (that is when the angle between the direction of the field and the direction of the normal is zero) Then
( ) ( ) ( )2
( cos ) 580 N C (cos 0 ) 016 m 038 m
35 N m C E
E
E AφΦ = = deg
Φ = sdot
2202009
20
Gaussrsquo Lawbull Electric Flux through a closed surface is proportional to the charge enclosed by that surface
encE
qε
Φ =
bull The fundamental constant ε0 is called the permittivity of free space It is related to Coulombrsquos law by ε0 = 885 x 10‐12 C2(Nm2)
bull Gauss Law is most useful when the electric field on the entire surface is constant
bull Take a point charge Construct a sphere centered on the point charge (all points on the surface equidistant from the charge) ndash the electric field is constant
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so
0ε
0
14
kπε
=
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so φ = 0 cos φ = 1
bull The direction of the field is radially outward from the charge (or inward towards the charge if q is negative)
0 0
cosenc encE
q qEA φε ε
Φ =rArr=
( )22
0 0
44
encq qE r Er
πε πε
= rArr =
Gaussrsquo Law for a plate of chargebull In this case we take our closed surface to be a box oriented such that two faces are
parallel to the plate of charge and the others are perpendicular to the plate
Si th l t i fi ld f l t f h i di l t th f f thbull Since the electric field from a plate of charge is perpendicular to the surface of the plate the sides that are perpendicular to the plate have no flux
Er
bull The charge enclosed by the box is the surface charge density multiplied by the area the box passes through the plate Aσthe plate
bull The area of the box with flux is 2A
Aσ
0 0 0E
q AEA Eσ σε ε ε
rArr rArrΦ = = =
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
11
Electric Field Lines
bull Go out from positive charges and into ti hnegative charges
Positive Negative Neutral
The Inverse Square Law
bull Many laws based on the geometry of three‐di i ldimensional space
1 22
2 24 4
g
sound light
Gm mFr
P LI Ir rπ π
= minus
= =
1 22e
q qF kr
=
2202009
12
The Electric Field
1828 ndash Four point charges have the same magnitude of 24 x 10‐12 C and are fixed to the corners of a square that is 40 cm on a side Three of the charges are positive d i ti D t i th it d f th t l t i fi ld th t i tand one is negative Determine the magnitude of the net electric field that exists
at the center of the square
A
B C
D-
++
+
EA
EC
EB
ED
The drawing at the right shows each of the field contributions at the center of the square (see black dot) Each is directed along a diagonal of the square Note that ED and EB point in opposite directions and therefore cancel since they have the same magnitude In contrast EA and EC point in the same direction toward corner A and therefore combine to give a net field thatcorner A and therefore combine to give a net field that is twice the magnitude of EA or EC In other words the net field at the center of the square is given by the following vector equation
2
Σ = + + +
= + + minus
= +
=
A B C D
A B C B
A C
A
E E E E EE E E EE EE
The Electric Field
1828 ndash (cont)
B C++
ED
The magnitude of the net field is A 22 2k q
E Er
Σ = =
A D- +
EA
EC
EB
In this result r is the distance from a corner to the center of the square which is one half of the diagonal distance d Using L for the length of a side of the square and taking advantage of the Pythagorean theorem we have With this substitution for r the magnitude of the net field becomes
r
2r L=
4k k
( )( ) ( )
( )
2 2
9 2 2 12
2
42
2
4 899 10 N m C 24 10 C
0040 m
54 NC
k q k qE
LL
E
E
minus
Σ = =
times sdot timesΣ =
Σ =
2202009
13
Electric Field Lines
1834 ndash Review Conceptual Example 12 before attempting to work this problem The magnitude of each of the charges in Figure 18‐21 is 86 x 10‐12 C The lengths of th id f th t l 3 00 d 5 00 Fi d th it d f ththe sides of the rectangles are 300 cm and 500 cm Find the magnitude of the electric field at the center of the rectangle in Figures 18‐21a and b
1 2
34
1
4
+ q
+ q + q
- qThe figure at the right shows the configuration given in text Figure 1821a The electric field at the center of the rectangle is the resultant of the electric fields at the center due to each of the four charges As discussed in Conceptual Example 11 the magnitudes of the electric field at the center due to each of the four charges are equal However the fields produced by the charges in corners 1 and 3 are in opposite directions Since they have the same magnitudes they combine to give directions Since they have the same magnitudes they combine to givezero resultant The fields produced by the charges in corners 2 and 4 point in the same direction (toward corner 2) Thus EC = EC2 + EC4 where EC is the magnitude of the electric field at the center of the rectangle and EC2 and EC4 are the magnitudes of the electric field at the center due to the charges in corners 2 and 4 respectively Since both EC2 and EC4 have the same magnitude we have EC = 2 EC2
Electric Field Lines1834 ndash (cont)
The distance r from any of the charges to the center of the rectangle can be found using the Pythagorean theorem
1 21
d500 cm
2 2(3 00 ) +(5 00 ) 5 83d
The figure at the right shows the configuration given in text Fi 18 21b All f h t ib t t
344
300 cm
θ
2 2(300 cm) +(500 cm) 583 cmd = =
2Therefore 292 cm 292 10 m2dr minus= = = times
9 2 2 1222
2 2 2 22 2(899 10 N m C )( 860 10 C)2 181 10 NC
(292 10 m)C Ckq
E Er
minus
minustimes sdot minus times
= = = = minus timestimes
- q - q
Figure 1821b All four charges contribute a non‐zero component to the electric field at the center of the rectangle As discussed in Conceptual Example 11 the contribution from the charges in corners 2 and 4 point toward corner 2 and the contribution from the charges in corners 1 and 3 point toward corner 1
1 2
34
1
4+ q + q
C
E 13 E 24
2202009
14
Electric Field Lines1834 ndash (cont)
Notice also the magnitudes of E24 and E13 are equal and from the first part of this problem we know that
E24 = E13 = 181 times 102 NC
1 21- q - q
E 13 E 24
The electric field at the center of the rectangle is the vector sum of E24 and E13 The x components of E24 and E13 are equal in magnitude and opposite in direction hence
(E13)x ndash (E24)x = 0
Therefore
F Fi 2 h th t
344+ q + q
C
C 13 24 13 13( ) ( ) 2( ) 2( )siny y yE E E E E θ= + = =
From Figure 2 we have that
500 cm 500 cmsin 0858583 cmd
θ = = =
( ) ( )( )2 2C 132 sin 2 181 10 NC 0858 311 10 NCE E θ= = times = times
The Electric Field
bull The electric field exists around all charged objects
bull This is shown byelectric fieldlines
bull Constructedid i ll hidentically to theforce lines
2202009
15
Drawing Electric Field Lines
bull Number of lines at a charge is proportional to the chargethe charge
Electric Field Lines
bull The charge with ‐3qmust have three times as many lines as the charge with +q and 15 times the lines for +‐ 2qthe charge with q and 15 times the lines for 2q
2202009
16
Parallel Plates of Charge
bull Adding fields from each point on surface causes field lines parallel to the plates to cancel
PositivelyCharged
NegativelyCharged
bull Away from the edges the field is parallel and perpendicular to the plate(s)
Electric Field amp Conductors
bull In conductors excess electrons will move when an electric field is present
bull For ldquosteady staterdquo (ie no moving charges) the net electric field must be zeror r r r
bull Therefore the charges move to the surface(s) of the conductor until they cancel the electric field inside the conductor
bull Field lines can start and end on a conductor at right angles to the surface of the conductor
0q rArr= = =F E F Er r r r
2202009
17
Electric Field Lines
1844 ndash Two particles are in a uniform electric field whose value is + 2500 NC The mass and charge of particle 1 are m1 = 14 x 10‐5 kg and q1 = ‐70 μC while the
di l f ti l 2 2 6 10 5 k d 18 C I iti llcorresponding values for particle 2 are m2 = 26 x 10‐5 kg and q2 = +18 μC Initially the particles are at rest The particles are both located on the same electric field line but are separated from each other by a distance d When released they accelerate but always remain at this same distance from each other Find d
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The drawing shows the two particles in the electric field Ex They are separated by a distance d If the particles are to stay at the same distance from each other after being released they must have the same acceleration so ax1 = ax2 According to Newtonrsquos second law the acceleration ax of each particle is equal to the net force ΣFx acting on it divided by its mass m or x xa F m= Σ
Electric Field Lines
1844 ndash (cont)
N t th t ti l 1 t b t th l ft f ti l 2 If ti l 1 t th i ht
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
Note that particle 1 must be to the left of particle 2 If particle 1 were to the right of particle 2 the particles would accelerate in opposite directions and the distance between them would not remain constant
1 2 1 1 2
1 21 2 1
11 1
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = minus
minusΣ= =
1 2 2 2 2
1 22 2 2
22 2
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = +
+Σ= =
Note the signs indicated the direction of the vector pointing from charge 2 to charge 1 for the force on charge 1
2202009
18
Electric Field Lines
1844 ndash (cont)
Th l ti f h h t b th if th di t b t th
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The acceleration of each charge must be the same if the distance between the charges remains unchanged
Solving for d
1 2
1 2 1 21 22 2
1 2
x x
x x
a a
q q q qq E k q E k
d dm m
=
minus +=
Solving for d
( )( )1 2 1 2
1 2 2 165 m
x
kq q m md
E q m q m+
= =minus
Electric Fluxbull Flux (from Latin ndash meaning to flow)
bull Electric flux measures amount of electric field passing through a surface
Er
Er
So to calculate find the normal to
φ
E EAΦ =
Flux through surface is Area of surface times the field flowing through that surface
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
( )cosE E AφΦ =
2202009
19
Electric Flux
( )cosE E AφΦ =
Er
So to calculate find the normal to
This can be expressed as the vector dot product
EΦ = sdotE Arr
cosE A φsdot =E Arr
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
cos
x x y y z z
E Aor
E A E A E A
φsdot =
sdot = + +
E A
E Arr
Electric Flux
1848 ndash A rectangular surface (016 m x 038m) is oriented in a uniform electric field of 580 NC What is the maximum possible electric flux through the surface
The maximum possible flux occurs when the electric field is parallel to the normal of the rectangular surface (that is when the angle between the direction of the field and the direction of the normal is zero) Then
( ) ( ) ( )2
( cos ) 580 N C (cos 0 ) 016 m 038 m
35 N m C E
E
E AφΦ = = deg
Φ = sdot
2202009
20
Gaussrsquo Lawbull Electric Flux through a closed surface is proportional to the charge enclosed by that surface
encE
qε
Φ =
bull The fundamental constant ε0 is called the permittivity of free space It is related to Coulombrsquos law by ε0 = 885 x 10‐12 C2(Nm2)
bull Gauss Law is most useful when the electric field on the entire surface is constant
bull Take a point charge Construct a sphere centered on the point charge (all points on the surface equidistant from the charge) ndash the electric field is constant
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so
0ε
0
14
kπε
=
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so φ = 0 cos φ = 1
bull The direction of the field is radially outward from the charge (or inward towards the charge if q is negative)
0 0
cosenc encE
q qEA φε ε
Φ =rArr=
( )22
0 0
44
encq qE r Er
πε πε
= rArr =
Gaussrsquo Law for a plate of chargebull In this case we take our closed surface to be a box oriented such that two faces are
parallel to the plate of charge and the others are perpendicular to the plate
Si th l t i fi ld f l t f h i di l t th f f thbull Since the electric field from a plate of charge is perpendicular to the surface of the plate the sides that are perpendicular to the plate have no flux
Er
bull The charge enclosed by the box is the surface charge density multiplied by the area the box passes through the plate Aσthe plate
bull The area of the box with flux is 2A
Aσ
0 0 0E
q AEA Eσ σε ε ε
rArr rArrΦ = = =
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
12
The Electric Field
1828 ndash Four point charges have the same magnitude of 24 x 10‐12 C and are fixed to the corners of a square that is 40 cm on a side Three of the charges are positive d i ti D t i th it d f th t l t i fi ld th t i tand one is negative Determine the magnitude of the net electric field that exists
at the center of the square
A
B C
D-
++
+
EA
EC
EB
ED
The drawing at the right shows each of the field contributions at the center of the square (see black dot) Each is directed along a diagonal of the square Note that ED and EB point in opposite directions and therefore cancel since they have the same magnitude In contrast EA and EC point in the same direction toward corner A and therefore combine to give a net field thatcorner A and therefore combine to give a net field that is twice the magnitude of EA or EC In other words the net field at the center of the square is given by the following vector equation
2
Σ = + + +
= + + minus
= +
=
A B C D
A B C B
A C
A
E E E E EE E E EE EE
The Electric Field
1828 ndash (cont)
B C++
ED
The magnitude of the net field is A 22 2k q
E Er
Σ = =
A D- +
EA
EC
EB
In this result r is the distance from a corner to the center of the square which is one half of the diagonal distance d Using L for the length of a side of the square and taking advantage of the Pythagorean theorem we have With this substitution for r the magnitude of the net field becomes
r
2r L=
4k k
( )( ) ( )
( )
2 2
9 2 2 12
2
42
2
4 899 10 N m C 24 10 C
0040 m
54 NC
k q k qE
LL
E
E
minus
Σ = =
times sdot timesΣ =
Σ =
2202009
13
Electric Field Lines
1834 ndash Review Conceptual Example 12 before attempting to work this problem The magnitude of each of the charges in Figure 18‐21 is 86 x 10‐12 C The lengths of th id f th t l 3 00 d 5 00 Fi d th it d f ththe sides of the rectangles are 300 cm and 500 cm Find the magnitude of the electric field at the center of the rectangle in Figures 18‐21a and b
1 2
34
1
4
+ q
+ q + q
- qThe figure at the right shows the configuration given in text Figure 1821a The electric field at the center of the rectangle is the resultant of the electric fields at the center due to each of the four charges As discussed in Conceptual Example 11 the magnitudes of the electric field at the center due to each of the four charges are equal However the fields produced by the charges in corners 1 and 3 are in opposite directions Since they have the same magnitudes they combine to give directions Since they have the same magnitudes they combine to givezero resultant The fields produced by the charges in corners 2 and 4 point in the same direction (toward corner 2) Thus EC = EC2 + EC4 where EC is the magnitude of the electric field at the center of the rectangle and EC2 and EC4 are the magnitudes of the electric field at the center due to the charges in corners 2 and 4 respectively Since both EC2 and EC4 have the same magnitude we have EC = 2 EC2
Electric Field Lines1834 ndash (cont)
The distance r from any of the charges to the center of the rectangle can be found using the Pythagorean theorem
1 21
d500 cm
2 2(3 00 ) +(5 00 ) 5 83d
The figure at the right shows the configuration given in text Fi 18 21b All f h t ib t t
344
300 cm
θ
2 2(300 cm) +(500 cm) 583 cmd = =
2Therefore 292 cm 292 10 m2dr minus= = = times
9 2 2 1222
2 2 2 22 2(899 10 N m C )( 860 10 C)2 181 10 NC
(292 10 m)C Ckq
E Er
minus
minustimes sdot minus times
= = = = minus timestimes
- q - q
Figure 1821b All four charges contribute a non‐zero component to the electric field at the center of the rectangle As discussed in Conceptual Example 11 the contribution from the charges in corners 2 and 4 point toward corner 2 and the contribution from the charges in corners 1 and 3 point toward corner 1
1 2
34
1
4+ q + q
C
E 13 E 24
2202009
14
Electric Field Lines1834 ndash (cont)
Notice also the magnitudes of E24 and E13 are equal and from the first part of this problem we know that
E24 = E13 = 181 times 102 NC
1 21- q - q
E 13 E 24
The electric field at the center of the rectangle is the vector sum of E24 and E13 The x components of E24 and E13 are equal in magnitude and opposite in direction hence
(E13)x ndash (E24)x = 0
Therefore
F Fi 2 h th t
344+ q + q
C
C 13 24 13 13( ) ( ) 2( ) 2( )siny y yE E E E E θ= + = =
From Figure 2 we have that
500 cm 500 cmsin 0858583 cmd
θ = = =
( ) ( )( )2 2C 132 sin 2 181 10 NC 0858 311 10 NCE E θ= = times = times
The Electric Field
bull The electric field exists around all charged objects
bull This is shown byelectric fieldlines
bull Constructedid i ll hidentically to theforce lines
2202009
15
Drawing Electric Field Lines
bull Number of lines at a charge is proportional to the chargethe charge
Electric Field Lines
bull The charge with ‐3qmust have three times as many lines as the charge with +q and 15 times the lines for +‐ 2qthe charge with q and 15 times the lines for 2q
2202009
16
Parallel Plates of Charge
bull Adding fields from each point on surface causes field lines parallel to the plates to cancel
PositivelyCharged
NegativelyCharged
bull Away from the edges the field is parallel and perpendicular to the plate(s)
Electric Field amp Conductors
bull In conductors excess electrons will move when an electric field is present
bull For ldquosteady staterdquo (ie no moving charges) the net electric field must be zeror r r r
bull Therefore the charges move to the surface(s) of the conductor until they cancel the electric field inside the conductor
bull Field lines can start and end on a conductor at right angles to the surface of the conductor
0q rArr= = =F E F Er r r r
2202009
17
Electric Field Lines
1844 ndash Two particles are in a uniform electric field whose value is + 2500 NC The mass and charge of particle 1 are m1 = 14 x 10‐5 kg and q1 = ‐70 μC while the
di l f ti l 2 2 6 10 5 k d 18 C I iti llcorresponding values for particle 2 are m2 = 26 x 10‐5 kg and q2 = +18 μC Initially the particles are at rest The particles are both located on the same electric field line but are separated from each other by a distance d When released they accelerate but always remain at this same distance from each other Find d
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The drawing shows the two particles in the electric field Ex They are separated by a distance d If the particles are to stay at the same distance from each other after being released they must have the same acceleration so ax1 = ax2 According to Newtonrsquos second law the acceleration ax of each particle is equal to the net force ΣFx acting on it divided by its mass m or x xa F m= Σ
Electric Field Lines
1844 ndash (cont)
N t th t ti l 1 t b t th l ft f ti l 2 If ti l 1 t th i ht
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
Note that particle 1 must be to the left of particle 2 If particle 1 were to the right of particle 2 the particles would accelerate in opposite directions and the distance between them would not remain constant
1 2 1 1 2
1 21 2 1
11 1
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = minus
minusΣ= =
1 2 2 2 2
1 22 2 2
22 2
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = +
+Σ= =
Note the signs indicated the direction of the vector pointing from charge 2 to charge 1 for the force on charge 1
2202009
18
Electric Field Lines
1844 ndash (cont)
Th l ti f h h t b th if th di t b t th
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The acceleration of each charge must be the same if the distance between the charges remains unchanged
Solving for d
1 2
1 2 1 21 22 2
1 2
x x
x x
a a
q q q qq E k q E k
d dm m
=
minus +=
Solving for d
( )( )1 2 1 2
1 2 2 165 m
x
kq q m md
E q m q m+
= =minus
Electric Fluxbull Flux (from Latin ndash meaning to flow)
bull Electric flux measures amount of electric field passing through a surface
Er
Er
So to calculate find the normal to
φ
E EAΦ =
Flux through surface is Area of surface times the field flowing through that surface
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
( )cosE E AφΦ =
2202009
19
Electric Flux
( )cosE E AφΦ =
Er
So to calculate find the normal to
This can be expressed as the vector dot product
EΦ = sdotE Arr
cosE A φsdot =E Arr
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
cos
x x y y z z
E Aor
E A E A E A
φsdot =
sdot = + +
E A
E Arr
Electric Flux
1848 ndash A rectangular surface (016 m x 038m) is oriented in a uniform electric field of 580 NC What is the maximum possible electric flux through the surface
The maximum possible flux occurs when the electric field is parallel to the normal of the rectangular surface (that is when the angle between the direction of the field and the direction of the normal is zero) Then
( ) ( ) ( )2
( cos ) 580 N C (cos 0 ) 016 m 038 m
35 N m C E
E
E AφΦ = = deg
Φ = sdot
2202009
20
Gaussrsquo Lawbull Electric Flux through a closed surface is proportional to the charge enclosed by that surface
encE
qε
Φ =
bull The fundamental constant ε0 is called the permittivity of free space It is related to Coulombrsquos law by ε0 = 885 x 10‐12 C2(Nm2)
bull Gauss Law is most useful when the electric field on the entire surface is constant
bull Take a point charge Construct a sphere centered on the point charge (all points on the surface equidistant from the charge) ndash the electric field is constant
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so
0ε
0
14
kπε
=
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so φ = 0 cos φ = 1
bull The direction of the field is radially outward from the charge (or inward towards the charge if q is negative)
0 0
cosenc encE
q qEA φε ε
Φ =rArr=
( )22
0 0
44
encq qE r Er
πε πε
= rArr =
Gaussrsquo Law for a plate of chargebull In this case we take our closed surface to be a box oriented such that two faces are
parallel to the plate of charge and the others are perpendicular to the plate
Si th l t i fi ld f l t f h i di l t th f f thbull Since the electric field from a plate of charge is perpendicular to the surface of the plate the sides that are perpendicular to the plate have no flux
Er
bull The charge enclosed by the box is the surface charge density multiplied by the area the box passes through the plate Aσthe plate
bull The area of the box with flux is 2A
Aσ
0 0 0E
q AEA Eσ σε ε ε
rArr rArrΦ = = =
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
13
Electric Field Lines
1834 ndash Review Conceptual Example 12 before attempting to work this problem The magnitude of each of the charges in Figure 18‐21 is 86 x 10‐12 C The lengths of th id f th t l 3 00 d 5 00 Fi d th it d f ththe sides of the rectangles are 300 cm and 500 cm Find the magnitude of the electric field at the center of the rectangle in Figures 18‐21a and b
1 2
34
1
4
+ q
+ q + q
- qThe figure at the right shows the configuration given in text Figure 1821a The electric field at the center of the rectangle is the resultant of the electric fields at the center due to each of the four charges As discussed in Conceptual Example 11 the magnitudes of the electric field at the center due to each of the four charges are equal However the fields produced by the charges in corners 1 and 3 are in opposite directions Since they have the same magnitudes they combine to give directions Since they have the same magnitudes they combine to givezero resultant The fields produced by the charges in corners 2 and 4 point in the same direction (toward corner 2) Thus EC = EC2 + EC4 where EC is the magnitude of the electric field at the center of the rectangle and EC2 and EC4 are the magnitudes of the electric field at the center due to the charges in corners 2 and 4 respectively Since both EC2 and EC4 have the same magnitude we have EC = 2 EC2
Electric Field Lines1834 ndash (cont)
The distance r from any of the charges to the center of the rectangle can be found using the Pythagorean theorem
1 21
d500 cm
2 2(3 00 ) +(5 00 ) 5 83d
The figure at the right shows the configuration given in text Fi 18 21b All f h t ib t t
344
300 cm
θ
2 2(300 cm) +(500 cm) 583 cmd = =
2Therefore 292 cm 292 10 m2dr minus= = = times
9 2 2 1222
2 2 2 22 2(899 10 N m C )( 860 10 C)2 181 10 NC
(292 10 m)C Ckq
E Er
minus
minustimes sdot minus times
= = = = minus timestimes
- q - q
Figure 1821b All four charges contribute a non‐zero component to the electric field at the center of the rectangle As discussed in Conceptual Example 11 the contribution from the charges in corners 2 and 4 point toward corner 2 and the contribution from the charges in corners 1 and 3 point toward corner 1
1 2
34
1
4+ q + q
C
E 13 E 24
2202009
14
Electric Field Lines1834 ndash (cont)
Notice also the magnitudes of E24 and E13 are equal and from the first part of this problem we know that
E24 = E13 = 181 times 102 NC
1 21- q - q
E 13 E 24
The electric field at the center of the rectangle is the vector sum of E24 and E13 The x components of E24 and E13 are equal in magnitude and opposite in direction hence
(E13)x ndash (E24)x = 0
Therefore
F Fi 2 h th t
344+ q + q
C
C 13 24 13 13( ) ( ) 2( ) 2( )siny y yE E E E E θ= + = =
From Figure 2 we have that
500 cm 500 cmsin 0858583 cmd
θ = = =
( ) ( )( )2 2C 132 sin 2 181 10 NC 0858 311 10 NCE E θ= = times = times
The Electric Field
bull The electric field exists around all charged objects
bull This is shown byelectric fieldlines
bull Constructedid i ll hidentically to theforce lines
2202009
15
Drawing Electric Field Lines
bull Number of lines at a charge is proportional to the chargethe charge
Electric Field Lines
bull The charge with ‐3qmust have three times as many lines as the charge with +q and 15 times the lines for +‐ 2qthe charge with q and 15 times the lines for 2q
2202009
16
Parallel Plates of Charge
bull Adding fields from each point on surface causes field lines parallel to the plates to cancel
PositivelyCharged
NegativelyCharged
bull Away from the edges the field is parallel and perpendicular to the plate(s)
Electric Field amp Conductors
bull In conductors excess electrons will move when an electric field is present
bull For ldquosteady staterdquo (ie no moving charges) the net electric field must be zeror r r r
bull Therefore the charges move to the surface(s) of the conductor until they cancel the electric field inside the conductor
bull Field lines can start and end on a conductor at right angles to the surface of the conductor
0q rArr= = =F E F Er r r r
2202009
17
Electric Field Lines
1844 ndash Two particles are in a uniform electric field whose value is + 2500 NC The mass and charge of particle 1 are m1 = 14 x 10‐5 kg and q1 = ‐70 μC while the
di l f ti l 2 2 6 10 5 k d 18 C I iti llcorresponding values for particle 2 are m2 = 26 x 10‐5 kg and q2 = +18 μC Initially the particles are at rest The particles are both located on the same electric field line but are separated from each other by a distance d When released they accelerate but always remain at this same distance from each other Find d
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The drawing shows the two particles in the electric field Ex They are separated by a distance d If the particles are to stay at the same distance from each other after being released they must have the same acceleration so ax1 = ax2 According to Newtonrsquos second law the acceleration ax of each particle is equal to the net force ΣFx acting on it divided by its mass m or x xa F m= Σ
Electric Field Lines
1844 ndash (cont)
N t th t ti l 1 t b t th l ft f ti l 2 If ti l 1 t th i ht
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
Note that particle 1 must be to the left of particle 2 If particle 1 were to the right of particle 2 the particles would accelerate in opposite directions and the distance between them would not remain constant
1 2 1 1 2
1 21 2 1
11 1
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = minus
minusΣ= =
1 2 2 2 2
1 22 2 2
22 2
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = +
+Σ= =
Note the signs indicated the direction of the vector pointing from charge 2 to charge 1 for the force on charge 1
2202009
18
Electric Field Lines
1844 ndash (cont)
Th l ti f h h t b th if th di t b t th
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The acceleration of each charge must be the same if the distance between the charges remains unchanged
Solving for d
1 2
1 2 1 21 22 2
1 2
x x
x x
a a
q q q qq E k q E k
d dm m
=
minus +=
Solving for d
( )( )1 2 1 2
1 2 2 165 m
x
kq q m md
E q m q m+
= =minus
Electric Fluxbull Flux (from Latin ndash meaning to flow)
bull Electric flux measures amount of electric field passing through a surface
Er
Er
So to calculate find the normal to
φ
E EAΦ =
Flux through surface is Area of surface times the field flowing through that surface
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
( )cosE E AφΦ =
2202009
19
Electric Flux
( )cosE E AφΦ =
Er
So to calculate find the normal to
This can be expressed as the vector dot product
EΦ = sdotE Arr
cosE A φsdot =E Arr
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
cos
x x y y z z
E Aor
E A E A E A
φsdot =
sdot = + +
E A
E Arr
Electric Flux
1848 ndash A rectangular surface (016 m x 038m) is oriented in a uniform electric field of 580 NC What is the maximum possible electric flux through the surface
The maximum possible flux occurs when the electric field is parallel to the normal of the rectangular surface (that is when the angle between the direction of the field and the direction of the normal is zero) Then
( ) ( ) ( )2
( cos ) 580 N C (cos 0 ) 016 m 038 m
35 N m C E
E
E AφΦ = = deg
Φ = sdot
2202009
20
Gaussrsquo Lawbull Electric Flux through a closed surface is proportional to the charge enclosed by that surface
encE
qε
Φ =
bull The fundamental constant ε0 is called the permittivity of free space It is related to Coulombrsquos law by ε0 = 885 x 10‐12 C2(Nm2)
bull Gauss Law is most useful when the electric field on the entire surface is constant
bull Take a point charge Construct a sphere centered on the point charge (all points on the surface equidistant from the charge) ndash the electric field is constant
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so
0ε
0
14
kπε
=
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so φ = 0 cos φ = 1
bull The direction of the field is radially outward from the charge (or inward towards the charge if q is negative)
0 0
cosenc encE
q qEA φε ε
Φ =rArr=
( )22
0 0
44
encq qE r Er
πε πε
= rArr =
Gaussrsquo Law for a plate of chargebull In this case we take our closed surface to be a box oriented such that two faces are
parallel to the plate of charge and the others are perpendicular to the plate
Si th l t i fi ld f l t f h i di l t th f f thbull Since the electric field from a plate of charge is perpendicular to the surface of the plate the sides that are perpendicular to the plate have no flux
Er
bull The charge enclosed by the box is the surface charge density multiplied by the area the box passes through the plate Aσthe plate
bull The area of the box with flux is 2A
Aσ
0 0 0E
q AEA Eσ σε ε ε
rArr rArrΦ = = =
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
14
Electric Field Lines1834 ndash (cont)
Notice also the magnitudes of E24 and E13 are equal and from the first part of this problem we know that
E24 = E13 = 181 times 102 NC
1 21- q - q
E 13 E 24
The electric field at the center of the rectangle is the vector sum of E24 and E13 The x components of E24 and E13 are equal in magnitude and opposite in direction hence
(E13)x ndash (E24)x = 0
Therefore
F Fi 2 h th t
344+ q + q
C
C 13 24 13 13( ) ( ) 2( ) 2( )siny y yE E E E E θ= + = =
From Figure 2 we have that
500 cm 500 cmsin 0858583 cmd
θ = = =
( ) ( )( )2 2C 132 sin 2 181 10 NC 0858 311 10 NCE E θ= = times = times
The Electric Field
bull The electric field exists around all charged objects
bull This is shown byelectric fieldlines
bull Constructedid i ll hidentically to theforce lines
2202009
15
Drawing Electric Field Lines
bull Number of lines at a charge is proportional to the chargethe charge
Electric Field Lines
bull The charge with ‐3qmust have three times as many lines as the charge with +q and 15 times the lines for +‐ 2qthe charge with q and 15 times the lines for 2q
2202009
16
Parallel Plates of Charge
bull Adding fields from each point on surface causes field lines parallel to the plates to cancel
PositivelyCharged
NegativelyCharged
bull Away from the edges the field is parallel and perpendicular to the plate(s)
Electric Field amp Conductors
bull In conductors excess electrons will move when an electric field is present
bull For ldquosteady staterdquo (ie no moving charges) the net electric field must be zeror r r r
bull Therefore the charges move to the surface(s) of the conductor until they cancel the electric field inside the conductor
bull Field lines can start and end on a conductor at right angles to the surface of the conductor
0q rArr= = =F E F Er r r r
2202009
17
Electric Field Lines
1844 ndash Two particles are in a uniform electric field whose value is + 2500 NC The mass and charge of particle 1 are m1 = 14 x 10‐5 kg and q1 = ‐70 μC while the
di l f ti l 2 2 6 10 5 k d 18 C I iti llcorresponding values for particle 2 are m2 = 26 x 10‐5 kg and q2 = +18 μC Initially the particles are at rest The particles are both located on the same electric field line but are separated from each other by a distance d When released they accelerate but always remain at this same distance from each other Find d
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The drawing shows the two particles in the electric field Ex They are separated by a distance d If the particles are to stay at the same distance from each other after being released they must have the same acceleration so ax1 = ax2 According to Newtonrsquos second law the acceleration ax of each particle is equal to the net force ΣFx acting on it divided by its mass m or x xa F m= Σ
Electric Field Lines
1844 ndash (cont)
N t th t ti l 1 t b t th l ft f ti l 2 If ti l 1 t th i ht
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
Note that particle 1 must be to the left of particle 2 If particle 1 were to the right of particle 2 the particles would accelerate in opposite directions and the distance between them would not remain constant
1 2 1 1 2
1 21 2 1
11 1
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = minus
minusΣ= =
1 2 2 2 2
1 22 2 2
22 2
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = +
+Σ= =
Note the signs indicated the direction of the vector pointing from charge 2 to charge 1 for the force on charge 1
2202009
18
Electric Field Lines
1844 ndash (cont)
Th l ti f h h t b th if th di t b t th
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The acceleration of each charge must be the same if the distance between the charges remains unchanged
Solving for d
1 2
1 2 1 21 22 2
1 2
x x
x x
a a
q q q qq E k q E k
d dm m
=
minus +=
Solving for d
( )( )1 2 1 2
1 2 2 165 m
x
kq q m md
E q m q m+
= =minus
Electric Fluxbull Flux (from Latin ndash meaning to flow)
bull Electric flux measures amount of electric field passing through a surface
Er
Er
So to calculate find the normal to
φ
E EAΦ =
Flux through surface is Area of surface times the field flowing through that surface
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
( )cosE E AφΦ =
2202009
19
Electric Flux
( )cosE E AφΦ =
Er
So to calculate find the normal to
This can be expressed as the vector dot product
EΦ = sdotE Arr
cosE A φsdot =E Arr
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
cos
x x y y z z
E Aor
E A E A E A
φsdot =
sdot = + +
E A
E Arr
Electric Flux
1848 ndash A rectangular surface (016 m x 038m) is oriented in a uniform electric field of 580 NC What is the maximum possible electric flux through the surface
The maximum possible flux occurs when the electric field is parallel to the normal of the rectangular surface (that is when the angle between the direction of the field and the direction of the normal is zero) Then
( ) ( ) ( )2
( cos ) 580 N C (cos 0 ) 016 m 038 m
35 N m C E
E
E AφΦ = = deg
Φ = sdot
2202009
20
Gaussrsquo Lawbull Electric Flux through a closed surface is proportional to the charge enclosed by that surface
encE
qε
Φ =
bull The fundamental constant ε0 is called the permittivity of free space It is related to Coulombrsquos law by ε0 = 885 x 10‐12 C2(Nm2)
bull Gauss Law is most useful when the electric field on the entire surface is constant
bull Take a point charge Construct a sphere centered on the point charge (all points on the surface equidistant from the charge) ndash the electric field is constant
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so
0ε
0
14
kπε
=
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so φ = 0 cos φ = 1
bull The direction of the field is radially outward from the charge (or inward towards the charge if q is negative)
0 0
cosenc encE
q qEA φε ε
Φ =rArr=
( )22
0 0
44
encq qE r Er
πε πε
= rArr =
Gaussrsquo Law for a plate of chargebull In this case we take our closed surface to be a box oriented such that two faces are
parallel to the plate of charge and the others are perpendicular to the plate
Si th l t i fi ld f l t f h i di l t th f f thbull Since the electric field from a plate of charge is perpendicular to the surface of the plate the sides that are perpendicular to the plate have no flux
Er
bull The charge enclosed by the box is the surface charge density multiplied by the area the box passes through the plate Aσthe plate
bull The area of the box with flux is 2A
Aσ
0 0 0E
q AEA Eσ σε ε ε
rArr rArrΦ = = =
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
15
Drawing Electric Field Lines
bull Number of lines at a charge is proportional to the chargethe charge
Electric Field Lines
bull The charge with ‐3qmust have three times as many lines as the charge with +q and 15 times the lines for +‐ 2qthe charge with q and 15 times the lines for 2q
2202009
16
Parallel Plates of Charge
bull Adding fields from each point on surface causes field lines parallel to the plates to cancel
PositivelyCharged
NegativelyCharged
bull Away from the edges the field is parallel and perpendicular to the plate(s)
Electric Field amp Conductors
bull In conductors excess electrons will move when an electric field is present
bull For ldquosteady staterdquo (ie no moving charges) the net electric field must be zeror r r r
bull Therefore the charges move to the surface(s) of the conductor until they cancel the electric field inside the conductor
bull Field lines can start and end on a conductor at right angles to the surface of the conductor
0q rArr= = =F E F Er r r r
2202009
17
Electric Field Lines
1844 ndash Two particles are in a uniform electric field whose value is + 2500 NC The mass and charge of particle 1 are m1 = 14 x 10‐5 kg and q1 = ‐70 μC while the
di l f ti l 2 2 6 10 5 k d 18 C I iti llcorresponding values for particle 2 are m2 = 26 x 10‐5 kg and q2 = +18 μC Initially the particles are at rest The particles are both located on the same electric field line but are separated from each other by a distance d When released they accelerate but always remain at this same distance from each other Find d
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The drawing shows the two particles in the electric field Ex They are separated by a distance d If the particles are to stay at the same distance from each other after being released they must have the same acceleration so ax1 = ax2 According to Newtonrsquos second law the acceleration ax of each particle is equal to the net force ΣFx acting on it divided by its mass m or x xa F m= Σ
Electric Field Lines
1844 ndash (cont)
N t th t ti l 1 t b t th l ft f ti l 2 If ti l 1 t th i ht
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
Note that particle 1 must be to the left of particle 2 If particle 1 were to the right of particle 2 the particles would accelerate in opposite directions and the distance between them would not remain constant
1 2 1 1 2
1 21 2 1
11 1
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = minus
minusΣ= =
1 2 2 2 2
1 22 2 2
22 2
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = +
+Σ= =
Note the signs indicated the direction of the vector pointing from charge 2 to charge 1 for the force on charge 1
2202009
18
Electric Field Lines
1844 ndash (cont)
Th l ti f h h t b th if th di t b t th
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The acceleration of each charge must be the same if the distance between the charges remains unchanged
Solving for d
1 2
1 2 1 21 22 2
1 2
x x
x x
a a
q q q qq E k q E k
d dm m
=
minus +=
Solving for d
( )( )1 2 1 2
1 2 2 165 m
x
kq q m md
E q m q m+
= =minus
Electric Fluxbull Flux (from Latin ndash meaning to flow)
bull Electric flux measures amount of electric field passing through a surface
Er
Er
So to calculate find the normal to
φ
E EAΦ =
Flux through surface is Area of surface times the field flowing through that surface
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
( )cosE E AφΦ =
2202009
19
Electric Flux
( )cosE E AφΦ =
Er
So to calculate find the normal to
This can be expressed as the vector dot product
EΦ = sdotE Arr
cosE A φsdot =E Arr
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
cos
x x y y z z
E Aor
E A E A E A
φsdot =
sdot = + +
E A
E Arr
Electric Flux
1848 ndash A rectangular surface (016 m x 038m) is oriented in a uniform electric field of 580 NC What is the maximum possible electric flux through the surface
The maximum possible flux occurs when the electric field is parallel to the normal of the rectangular surface (that is when the angle between the direction of the field and the direction of the normal is zero) Then
( ) ( ) ( )2
( cos ) 580 N C (cos 0 ) 016 m 038 m
35 N m C E
E
E AφΦ = = deg
Φ = sdot
2202009
20
Gaussrsquo Lawbull Electric Flux through a closed surface is proportional to the charge enclosed by that surface
encE
qε
Φ =
bull The fundamental constant ε0 is called the permittivity of free space It is related to Coulombrsquos law by ε0 = 885 x 10‐12 C2(Nm2)
bull Gauss Law is most useful when the electric field on the entire surface is constant
bull Take a point charge Construct a sphere centered on the point charge (all points on the surface equidistant from the charge) ndash the electric field is constant
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so
0ε
0
14
kπε
=
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so φ = 0 cos φ = 1
bull The direction of the field is radially outward from the charge (or inward towards the charge if q is negative)
0 0
cosenc encE
q qEA φε ε
Φ =rArr=
( )22
0 0
44
encq qE r Er
πε πε
= rArr =
Gaussrsquo Law for a plate of chargebull In this case we take our closed surface to be a box oriented such that two faces are
parallel to the plate of charge and the others are perpendicular to the plate
Si th l t i fi ld f l t f h i di l t th f f thbull Since the electric field from a plate of charge is perpendicular to the surface of the plate the sides that are perpendicular to the plate have no flux
Er
bull The charge enclosed by the box is the surface charge density multiplied by the area the box passes through the plate Aσthe plate
bull The area of the box with flux is 2A
Aσ
0 0 0E
q AEA Eσ σε ε ε
rArr rArrΦ = = =
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
16
Parallel Plates of Charge
bull Adding fields from each point on surface causes field lines parallel to the plates to cancel
PositivelyCharged
NegativelyCharged
bull Away from the edges the field is parallel and perpendicular to the plate(s)
Electric Field amp Conductors
bull In conductors excess electrons will move when an electric field is present
bull For ldquosteady staterdquo (ie no moving charges) the net electric field must be zeror r r r
bull Therefore the charges move to the surface(s) of the conductor until they cancel the electric field inside the conductor
bull Field lines can start and end on a conductor at right angles to the surface of the conductor
0q rArr= = =F E F Er r r r
2202009
17
Electric Field Lines
1844 ndash Two particles are in a uniform electric field whose value is + 2500 NC The mass and charge of particle 1 are m1 = 14 x 10‐5 kg and q1 = ‐70 μC while the
di l f ti l 2 2 6 10 5 k d 18 C I iti llcorresponding values for particle 2 are m2 = 26 x 10‐5 kg and q2 = +18 μC Initially the particles are at rest The particles are both located on the same electric field line but are separated from each other by a distance d When released they accelerate but always remain at this same distance from each other Find d
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The drawing shows the two particles in the electric field Ex They are separated by a distance d If the particles are to stay at the same distance from each other after being released they must have the same acceleration so ax1 = ax2 According to Newtonrsquos second law the acceleration ax of each particle is equal to the net force ΣFx acting on it divided by its mass m or x xa F m= Σ
Electric Field Lines
1844 ndash (cont)
N t th t ti l 1 t b t th l ft f ti l 2 If ti l 1 t th i ht
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
Note that particle 1 must be to the left of particle 2 If particle 1 were to the right of particle 2 the particles would accelerate in opposite directions and the distance between them would not remain constant
1 2 1 1 2
1 21 2 1
11 1
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = minus
minusΣ= =
1 2 2 2 2
1 22 2 2
22 2
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = +
+Σ= =
Note the signs indicated the direction of the vector pointing from charge 2 to charge 1 for the force on charge 1
2202009
18
Electric Field Lines
1844 ndash (cont)
Th l ti f h h t b th if th di t b t th
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The acceleration of each charge must be the same if the distance between the charges remains unchanged
Solving for d
1 2
1 2 1 21 22 2
1 2
x x
x x
a a
q q q qq E k q E k
d dm m
=
minus +=
Solving for d
( )( )1 2 1 2
1 2 2 165 m
x
kq q m md
E q m q m+
= =minus
Electric Fluxbull Flux (from Latin ndash meaning to flow)
bull Electric flux measures amount of electric field passing through a surface
Er
Er
So to calculate find the normal to
φ
E EAΦ =
Flux through surface is Area of surface times the field flowing through that surface
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
( )cosE E AφΦ =
2202009
19
Electric Flux
( )cosE E AφΦ =
Er
So to calculate find the normal to
This can be expressed as the vector dot product
EΦ = sdotE Arr
cosE A φsdot =E Arr
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
cos
x x y y z z
E Aor
E A E A E A
φsdot =
sdot = + +
E A
E Arr
Electric Flux
1848 ndash A rectangular surface (016 m x 038m) is oriented in a uniform electric field of 580 NC What is the maximum possible electric flux through the surface
The maximum possible flux occurs when the electric field is parallel to the normal of the rectangular surface (that is when the angle between the direction of the field and the direction of the normal is zero) Then
( ) ( ) ( )2
( cos ) 580 N C (cos 0 ) 016 m 038 m
35 N m C E
E
E AφΦ = = deg
Φ = sdot
2202009
20
Gaussrsquo Lawbull Electric Flux through a closed surface is proportional to the charge enclosed by that surface
encE
qε
Φ =
bull The fundamental constant ε0 is called the permittivity of free space It is related to Coulombrsquos law by ε0 = 885 x 10‐12 C2(Nm2)
bull Gauss Law is most useful when the electric field on the entire surface is constant
bull Take a point charge Construct a sphere centered on the point charge (all points on the surface equidistant from the charge) ndash the electric field is constant
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so
0ε
0
14
kπε
=
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so φ = 0 cos φ = 1
bull The direction of the field is radially outward from the charge (or inward towards the charge if q is negative)
0 0
cosenc encE
q qEA φε ε
Φ =rArr=
( )22
0 0
44
encq qE r Er
πε πε
= rArr =
Gaussrsquo Law for a plate of chargebull In this case we take our closed surface to be a box oriented such that two faces are
parallel to the plate of charge and the others are perpendicular to the plate
Si th l t i fi ld f l t f h i di l t th f f thbull Since the electric field from a plate of charge is perpendicular to the surface of the plate the sides that are perpendicular to the plate have no flux
Er
bull The charge enclosed by the box is the surface charge density multiplied by the area the box passes through the plate Aσthe plate
bull The area of the box with flux is 2A
Aσ
0 0 0E
q AEA Eσ σε ε ε
rArr rArrΦ = = =
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
17
Electric Field Lines
1844 ndash Two particles are in a uniform electric field whose value is + 2500 NC The mass and charge of particle 1 are m1 = 14 x 10‐5 kg and q1 = ‐70 μC while the
di l f ti l 2 2 6 10 5 k d 18 C I iti llcorresponding values for particle 2 are m2 = 26 x 10‐5 kg and q2 = +18 μC Initially the particles are at rest The particles are both located on the same electric field line but are separated from each other by a distance d When released they accelerate but always remain at this same distance from each other Find d
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The drawing shows the two particles in the electric field Ex They are separated by a distance d If the particles are to stay at the same distance from each other after being released they must have the same acceleration so ax1 = ax2 According to Newtonrsquos second law the acceleration ax of each particle is equal to the net force ΣFx acting on it divided by its mass m or x xa F m= Σ
Electric Field Lines
1844 ndash (cont)
N t th t ti l 1 t b t th l ft f ti l 2 If ti l 1 t th i ht
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
Note that particle 1 must be to the left of particle 2 If particle 1 were to the right of particle 2 the particles would accelerate in opposite directions and the distance between them would not remain constant
1 2 1 1 2
1 21 2 1
11 1
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = minus
minusΣ= =
1 2 2 2 2
1 22 2 2
22 2
x x
xxx
q qF q E k
dq q
q E kF dam m
Σ = +
+Σ= =
Note the signs indicated the direction of the vector pointing from charge 2 to charge 1 for the force on charge 1
2202009
18
Electric Field Lines
1844 ndash (cont)
Th l ti f h h t b th if th di t b t th
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The acceleration of each charge must be the same if the distance between the charges remains unchanged
Solving for d
1 2
1 2 1 21 22 2
1 2
x x
x x
a a
q q q qq E k q E k
d dm m
=
minus +=
Solving for d
( )( )1 2 1 2
1 2 2 165 m
x
kq q m md
E q m q m+
= =minus
Electric Fluxbull Flux (from Latin ndash meaning to flow)
bull Electric flux measures amount of electric field passing through a surface
Er
Er
So to calculate find the normal to
φ
E EAΦ =
Flux through surface is Area of surface times the field flowing through that surface
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
( )cosE E AφΦ =
2202009
19
Electric Flux
( )cosE E AφΦ =
Er
So to calculate find the normal to
This can be expressed as the vector dot product
EΦ = sdotE Arr
cosE A φsdot =E Arr
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
cos
x x y y z z
E Aor
E A E A E A
φsdot =
sdot = + +
E A
E Arr
Electric Flux
1848 ndash A rectangular surface (016 m x 038m) is oriented in a uniform electric field of 580 NC What is the maximum possible electric flux through the surface
The maximum possible flux occurs when the electric field is parallel to the normal of the rectangular surface (that is when the angle between the direction of the field and the direction of the normal is zero) Then
( ) ( ) ( )2
( cos ) 580 N C (cos 0 ) 016 m 038 m
35 N m C E
E
E AφΦ = = deg
Φ = sdot
2202009
20
Gaussrsquo Lawbull Electric Flux through a closed surface is proportional to the charge enclosed by that surface
encE
qε
Φ =
bull The fundamental constant ε0 is called the permittivity of free space It is related to Coulombrsquos law by ε0 = 885 x 10‐12 C2(Nm2)
bull Gauss Law is most useful when the electric field on the entire surface is constant
bull Take a point charge Construct a sphere centered on the point charge (all points on the surface equidistant from the charge) ndash the electric field is constant
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so
0ε
0
14
kπε
=
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so φ = 0 cos φ = 1
bull The direction of the field is radially outward from the charge (or inward towards the charge if q is negative)
0 0
cosenc encE
q qEA φε ε
Φ =rArr=
( )22
0 0
44
encq qE r Er
πε πε
= rArr =
Gaussrsquo Law for a plate of chargebull In this case we take our closed surface to be a box oriented such that two faces are
parallel to the plate of charge and the others are perpendicular to the plate
Si th l t i fi ld f l t f h i di l t th f f thbull Since the electric field from a plate of charge is perpendicular to the surface of the plate the sides that are perpendicular to the plate have no flux
Er
bull The charge enclosed by the box is the surface charge density multiplied by the area the box passes through the plate Aσthe plate
bull The area of the box with flux is 2A
Aσ
0 0 0E
q AEA Eσ σε ε ε
rArr rArrΦ = = =
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
18
Electric Field Lines
1844 ndash (cont)
Th l ti f h h t b th if th di t b t th
Ex
d
+x
q1 = minus70 μC
m1 = 14 times 10minus5 kg
q2 = +18 μC
m2 = 26 times 10minus5 kg
The acceleration of each charge must be the same if the distance between the charges remains unchanged
Solving for d
1 2
1 2 1 21 22 2
1 2
x x
x x
a a
q q q qq E k q E k
d dm m
=
minus +=
Solving for d
( )( )1 2 1 2
1 2 2 165 m
x
kq q m md
E q m q m+
= =minus
Electric Fluxbull Flux (from Latin ndash meaning to flow)
bull Electric flux measures amount of electric field passing through a surface
Er
Er
So to calculate find the normal to
φ
E EAΦ =
Flux through surface is Area of surface times the field flowing through that surface
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
( )cosE E AφΦ =
2202009
19
Electric Flux
( )cosE E AφΦ =
Er
So to calculate find the normal to
This can be expressed as the vector dot product
EΦ = sdotE Arr
cosE A φsdot =E Arr
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
cos
x x y y z z
E Aor
E A E A E A
φsdot =
sdot = + +
E A
E Arr
Electric Flux
1848 ndash A rectangular surface (016 m x 038m) is oriented in a uniform electric field of 580 NC What is the maximum possible electric flux through the surface
The maximum possible flux occurs when the electric field is parallel to the normal of the rectangular surface (that is when the angle between the direction of the field and the direction of the normal is zero) Then
( ) ( ) ( )2
( cos ) 580 N C (cos 0 ) 016 m 038 m
35 N m C E
E
E AφΦ = = deg
Φ = sdot
2202009
20
Gaussrsquo Lawbull Electric Flux through a closed surface is proportional to the charge enclosed by that surface
encE
qε
Φ =
bull The fundamental constant ε0 is called the permittivity of free space It is related to Coulombrsquos law by ε0 = 885 x 10‐12 C2(Nm2)
bull Gauss Law is most useful when the electric field on the entire surface is constant
bull Take a point charge Construct a sphere centered on the point charge (all points on the surface equidistant from the charge) ndash the electric field is constant
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so
0ε
0
14
kπε
=
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so φ = 0 cos φ = 1
bull The direction of the field is radially outward from the charge (or inward towards the charge if q is negative)
0 0
cosenc encE
q qEA φε ε
Φ =rArr=
( )22
0 0
44
encq qE r Er
πε πε
= rArr =
Gaussrsquo Law for a plate of chargebull In this case we take our closed surface to be a box oriented such that two faces are
parallel to the plate of charge and the others are perpendicular to the plate
Si th l t i fi ld f l t f h i di l t th f f thbull Since the electric field from a plate of charge is perpendicular to the surface of the plate the sides that are perpendicular to the plate have no flux
Er
bull The charge enclosed by the box is the surface charge density multiplied by the area the box passes through the plate Aσthe plate
bull The area of the box with flux is 2A
Aσ
0 0 0E
q AEA Eσ σε ε ε
rArr rArrΦ = = =
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
19
Electric Flux
( )cosE E AφΦ =
Er
So to calculate find the normal to
This can be expressed as the vector dot product
EΦ = sdotE Arr
cosE A φsdot =E Arr
So to calculate find the normal to the surface
Then find the angle between the field and that normal vector φ
cos
x x y y z z
E Aor
E A E A E A
φsdot =
sdot = + +
E A
E Arr
Electric Flux
1848 ndash A rectangular surface (016 m x 038m) is oriented in a uniform electric field of 580 NC What is the maximum possible electric flux through the surface
The maximum possible flux occurs when the electric field is parallel to the normal of the rectangular surface (that is when the angle between the direction of the field and the direction of the normal is zero) Then
( ) ( ) ( )2
( cos ) 580 N C (cos 0 ) 016 m 038 m
35 N m C E
E
E AφΦ = = deg
Φ = sdot
2202009
20
Gaussrsquo Lawbull Electric Flux through a closed surface is proportional to the charge enclosed by that surface
encE
qε
Φ =
bull The fundamental constant ε0 is called the permittivity of free space It is related to Coulombrsquos law by ε0 = 885 x 10‐12 C2(Nm2)
bull Gauss Law is most useful when the electric field on the entire surface is constant
bull Take a point charge Construct a sphere centered on the point charge (all points on the surface equidistant from the charge) ndash the electric field is constant
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so
0ε
0
14
kπε
=
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so φ = 0 cos φ = 1
bull The direction of the field is radially outward from the charge (or inward towards the charge if q is negative)
0 0
cosenc encE
q qEA φε ε
Φ =rArr=
( )22
0 0
44
encq qE r Er
πε πε
= rArr =
Gaussrsquo Law for a plate of chargebull In this case we take our closed surface to be a box oriented such that two faces are
parallel to the plate of charge and the others are perpendicular to the plate
Si th l t i fi ld f l t f h i di l t th f f thbull Since the electric field from a plate of charge is perpendicular to the surface of the plate the sides that are perpendicular to the plate have no flux
Er
bull The charge enclosed by the box is the surface charge density multiplied by the area the box passes through the plate Aσthe plate
bull The area of the box with flux is 2A
Aσ
0 0 0E
q AEA Eσ σε ε ε
rArr rArrΦ = = =
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
20
Gaussrsquo Lawbull Electric Flux through a closed surface is proportional to the charge enclosed by that surface
encE
qε
Φ =
bull The fundamental constant ε0 is called the permittivity of free space It is related to Coulombrsquos law by ε0 = 885 x 10‐12 C2(Nm2)
bull Gauss Law is most useful when the electric field on the entire surface is constant
bull Take a point charge Construct a sphere centered on the point charge (all points on the surface equidistant from the charge) ndash the electric field is constant
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so
0ε
0
14
kπε
=
bull Since the field is radial it is perpendicular to the concentric spherical surface everywhere so φ = 0 cos φ = 1
bull The direction of the field is radially outward from the charge (or inward towards the charge if q is negative)
0 0
cosenc encE
q qEA φε ε
Φ =rArr=
( )22
0 0
44
encq qE r Er
πε πε
= rArr =
Gaussrsquo Law for a plate of chargebull In this case we take our closed surface to be a box oriented such that two faces are
parallel to the plate of charge and the others are perpendicular to the plate
Si th l t i fi ld f l t f h i di l t th f f thbull Since the electric field from a plate of charge is perpendicular to the surface of the plate the sides that are perpendicular to the plate have no flux
Er
bull The charge enclosed by the box is the surface charge density multiplied by the area the box passes through the plate Aσthe plate
bull The area of the box with flux is 2A
Aσ
0 0 0E
q AEA Eσ σε ε ε
rArr rArrΦ = = =
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
21
Gaussrsquo Law for a line of charge
1854 ndash A long thin straight wire of length L has a positive charge Q distributed uniformly along it Use Gaussrsquo law to find the electric field created by this wire at a di l di tradial distance r
As the hint suggests we will use a cylinder as our Gaussian surface The electric field must point radially outward from the wire Because of this symmetry the field must be the same strength pointing radially outward on the surface of the cylinder regardless of the point on the cylinder thus we are justified in using Gaussrsquo Law
Long straight wire Gaussian cylinderGausssiancylinderE
++++++ ++
12
3
L
r
End view
Gaussrsquo Law for a line of charge
1854 ndash (cont) Let us define the flux through the surface Note that the field is parallel to the
d th th l b t th l f th f ( hi h i ll l tendcaps thus the angle between the normal from the surface (which is parallel to the wire) is perpendicular to the field thus φ = π2 so cos(φ)=0 for the ends For the side if the cylinder the flux is
Thus from Gaussrsquo Law
( )2E sideEA E rLπΦ = =++++++ ++
Long straight wire Gaussian cylinder
12
3
L
r
( )2encE
q QE rLπΦ = =rArr
It is customary to refer to charge density λ = QL
End view
GausssiancylinderE
( )0 0
02QE
rL
ε ε
πε=
02E
rλπε
=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
22
Work
bull Let us take an object with charge q in a constant electric field E
bull If we release this charge the electric field causes a force to move the charge
bull Let us define the field as so there is only one dimension to consider Let us also set the initial velocity and position to be both zero
bull Let the object move through a distance d The equations of motion are
qqm
= rArr =F E a E
21x at at= =υ
ˆE=E x
bull So after a time t the object moves 2
x at atυ
x d=
2122
qE d m qEdm
= rArr =υ υ
21 2 2
d at t d a= rArr =
2 2
t a
d ada a
=
= rArr =
υ
υυ
Conservation of Energy
bull The left hand side is the kinetic energy the particle has after moving a distance d
212
m qEd=υ
bull That means the right hand side must represent some form of potential energy the object had prior to being released
bull Types of Energy already covered
ndash Translational kinetic energy
ndash Rotational kinetic energy
ndash Gravitational potential energy
ndash Elastic (spring) potential energy
bull Electric Potential Energy
ndash Cutnell amp Johnson use (EPE) however it is often more convenient to use U
2 2212
1 )12
(2totalE kx Em h PI g Emω= + + + +υ
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
23
Electric Potential Energy
bull Like gravitational potential energy EPE is defined with respect to the energy at some location
bull Importance lies in the difference between electric potential energy between two points
Example Problem
194 ndash A particle has a charge of +15 μC and moves from point A to point B a distance of 020 m The particle experiences a constant electric force and its motion is l th li f ti f th f Th diff b t th ti l rsquo l t ialong the line of action of the force The difference between the particlersquos electric potential energy at A and B is UA ndash UB = +90 x 10‐4 J Find the magnitude and direction of the electric force that acts on the particle and the magnitude and direction of the electric field that the particle experiences The work done must be equal to the change in potential energy
(from A to B)4
390 10 J 45 10 N
A B A B
A B
Work U U F x U U
U UFminus
minus
= minus Δ = minus
minus times= = = times
rArr
(from A to B)
But we know that for a particle in an electric field F = qE
( )45 10 N
020mF
xtimes
Δ
( )( )4
36
90 10 J 30 10 N C15 10 C 020 m
A B A BU U U UqE Ex q x
Eminus
minus
minus minus= =
Δ Δ
= = timestimes
rArr
times
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
24
Electric Potential
bull The electric potential energy of an object depends on the charge of an object
( )EPEbull Once again we can find a measurement independent of the objects charge
bull The SI unit for V is measured in Volts ( = JoulesCoulomb)
bull A common unit of energy is the electron Volt
bull Specifically How much potential energy does an electron get if moved through a
( )EPEV
q=
potential of 1 Volt
191eV 1602 10 Jminus= times
Electric Potential
bull Like the Electric field the electric potential is independent of any test charge placed to measure a force
bull Unlike the electric field the potential is a scaler not a vector
bull By using calculus methods it can be shown that the electric potential due to a point charge is
N h h l f h l i i l d d h i f h h
kqVr
=
bull Note that the value of the electric potential depends on the sign of the charge
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
25
Electric Potentialfor a point charge
bull Example What was the potential difference due to a charge qA = 100 nC at the distances r0 = 100 cm and r1 = 200 cm qA 100 nC at the distances r0 100 cm and r1 200 cm
0 10 1
0 1
1 1
A A
A
kq kqV V Vr r
V kqr r
Δ = minus = minus
⎛ ⎞Δ = minus⎜ ⎟
⎝ ⎠
( )( )9 2 2 5 1 1899 10 N m C 10 C01m 02 m
450V
V
V
minus ⎛ ⎞Δ = times sdot minus⎜ ⎟⎝ ⎠
Δ =
Electric Potentialfor a point charge
bull Example A charge qA is held fixed A second charge qB is placed a distance r0 away and is released What is the velocityplaced a distance r0 away and is released What is the velocity of qB when it is a distance r1 away
bull We use the conservation of energy
2 2 20 0 1 1
1 1 12 2 2
Energy m U m U m U= + rArr + = +υ υ υ
( )21 0 1 0 1
12 Bm U U q V V= minus = minusυ
( )1 0 12q V Vm
= minusυ
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
26
Electric Potentialfor a point charge
( )1 0 12q V Vm
= minusυ
bull Find the electric potential at r0 and r1
0 10 1
A Akq kqV Vr r
= =
122 1 1A A A Bkq kq kq qq
m r r m r r⎛ ⎞ ⎛ ⎞
= minus = minus⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
υ
m
bull Or
0 1 0 1m r r m r r⎝ ⎠ ⎝ ⎠
1 2 q V m= Δυ
Electric Potentialfor a point charge
bull For a single charge we set the potential at infinity to zero
kqV =
bull So what is the potential energy contained in two charges
bull Bring in the charges from infinity
bull The first charge is free as it does not experience any forces
bull The second moves through the field created by the first
bull So the energy in the system must be
Vr
=
2
1 1 1
12 12
rU q V V V Vkq kq kqVr r
infinΔ = Δ Δ = minus
Δ = minus =infin
1 2
12
kq qUr
=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
27
Potential Energyof a collection of charges
bull Letrsquos add a third charge to this group
2 1 3 1 3 2
1 21 2
13 23
U q V q V q Vkq kqV Vr r
Δ = Δ + Δ + Δ
Δ = Δ =
1 3 2 31 2
12 13 23
kq q kq qkq qUr r r
Δ = + +
bull Each successive charge experiences the field (ie potential) created by the previous charges Be careful not to double count
Example Problem
1913 ndash An electron and a proton are initially very far apart (effectively an infinite distance apart) They are then brought together to form a hydrogen atom in which th l t bit th t t di t f 5 29 10 11 Wh t i ththe electron orbits the proton at an average distance of 529 x 10‐11 m What is the change in the electric potential energyThe first charge comes in for free the second charge experiences the field of the first so
( )( )
2
1 1 1
12 12
9 2 2 198 99 10 N m C 1 602 10 C
rU q V V V Vkq kq kqVr r
infin
minus
Δ = Δ Δ = minus
Δ = minus =infin
times times( ) ( )( )19
11
18
899 10 N m C 1602 10 C1602 10 C
529 10 m435 10 J
U
U
minusminus
minus
times sdot timesΔ = minus times
timesΔ = minus times
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
28
Electric Field and Potential
bull Let a charge be moved by a constant electric field from points A to B (let Δd be that distance)
W F d W qE d= Δ rArr = Δ
bull Now we know that the work done must be equal to the negative change in electric potential energy
bull But potential energy is simply
W F d W qE d= Δ rArr = Δ
ABU qE dminusΔ = Δ
bull Thus
ABq V qE dminus Δ = Δ
VEd
Δ= minus
Δ
Electric Field and Potential
bull This implies that the field lines are perpendicular to lines (surfaces) of equal
VEd
Δ= minus
Δ
potential
bull This equation only gives the component of the electric field along the displacement Δd
bull To truly find the electric vector field you must find the components in each direction
V V VΔ Δ Δx y z
V V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
29
Electric Field and Potential
bull If the potentials at the following points (x y) areV(11) = 30 V V(091)=15V and V(109)=40 V What is theV(11) 30 V V(091) 15V and V(109) 40 V What is the (approximate) electric field at (11)
x y zV V VE E Ex y z
Δ Δ Δ= minus = minus = minus
Δ Δ Δ
30V 15V 150 V m10m 09mxE minus
= minus = minusminus
30V 40V 100 V m10m 09myE minus
= minus =minus
( ) ( )2 2
1
150V 100V 180V
100Vtan 34150VE
E
θ minus
= minus + =
⎛ ⎞= = minus⎜ ⎟minus⎝ ⎠o
x
E
(11)
y
Example
Problem 1934 ndash Determine the electric field between each of theelectric field between each of the points
VEx
Δ= minus
Δ
50V 50V 0 V m020m 00mABE minus
= minus =minus
30V 50V 100 V m040m 020mBCE minus
= minus =minus
10V 30V 50 V m080m 040mCDE minus
= minus =minus
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
30
Equipotential Surfaces
bull Electric field lines are perpendicular to equipotential surfaces
bull Electric field lines are perpendicular to surfaces of conductors
bull Thus surfaces of conductors are equipotential surfaces
bull The interior of conductors have no electric field and are therefore equipotential
bull The Electric potential of a conductor is constant throughout the conductor
Example1968 ndash An electron is released at the negative plate of a
parallel plate capacitor and accelerates to the positive plate (see the drawing) (a) As the electron gains kinetic energy does its electric potential energy increase or decrease Why decrease Why
(b) The difference in the electronrsquos electric potential energy between the positive and negative plates is Upos ndash Uneg How is this difference related to the charge on the electron (minuse) and to the difference in the electric potential between the platesThe change in the electronrsquos electric potential energy is equal to the charge on the
The electric potential energy decreases The electric force F is a conservative force so the total energy (kinetic energy plus electric potential energy) remains constant as the electron moves across the capacitor Thus as the electron accelerates and its kinetic energy increases its electric potential energy decreases
The change in the electron s electric potential energy is equal to the charge on the electron (ndashe) times the potential difference between the plates
(c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
31
Example1968 ndash (cont) (c) How is the potential difference related to
the electric field within the capacitor and the displacement of the positive plate relative to the negative plate
Problem The plates of a parallel plate capacitor are separated by a distance of 12 cm
The electric field E is related to the potential difference between the plates and the displacement Δs by
Note that (Vpos ndash Vneg) and Δs are positive numbers so the electric field is a negative number denoting that it points to the left in the drawing
( )pos negV VE
s
minus= minus
Δ
and the electric field within the capacitor has a magnitude of 21 x 106 Vm An electron starts from rest at the negative plate and accelerates to the positive plate What is the kinetic energy of the electron just as the electron reaches the positive plateUse conservation of energy
Total energy at Total energy atpositive plate negative plate
pos pos neg n pos negeg pos negeV eVKE U KE U KE KErArr + = ++ = +1442443 1442443
Example1968 ndash (cont)
But the electron starts from rest on the positive plate so h ki i h i i l i
pospos neg negKE KEeV eV+ = +
the kinetic energy at the positive plate is zero
But so( )pos negV VE
minus
( )
pos neg
pos
neg
neg neg
pos n gneg e
KE
KE
eV eV
eV eV
e V VKE
= +
minus
minus
=
=
But so( )E
s= minus
Δ
( ) ( )( )( )19 6
15
1602 10 C 216 10 V m 0012m
40 10 J
neg
neg
KE
K
s
E
e E minus
minus
minus Δ =
times
= minus times
=
minus times
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
32
Lines of Equal Potential
bull Lines of equal elevation on amap show equal potentialmap show equal potentialenergy (mgh)
Example
1929 ndash Two equipotential surfaces surround a +15 x 10‐8 C point charge How far is the 190‐V surface from the 750‐V surfaceTh l t i t ti l V t di t f i t h i V k ThThe electric potential V at a distance r from a point charge q is V = kqr The potential is the same at all points on a spherical surface whose distance from the charge is r =kqV The radial distance r75 from the charge to the 750‐V equipotential surface is r75 = kqV75 and the distance to the 190‐V equipotentialsurface is r190 = kqV190 The distance between these two surfaces is
75 19075 190 75 190
1 1kq kqr r kqV V V V
⎛ ⎞minus = minus = minus⎜ ⎟⎜ ⎟
⎝ ⎠
( )2
9 875 190 2
N m 1 1899 10 150 10 C 11 m750 V 190 VC
r r minus⎛ ⎞sdot ⎛ ⎞minus = times + times minus =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
33
Capacitors
bull A capacitor consists of two conductors at different potentials being used to store charge
bull Commonly these take the shape of parallel plates concentric spheres or concentric cylinders but any conductors will do
bull q = CV where C is the capacitance measured in SI units of Farads (Coulomb Volt or C2 J ) V is the voltage difference between the conductors and q is the charge on each conductor (one is positive the other negative)
bull Capacitance is depends only upon the geometry of the conductors
Parallel Plate Capacitor
bull Field in a parallel plate capacitor is0
qEAε
=
bull But the a constant electric field can be defined in terms of electric potential
bull Substituting into the equation for capacitance
VE V Edd
= minus rArr =
0
0
0
q CV CEd
Aqq C d CA d
εε
= =
⎛ ⎞= rArr =⎜ ⎟
⎝ ⎠
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
34
Other Capacitors
bull Two concentric spherical shells
4πε
bull Cylindrical capacitor has a capacitance perunit length 2C πε
041 1
C
a b
πε=⎛ ⎞minus⎜ ⎟⎝ ⎠
02
ln
CbLa
πε=
⎛ ⎞⎜ ⎟⎝ ⎠
Energy stored in a capacitor
bull A battery connected to two plates of a capacitor does work charging the capacitorcharging the capacitor
bull Each time a small amount of charge is added to the capacitor the charge increases so the voltage between the plates increases
V
qΔq
qc
q
V =qc
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
35
Energy stored in a capacitor
bull Potential Energy is U = qV
bull We see that adding up each small increase in charge Δq x V is just the area of the triangle
Such that2
2
12
12 2
U qV q CV
qU U CVV
= =
= =
VSo the energy density (i e energy per unit
qΔq
qc
q
V =qc ( )
( )
20
20
1212
AU Edd
U E Ad
κε
κε
⎛ ⎞= ⎜ ⎟⎝ ⎠
=
So the energy density (ieenergy per unit volume) must be
Ad is the area of the capacitor2
012
Uu EAd
κε= =
Example
bull How much energy is stored in a parallel plate capacitor made of 100 cm diameter circles kept 20 mm apart when aof 100 cm diameter circles kept 20 mm apart when a potential difference of 120 V is applied
bull Find the capacitance2
0 0A rCd dε ε π
= =
bull The energy stored is
( ) ( ) ( )( )
2 212 2 22 22 0
885 10 C (N m ) 005m 120V12 2 2 0002m
r VU CVd
πε πminussdot sdot
= = =
9209 10 JU minus= times
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
36
Electric Polarization in Matter
bull Electric charge distribution in some molecules is not uniformis not uniform
bull Example Water ndash both Hatoms are on one side
bull In the presence of anElectric Field water willalign opposite to the field
Electric Polarization in Matter
bull In the absence of a field l l ll bMolecules will be
oriented randomly
bull In an electric field they line up opposite to theline up opposite to the field
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
37
Electric Polarization in Matter
bull The resulting electric field is the sum of the external field and the fields generated by the molecules
bull The electric field strength through the matter is reduced
bull The reduction of the field depends on the energy in the molecules the strength of the field each one makes and the density of thefield each one makes and the density of the molecules
bull This can be measured and the material is said to have a dielectric constant ‐ κ
Dielectrics
bull This dielectric constant changes the value of the permittivity in space (ε)
q qbull Coulombs law in a dielectric is
bull Where ε is related to ε0 by a constant κ
bull The dielectric constant for a vacuum is κ = 1
1 22
12
ˆ4q q
rπε=F r
0ε κε=
bull All other materials have a dielectric constant value greater than 1
bull Water at about 20˚C has κ = 804
bull Air at about 20˚C has κ = 100054
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
38
Dielectrics
0ε κε=
bull How does this affect the capacitance of parallel plates
0
0
d
d
AACd d
C C
κεε
κ
= =
=
Dielectric Breakdown
bull If the field in the dielectric becomes too t (i lt i t t) th t i lstrong (ie voltage is too great) the material begins to conduct current ndash this is called dielectric breakdown
bull When this happens the charge flows throughWhen this happens the charge flows through the material to the other plate and neutralizes the capacitor
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=
2202009
39
Example1957 ndash Two hollow metal spheres are concentric with each other The inner sphere
has a radius of 01500 m and a potential of 850 V The radius of the outer sphere is 01520 m and its potential is 820 V If the region between the spheres is filled with Teflon find the electric energy contained in this spaceThe electric energy stored in the region between the metal spheres is
The volume between the spheres is the difference in the volumes of the spheres
2 20 0
1 1 ( )2 2
u E E VolumU eκε κεrArr ==
( )3 3 3 32 2 2 2
4 4 4( )3 3 3
Volume r r r rπ π π= minus = minus
Since E = ΔVΔs and Δs = r2 ndashr1 so E = ΔV(r2 ndashr1)
( )2
3 3 80 2 2
2 1
1 4 12 10 J2 3
V r rr
Ur
κε π minus⎛ ⎞Δ ⎡ ⎤minus = times⎜ ⎟ ⎢ ⎥minus ⎣ ⎦⎝ ⎠=