Chapter 22: Electric Potential Electric Potential Energy Review of work, potential and kinetic energy Consider a force acts on a particle moving from

Chapter 22: Electric Potential

Electric Potential Energy Review of work, potential and kinetic energy

• Consider a force acts on a particle moving from point a to point b. The work done by the force WAB is given by:

B

A

B

ABA dFdFW

cos

path thealongpoint each at andbetween angle the:

path sparticle' thealongnt displaceme malinfinitesian :

dF

d

• If the force is conservative, namely when the work done by the force depends only on the initial and final position of the particle but not on the path taken along the particle’s path, the work done by the force F can always be expressed in terms of a potential energy U.

Electric Potential Energy

Review of work, potential and kinetic energy

• In case of a conservative force, the work done by the force can be expressed in terms of a potential energy U:

iU

UUUUUUW

i

BAABABBABA

point at energy potential the:

)(

• The change in kinetic energy K of a particle during any displacement is equal to the total work done on the particle:

ABBA KKKW

• If the force is conservative, then

BBAA

ABABABBA

UKUK

UUUKKKW

)(


Electric potential energy in a uniform field

• Consider a pair of charged parallel metal plates that generate a uniform downward electric field E and a test charge q0 >0

+ + ++ ++ +

- -- - - - --- -

++ +

d E

0q

A

B

0

location charge test thedependnot doesit ;

0

0

EdqFdW

EqF

BAconservativeforcethe force is in the same direction as the

net displacement of the test charge

• In general a force is a vector:

0,;),,( 0 zxyzyx FFEqFFFFF

Note that this force is similar to the force due to gravity:

0,;),,( ,,,,,, zgxgygzgygxgg FFmgFFFFF

m

gFg


Electric potential energy in a uniform field (cont’d)

• In analogy to the gravitational force, a potential can be defined as:

)..(0 mgyUfcEyqU 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

)()()( 000

ABBA

BAABABABBA

Uyy

yyEqEyqEyqUUUW

• U increases (decreases) if the test charge moves in the direction opposite to (the same direction as) the electric force

+

+

A

B

EqF

0E

UAB <0

+

+

B

AEqF

0

E

UAB>0

-

-

A

BE

UAB >0EqF

0-

-

B

AEqF

0E

UAB<0


Electric potential energy of two point charges

• The force on the test charge at a distance r

+q

q0

rb

r

ra

20

04

1

r

qqFr

• The work done on the test charge

B

A

B

A

r

rBA

r

r rBA rr

qqdr

r

qqdrFW )

11(

44

1

0

020

0

attractive0

repulsive0

0

0

qq

qq

E

E

a

b


Electric potential energy of two point charges (cont’d)

• In more general situation

A

B

r

d

rdF

E

B

A

B

A

B

A

r

r

r

r

r

rBA drr

qqdFdFW

20

04

1cos

tangent to the path

dr

BABA

UUrr

qq

11

4 0

0

Natural and consistent definition of theelectric potentialr

qqU 0

04

1


Electric potential energy of two point charges (cont’d)

• Definition of the electric potential energy

r

qqU 0

04

1

• Reference point of the electric potential energy

Potential energy is always defined relative to a reference pointwhere U=0. When r goes to infinity, U goes to zero. Thereforer= is the reference point. This means U represents the workto move the test charge from an initial distance r to infinity.

If q and q0 have the same sign, this work is POSITIVE ; otherwiseit is NEGATIVE.

qq0>0qq0<0

U U

0

0


Electric potential energy with several point charges

• A test charge placed in electric field by several particles

charge test theand charge ebetween th distance the:

4...

4 0

0

2

2

1

1

0

0

ir

r

qq

r

q

r

qqU

i

ii

i

• Electric potential energy to assemble particles in a configuration

jir

r

qqU

ij

jiij

ji

and charge ebetween th distance the:

4

1

0

Electric Potential

Example : A system of point charges

+

q1=-e q2=+e q3=+e

+-x=0 x=a x=2a

a

e

a

e

a

ee

r

q

r

qqUW

0

2

023

2

13

1

0

3

8244

a

e

a

ee

a

ee

a

ee

r

qq

r

qq

r

qq

r

qqU

ji ij

ji

00

23

32

13

31

12

21

00

8

))((

2

))(())((

4

1

4

1

4

1

Work done to take q3 from x=2a to x=infinity

Work done to take q1,q2 and q3 to infinity


Two interpretations of electric potential energy

• Work done by the electric field on a charged particle moving in the field

• Work needed by an external force to move a charged particle slowly from the initial to the final position against the electric force

Work done by the electric force when the particle moves from A to B

ABBAelectric

BA rdUUW

nt displaceme;

Work done by the external force when the particle moves from B to A

BAext

AB

ABBAelectricext

UUW

rdrdFF

ntdisplaceme;

Electric Potential

Electric potential or potential

• Electric potential V is potential energy per unit charge

VqUq

UV 0

0

or 1 V = 1 volt = 1 J/C = 1 joule/coulomb

ABBABABA VqVVqUUW 00 )(

potential of A with respect to B

work done by the electric forcewhen a unit charge moves fromA to Bwork needed to move a unitcharge slowly from b to aagainst the electric force

Electric Potential

Electric potential or potential (cont’d)

• Electric potential due to a single point charge

r

q

q

UV

00 4

1

• Electric potential due to a collection of point charges

i i

i

r

q

q

UV

00 4

1

• Electric potential due to a continuous distribution of charge

r

dq

q

UV

00 4

1

Electric Potential

From E to V

• Sometimes it is easier to calculate the potential from the known electric field

B

A

B

ABA dEqdFW

0

B

A

B

ABA dEdEVV

cos

A

BBA dEVV

The unit of electric field can be expressed as:1 V/m = 1 volt/meter = 1 N/C = 1 newton / coulomb

Electric Potential

Example : f

i

if sdEVV

V f Vi E drR

kq1

r2 drkq1

r R

2r

kqE

R

kqV

04

1

k

Replace R with r

r

qV

04

1

0 Vi kq

R

Electric Potential

Example:

+ -m, q0

q1 q2

A B

BBAA UKUK on conservatiEnergy

= 0

qVUmK ,2

1 2

BA VqmVq 02

0 2

10

m

VVq BA )(2 0

Electric Potential

Unit: electron volt (useful in atomic & nuclear physics)

• If the charge q equals the magnitude e of the electron charge 1.602 x 10-19 C and the potential difference VAB= 1 V, the change in energy is:

• Consider a particle with charge q moves from a point where the potential is VA to a point where it is VB , the change in the potential energy U is:

ABBAABAB qVqVVVqUU )(

eV 1

J 10602.1V)1)(C10602.1( 1919

BA UU

meV, keV, MeV, GeV, TeV,…

Calculating Electric Potential

Example: A charged conducting sphere

+++

+ ++

+

+

E

V

0

0

R

r

r

0E

204

1

R

qE

204

1

r

qE

R

qV

04

1

r

qV

04

1

Using Gauss’s law we calculated the electricfield.Now we use this result to calculate the potentialand we take V=0 at infinity.

4

1:

0 r

qVrR

the same as the potential

due to a point charge

4

1:

0 R

qVrR

4

1:

0 R

qVrR

inside of the conductorE is zero. So the potentialstays constant and isthe same as at the surface

Equipotential Surface

Equipotential surface

• No point can be at two different potentials, so equipotential surfaces for different potentials can never touch or intersect

• An equipotential surface is a 3-d surface on which the electric potential V is the same at every point

• Because potential energy does not change as a test charge moves over an equipotential surface, the electric field can do no work

• E is perpendicular to the surface at every point

• Field lines and equipotential surfaces are always mutually perpendicular


Examples of equipotential surface


Equipotentials and conductors

• 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


Equipotentials and conductors (cont’d)

• Consider a conductor with a cavity without any charge inside the cavity- The conducting cavity surface is an equipotential surface A- Take point P in the cavity at a different potential and it is on a different equipotential surface B- The field goes from surface B to A or A to B- Draw a Gaussian surface which surrounds the surface B inside cavity

conductor

surface of cavity

Guassian surface

equipotentialsurface through P

PA

B

- The net flux that goes through this Gaussian surface is not zero because the electric field is perpendicular to the surface- Gauss’s law says this flux is zero as there is no charge inside- Then the surfaces A and B are at the same potential

• In an electrostatic situation, if a conductor contains a cavity and if no charge is present inside the cavity, there can be no net charge anywhere on the surface of the cavity


Electrostatic shielding

Potential Gradient Potential gradient

• Potential difference and electric field

B

ABA dEVV

• Potential difference and electric field

A

B

B

ABA dVdVVV

B

A

B

AdEdV

dzEdyEdxEdEdV zyx

kdzjdyidxd

kEjEiEE zyx

ˆˆˆ

ˆˆˆ

Potential Gradient Potential gradient (cont’d)

• E from V

z

VE

y

VE

x

VE zyx

...)()(

lim0 x

xfxxf

x

fx

kz

Vj

y

Vi

x

VE ˆˆˆ

• Gradient of a function f

fz

ky

jx

if

ˆˆˆ

VE

r

VEr

If E is radial with respect toa point or an axis

Potential Gradient Potential gradient (cont’d)

Exercises Exercise 1

Exercises Exercise 1 (cont’d)








Exercises Exercise 5: An infinite line charge + a conducting cylinder

Signal wire

Outer metal braid

r r

line charge density

Q -Q

rE

E

r

02

1

direction radialin is

a

b

b

a

b

ar

b

aba

r

r

r

drdrEdEVV

ln2

2

0

0

Documents

Chapter 22: Electric Potential Electric Potential Energy Review of work, potential and kinetic energy Consider a force acts on a particle moving from