Electric PotentialFlashback to PHY113…

Work done by a conservative force is independent of the path of the object.Gravity and elastic forces are examples.This leads to the concept of potential energy and helps us avoid tackling problems using only forces.

Electrostatic forces are also conservative.

UdxFWf

i

x

xs Δ−== ∫

Ev

A

Electric Potential Energy

q0

sEsF

EFvvvv

vv

dqdU

q

pathpath∫∫ −=−=Δ

=

.. 0

0

…but F is a conservative force

∫−=ΔB

A

dqU sE wv.0

…so the path we take does not matter

B

q0

Electric PotentialWork, ΔU, is dependent on the magnitude of the test charge, q0.We’d like to have a quantity independent of the test charge and only an attribute of the electric field.

0qUV =

Electric Potential(J/C=V)

∫−=Δ

=ΔB

A

dqUV sE vv

.0

Potential Difference between A and B

∫∞

−=P

P dV sE.

Electric Potential at P

Unit Pit Stop

Potential

Energy

Electric Field

CJV =

mNJ ⋅=

mVmN

JJCV

CNCN =⎟

⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛=

..

( ) ( )( ) JCJCVeeV 1919 106.11106.111 −− ×=×==Electron-Volt

Electric Potential in a Uniform Field

∫ ∫∫ −=°−=−=ΔB

A

B

A

B

A

dsEdsEdV 0cos. sE

EdV −=Δ EdqU 0−=Δ

Electric field lines point to decreasing potential.

A positive charge will lose potential energy and gain kinetic energy when moving in the direction of the field.

Equipotential Surfaces

B

A

s

E

Cθ

sEsEsE ... −=−=−=Δ ∫∫B

A

B

A

ddV

θcos90cos0cos

)()(

EsVEsEsV

V

CBAC

CBAC

−=Δ°−°−=Δ

−−=Δ E.sE.s

θcos0cos

)()(

0

0

00

EsqUEsqU

qqU

AC

CBAC

−=Δ°−=Δ

−−=Δ E.sE.s

D

θcos)()(... EsddV CDAC

D

A

D

AAD −=−−=−=−=−=Δ ∫∫ E.sE.ssEsEsE

No work is done moving a charged particle perpendicular to a field (along equipotential surfaces)

Equipotential Surfaces

Uniform Field Point Charge Electric Dipole

Electric Potential of a Point Charge

∫−=−B

AAB dVV sE.

drrqkds

rqkd

rqkd eee 222 cos.ˆ. === θsrsE

⎥⎦

⎤⎢⎣

⎡−=−=−

−=−

∫

∫

ABe

r

reAB

rAB

rrqk

rdrqkVV

drEVVB

A

112

0=∞=AV

rqkV e=

Two Point Charges

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

2

2

1

1

rq

rqkV eP

A System of Point Charges

12

21

rqqkU e=

⎟⎟⎠

⎞⎜⎜⎝

⎛++=

31

13

23

32

12

21

rqq

rqq

rqqkU e

Concept QuestionTwo test charges are brought separately into the vicinity of a charge +Q. First, test charge +q is brought to a point a distance r from +Q. Then this charge is removed and test charge –q is brought to the same point. The electrostatic potential energy of which test charge is greater:

1. +q2. –q3. It is the same for both.

Getting From V to E

∫−=B

A

dV sE.

sE ddV .−=

dxdVEx −=E

E

drdVEr −=

In general:xVEx ∂∂

−=zVEz ∂∂

−=yVEy ∂∂

−=

Electric Potential of a Dipole

22

xqakV e≈ (x >> a)

3

4xqak

dxdVE e

x =−=

222

axqakV e

P −=

222

xaqxkV e

P −=

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

−

+−=−= 222

22

2xaxaqk

dxdVE ex

Between the Charges

Electric Potential Due to Continuous Charge Distributions

Start with an infinitesimal charge, dq.

rdqkdV e=

Then integrate over the whole distribution

∫=r

dqkV e

Electric Potential Due to a Uniformly Charged Ring

22 axQkV e

+=

( ) 2/322 axQxkE e

+=

Electric Potential Due to a Finite Line of Charge

⎟⎟⎠

⎞⎜⎜⎝

⎛ ++=

aall

lQkV e

22

ln

Electric Potential Due to a Uniformly Charged Sphere

2rQkE er =

∫∫∞∞

−=−=r

e

r

rB rdrQkdrEV 2

rQkV eB =

RQkV eC =r

RQkE e

r 3=

( )2232

rRRQkdrEVV e

r

RrCD −=−=− ∫

⎟⎟⎠

⎞⎜⎜⎝

⎛−= 2

2

32 R

rRQkV e

D

r > R

r < R

Potential Due to a Charged Conductor

Charges always reside at the outer surface of the conductor.The field lines are always perpendicular to surface.Then E.ds=0 on the surface at any point.Which means, VB-VA=0along the surface.The surface is an equipotential surface.Finally, since E=0 inside the conductor, the potential V is constant and equal to the surface value.

Connected Charged Conducting Spheres

1

2

2

1

rr

EE

=

2

1

2

1

rr

qq

=

Cavity Within a Conductor

0. =−=− ∫B

AAB dVV sE

We can always find a path where E.ds

is

non-zero.But, since ΔV=0 for all paths, E must be zero everywhere in the cavity.

A cavity without any charges enclosed by a conducting wall is field free.

SummaryCharges can have different electric potential energy at different points in an electric field.Electric potential is the electric potential energy per unit charge.All points inside a conductor are at the same potential.

For Next ClassReading Assignment

Chapter 26 – Capacitance and Dielectrics

WebAssign: Assignment 3