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Chapter 16: Electric Energy and Capacitance
Potential Difference and Electric Potential Work and electric potential energy
• Consider a small positive charge placed in a uniform electric field E.
+ + ++ ++ +
- -- - - - --- -
++ +
d Eq
A
B
0
location charge test thedependnot doesit ;
qEdFdW
qEF
ABconservativeforcethe force is in the same direction as the
net displacement of the test charge
qEdWPE AB
The work done by a conservative force can be reinterpreted as thenegative of the change in a potential energy associated with that force.
SI unit : joule (J)
Homework assignment: 9,18,28,34,44,51 (Six problems!)
Electric Potential Energy
• In analogy to the gravitational force, a potential can be defined as:
)..( mgyPEfcqEyPE g • When the test charge moves from height yA to height yB , the work done on the charge by the field is given by:
decreases. potential theand 0 , If
)()(
PEyy
yqEqEyqEyPEPEWPE
BA
ABABAB
• U increases (decreases) if the test charge moves in the direction opposite to (the same direction as) the electric force
+
+
A
B
EqF
E
PE<0
+
+
B
AEqF
E
PE>0
-
-
A
BE
PE>0EqF
-
-
B
AEqF
E
PE<0
Work and electric potential energy (cont’d)
Electric Potential Energy
• In analogy to the gravitational force, a potential can be defined as:
)..( mgyPEfcqEyPE g • When the test charge moves from height yA to height yB , the work done on the charge by the field is given by:
decreases. potential theand 0 , If
)()(
PEyy
yqEqEyqEyPEPEWPE
BA
ABABAB
Work and electric potential energy (cont’d)
• Define an electric potential difference as:
q
PEVVV AB
Then for a special case of a uniform electric field:
xEVxEq
PExx
SI unit : joule per coulomb or volt (J/C or V)
SI unit : J/C=V=N/C
The change in electric potential energy as a charge q moves from A to Bdivided by charge q.
Electric Potential Energy
Example 16.1 : Potential energy differences in an electric field
• A proton is released from rest at x=-2.00 cm in a constant electric field with magnitude 1.50x103 V/C pointing the positive x-direction.
(a)Calculate the change in the electric potential energy associated with the proton when it reaches x=5.00 cm.
eqxxqExqEPE ifxx : J 1068.1)( 17
(b) Find the change in electric potential energy associated with an electron fired from x=-2.00 cm and reaching x=12.0 m.
eqxxqExqEPE ifxx : J 1036.3)( 17
(b) Find the change in electric potential energy associated with an electron fired from x=3.00 cm to x=7.00 cm if the direction of the electric field is reversed.
eqxxqExqEPE ifxx : J 1060.9)( 18
Electric Potential Energy
Example 16.2 : Dynamics of charged particles
• Continuation of Example 16.1.
(a)Find the speed of the proton at x=0.0500 m in part (a) of Example 16.1.
002
10 2
PEmvPEKE
m/s 1042.122 52 PEm
vPEm
v
(b) Find the electron’s initial speed, given that its speed has fallen by half at x=0.120 m.
02
1
2
10 22
PEmvmvPEKE if
PEmvPEmvvm iii 222
8
30
2
1)
2
1(
2
1
m/s 1092.93
8 6
m
PEvi
Electric Potential Energy
Example 16.3 : TV tubes and atom smasher
• Charged particles are accelerated through potential difference. Suppose a proton is injected at a speed of 1.00x106 m/s between two plates 5.00 cm apart. The proton subsequently accelerates across the gap and exits through the opening.
(a)What must the electric potential difference be if the exist speed is to be 3.00x106 m/s?
0 VqKEPEKE
V 1018.4 21
21
4
22
q
mvmv
q
KEV
if
(b) What is the magnitude of the electric field between the plates?
N/C 1036.8 5
x
VE
Electric Potential and Potential Energy
Electric potential by a point charge
• The electric field of a point charge extends throughout space, so does its electric potential.
• The zero point can be anywhere but it is convenient to choose the point at infinity. Then it can be shown that the electric potential due to a point charge q at a distance r is given by :
r
qV
4
1
• The superposition principle : the total electric potential at some point P due to several point charges is the algebraic sum of the electric potentials due to the individual charges.
Electric Potential and Potential Energy
Electric potential energy of a pair of point charges
• If V1 is the electric potential due to charge q1 at a point P, then the work required to bring charge q2 from infinity to P without acceleration is q2V1. This work is, by definition, equal to the potential energy PE of the two-particle system when the particles are separated by a distance r.
r
qqVqPE 21
012 4
1
If q1q2>0, PE>0
If q1q2<0, PE<0
Electric potential energy with several point charges
0
0
0
2
2
1
1
0
0
q charge and chargebetween distance the:
4...
4
ii
ii
i
qr
r
r
q
r
qqU
From the superposition principle
Electric Potential and Potential Energy
Example 16.4 : Finding the electric potential
• A 5.00-C point charge is at origin and a point charge q2=-2.00C is on the x-axis at (3.00,0) m.
+ -
P(0,4.00)
(3.00,0)
y(m)
x(m)q1 q2
0
r2r1(a)Find the electric potential at point P due to these charges.
V 1012.1m 00.4
C 1000.5mN 1099.8 4
6
2
29
1
11
Cr
qkV e
V 10360.0m 00.5
C 1000.2mN 1099.8 4
6
2
29
2
22
Cr
qkV e
V 1060.7 321 VVVP
(b) Find the work needed to bring the 4.00-C charge from infinity to P.
J 1004.3)010C)(7.60 1000.4()( 23633
VVqVqPEW P
Potentials and Charged Conductors Surface electric potential of a conductor in electrostatic equilibrium
• Work done to move a charge from point A to point B
PEW )( AB VVqPE
)( AB VVqW
No net work is needed to move a charge between two points that are at the same electric potential.
• We will learn that when a charged conductor is in electrostatic equilibrium:
-A net charge placed on it resides entirely on its surface.-The electric field just outside its surface is perpendicular to the surface and that the field inside the conductor is zero.-All points on its surface are at the same potential.
Equipotential surface
• E = 0 everywhere inside a conductor- At any point just inside the conductor the component of E tangent to the surface is zero- The tangential component of E is also zero just outside the surface
• When all charges are at rest, the surface of a conductor is always an equipotential surface
conductor0E
E
//EE
vacuum If it were not, a charge could move around arectangular path partly inside and partly outsideand return to its starting point with a net amountof work done on it.
• When all charges are at rest, the electric field just outside a conductor must be perpendicular to the surface at every point
Potentials and Charged Conductors
Examples of equipotential surface
Potentials and Charged Conductors
The equipotentials are perpendicular to the electric field linesat every point.
Electron volt
Potentials and Charged Conductors
The electron volt is defined as the kinetic energy that an electrongains when accelerated through a potential difference of 1V.
1 V = 1 J/C1 eV = 1.60 x 10-19 C.V = 1.60 x 10-19 J
In atomic, nuclear and particle physics, the electron volt is usedcommonly to express energies.
- Electrons in normal atoms have energies of tens of eV’s.- Excited electrons in atoms that emit x-rays have energies of thousands of eV’s ( keV = 103 eV).- High energy gamma rays emitted by the nucleus have energies of millions of eV’s (MeV = 106 eV).- The world most energetic accelerator near Chicago accelerates protons/anti-proton up to Tera eV’s (TeV = 1012 eV )
Capacitance Capacitor
• Any two conductors separated by an insulator (or a vacuum) form a capacitor• In practice each conductor initially has zero net charge and electrons are transferred from one conductor to the other (charging the conductor)
• Then two conductors have charge with equal magnitude and opposite sign, although the net charge is still zero
• When a capacitor has or stores charge Q , the conductor with the higher potential has charge +Q and the other -Q if Q>0
Capacitors and Capacitance Capacitance
• One way to charge a capacitor is to connect these conductors to opposite terminals of a battery, which gives a fixed potential difference Vab between conductors ( a-side for positive charge and b-side for negative charge). Then once the charge Q and –Q are established, the battery is disconnected.• If the magnitude of the charge Q is doubled, the electric field becomes twice stronger and Vab=V is twice larger.
• Then the ratio QV is still constant and it is called the capacitance C.
• When a capacitor has or stores charge Q , the conductor with the higher potential has charge +Q and the other -Q if Q>0
ltcoulomb/vo 1C/V 1 farad 1F 1units
V
QC
-Q Q
ab
Parallel-plate capacitor in vacuum
• Charge density:A
q
• Electric field:A
qE
00
• Potential diff.: A
qdEdV
0
1
• Capacitance:d
A
V
qC 0
• The capacitance depends only on the geometry of the capacitor. • It is proportional to the area A.• It is inversely proportional to the separation d• When matter is present between the plates, its properties affect the capacitance.
Parallel-Plate Capacitance
Parallel-Plate Capacitance Units
1 F = 1 C2/N m (Note [C2/N m2)
0 = 8.85 x 10-12 F/m
1 F = 10-6 F, 1 pF = 10-12 F
Example : Size of a 1-F capacitor
F 0.1 , mm 1 Cd
2812
3
0
m 101.1F/m 1085.8
m) 100.1F)(0.1(
Cd
A
Example : Properties of a parallel capacitor
kV 10.0 V 000,10 , m 00.2 , mm 00.5
in vacuumcapacitor palte-parallelA 2 VAd
F 0.00354F 1054.3
m1000.5
)m F/m)(2.00 1085.8(
5
3
212
0
d
AC
C 35.4C1054.3
V) 10C/V)(1.00 1054.3(5
49
abCVQ
N/C 1000.2
)m 00.2)(mN/C 1085.8(
C 1054.3
6
22212
5
00
A
QE
Parallel-Plate Capacitance
Combinations of Capacitors Symbols for circuit elements and circuits
Capacitors in parallel
a
b
VVab 1Q 1C 2C
VCQVCQ 2211
VCCQQQ )( 2121
21 CCV
Q
The parallel combination is equivalent to a single capacitor with thesame total charge Q=Q1+Q2 and potential difference.
i ieqeq CCCCC 21
2Q
Combinations of Capacitors
Capacitors in series
a
b
cVVab 1VVac
2VVcb
Q
Q
1C
2C
22
11 C
QVV
C
QVV cbac
2121
11
CCQVVVVab
21
11
CCQ
V
The equivalent capacitance Ceq of the series combination is defined asthe capacitance of a single capacitor for which the charge Q is the sameas for the combination, when the potential difference V is the same.
21
1111
CCCQ
V
CV
QC
eqeqeq
Combinations of Capacitors
iieq CC
11
Capacitor networks
Combinations of Capacitors
Capacitor networks (cont’d)
Combinations of Capacitors
Capacitor networks 2C
A
B
A
B
C C
CCC
CCC
C C
C C
C C C3
1
A
B
C C
C C
C C3
4A
B
C41
15
Combinations of Capacitors
Energy Stored in a Charged Capacitor
Work done to charge a capacitor
• Consider a process to charge a capacitor up to Q with the final potential difference V.
C
QV
• Let qi and (v)i be the charge and potential difference at an intermediate stage during the charging process.
C
qv ii )(
• At this stage the work (W)i required to transfer an additional element of charge q is:
C
qqqvW i
ii
)()(
• The total work needed to increase the capacitor charge q from zero to Q is:
VQC
qqW
N
i
i
N
2
1lim
1
Q
V
q
v)i
qi
• The energy stored:
C
QVCVQ
22
1
2
1 storedEnergy
22
Capacitor with Dielectrics
Dielectric materials
• Experimentally it is found that when a non-conducting material (dielectrics) between the conducting plates of a capacitor, the capacitance increases for the same stored charge Q.• Define the dielectric constant as:
0C
C
• When the charge is constant, VVCCVCVCQ // 0000
)(00
d
VE
EE
VV
Material Material
vacuum 1air(1 atm) 1.00059
Teflon 2.1
Polyethelene 2.25
Mica 3-6
Mylar 3.1
Plexiglas 3.40Water 80.4
C0 : capacitance w/o dielectric
C : capacitance w/ dielectric
000 ,
C
C
d
AC
d
A
d
AC 0
0
Dielectrics
Induced charge and polarization• Consider a two oppositely charged parallel plates with vacuum between the plates.• Now insert a dielectric material of dielectric constant .
constant is Qwhen /0 EE • Source of change in the electric field is redistribution of positive and negative charge within the dielectric material (net charge 0). This redistribution is called a polarization and it produces induced charge and field that partially cancels the original electric field.
0
000
EEEE ind
0ty permittivi thedefine and 1
1
ind
E 22000 2
1
2
1EEu
d
A
d
ACC
Dielectrics
Molecular model of induced charge
Dielectrics
Molecular model of induced charge (cont’d)
