Chapter 21 Electric Potential

++ + +

2

Potential difference(voltage)

air → conductor

discharge

charged

Conservative force

Conservation of energy

Potential (energy)

Coulomb force is conservative

3

1 22

m mF Gr

Comparing : 1 22~ Q QF k

r

Electrostatic / Coulomb force is conservative

Work doesn’t depend on the path:

1 2L L

W qE dl qE dl

0E dl

The circulation theorem of electric field

( 0 )E

Electric potential

4

Conservative force → potential energy U

+

-

U depends on the charge!

Define electric potential:

/V U q

Potential difference / voltage:

a b abab a b

U U WV V Vq q

Electric potential & electric field

5

Difference in potential energy between a and b:b

b a aU U qE dl

If position b is the zero point:

ab

b

ba b a aV V V E dl

Relationship between electric field and potential

0V

a aV E dl

Unit of electric potential: Volt (V)

Zero point

6

Electric potential:

It depends on the chosen of zero point

0V

a aV E dl

Usually let V = 0 at infinity: a aV E dl

For the field created by a point charge :

Q.a

204a a

QV drr

04 a

Qr

Properties of electric potential

7

1) Va → potential energy per unit (+) charge

b

ba b a aV V V E dl

3) Va and Vba are scalars, differ from E vector

0V

a aV E dl

2) Va depends on the zero point, Vba does not

0,

V

a a aU qV q E dl

b

ab ab aW qV q E dl

Calculation of electric potential

8

Potential of point charge:

04QV

r (V = 0 at

infinity)System of several point charges:

04i

i

QVr

0

or 4

dQVr

If the electric field is known:0V

a aV E dl

dQ

r.A

Potential of electric dipole

9

Example1: (a) Determine the potential at o, a, b. (b) To move charge q from a to b, how much work must be done?

Q Qo ..

a.h

l lb

Solution: (a)0 0

04 4o

Q QVr r

, 0aV

0 04 8bQ QV

l l

08Q

l

(b)08ba b a

QV V Vl

0

8baQqW qV

l

Charged conductor sphere

10

Example2: Determine the potential at a distance r from the center of a uniformly charged conductor sphere (Q, R) for (a) r > R; (b) r = R; (c) r < R.

Solution: All charges on surface: ++

+

++

++

++

+ Q2

0

( ),4

QE r Rr

0 ( )E r R

(a) V at r > R:

204a r

QV drr

04Q

r Same as point charge

.a

11

++

+

++

++

++

+ Q

(b) V at r = R:

20 04 4b R

Q QV drr R

(c) V at r < R: b

c bcV E dl V

04bQV

R

.b

.c

Discussion:

(1) Conductor: equipotential body

(2) Charges on holey conductor

+ + +

--- ?

Same potential at any point!

Uniformly charged rod

12

Example3: A thin rod is uniformly charged (λ, L). Determine the potential for points along the line outside of the rod.

Solution: Choose infinitesimal charge dQ

xa

dL

dx

dQ0

ln4

d Ld

aV d L

d

04

dxx

.b

Potential at point b?

Charged ring

13

Example4: A thin ring is uniformly charged (Q, R). Determine the potential on the axis.

Solution:

R

xQ

. a

r

dQ

aV Ring

04dQ

r 04Q

r

or:2 2

0

4

aQVx R

Discussion:

(1) Not uniform? (2) x >> R (3) other shapes

Conical surface

14

Question: Charges are uniformly distributed on a conical surface (σ, h, R). Determine the electric potential at top point a. (V = 0 at infinity)

h

R

a

Electrostatic induction

15

Question: Put a charge q nearby a conductor sphere initially carrying no charge. Determine the electric potential on the sphere.

R

b

qo

----

-

-

++

+

+

+

+

Q

Connect to the ground?

*Breakdown voltage

16

Air can become ionized due to high electric field

Air → conducting → charge flows

Breakdown of air: 63 10 /E V m

Breakdown voltage for a spherical conductor?

Q

R04 R

QV E RR

5cm 15000VR V

Equipotential surfaces

17

Visualize electric potential: equipotential surfaces

Points on surface → same potential

(1) Surfaces ⊥ field at any point.

(3) Move along any field line, V ↘

(4) Surfaces dense → strong field

(2) Move on surface, no work done

Some examples

18

(1) Equipotential surfaces for dipole system:

(3) Parallel but not uniform field lines?

(2) Conductors

E determined from V

19

To determine electric field from the potential, useb

b a aV V E dl

In differential form: dV E dl

lE dl

Component of E in the direction of dl :

/lE dV dl

Total electric field: V V VE i j kx y z

Gradient of V

20

Gradient of V :

Gradient → partial derivative in direction V changes most rapidly

dVV ndn

n

V

V+dV

1 2

dl

dna b

E

ldVEdl

dVEdn

V V VE V i j kx y z

a

V E dl

Determine E from V

21

Example5: The electric potential in a region of space varies as V = x2yz. Determine E.

Solution:x

VEx

2 ,xyz

yVEy

2 ,x z

2 2 2E xyzi x zj x yk

zVEz

2x y

Determine V from E

22

Solution: 2xVE xyx

2V xydx 2 ( )x y C y

Example6: The electric field in space varies as . Determine V.2 22 ( )E xyi x y j

2 ( )y

V dC yE xy dy

2 2x y

313

y C

2 31 3

V x y y C ( C is constant )