Chapter 26 Electric Potential

第二十六章 電位

Lightning

Just before lightning

Electric potential energy

Electric potential energy

f iU U U W

where W is the work done by the static electric field.

For convenience we often define

fU U W

Where is the work done by the field to move the charge particle from infinity to its current position and .

W

0U

Electric potential

f iU U U qE s

/V U q E s

In general we have

f

i

s

sV E ds

Work done by an applied force

f i appK K K W W

If ΔK = 0, then appW W q V

Equipotential surfaces

Surfaces in space on which V is constant.

Equipotential surfaces

If ΔV is chosen to be the same for all adjacent equipotential surfaces, then the electric filed is inversely proportional to the separations of the equipotential surfaces.

Calculating the potential from the field

f

i

s

f i sV V V E ds

f

i

s

f i sV V E ds

or

Vi can be assigned to any convenient value such as 0.

Potential due to a point charge

20

20

0

0 0

1ˆ

4

1

4

1( )

4

1 1( )

4 4

f

i

f

i

f

i

f

i

s

f i s

r

i r

r

i r

r

i r

if i

V V E ds

qV r dr

r

qV dr

r

qV

r

q qV

r r

0

1( )

4

qV r

r If

0

1

4ii

qV

r

Potential due to a point charge

0

1( )

4

qV r

r

Potential due to an electric dipole

0

1( )

4

q qV V V

r r

P

r r

p

0

( )4

r rqV

r r

If the point of interest P is far away from the dipole, then

20

cos( )

4

q dV

r

20

1 cos( )

4

pV

r

20

ˆ1( )

4

p rV

r

r̂

Potential due to an electric dipole

Induced dipole moment

Potential due to a group of point charges

1 0

1

4

n ni

ii i i

qV V

r

Example

Potential due to continuous charge distribution

0

1

4

dqdV

r

0

1

4

dqV dV

r

Line of charge

2 2 1/ 200

2 2 1/ 2

00

1

4 ( )

ln( ( ) )4

l

l

dxV

x a

x x a

Charged disk

2 2 1/ 200

2 2 1/ 2

00

1 (2 )

4 ( )

( )2

a

a

r drV

r x

r x

Calculating the field from the potential

0 0 cosq dV q E ds

cosdV

Eds

x

VE

x

ˆ ˆ ˆ( )

E V

V V Vx y z

x y z

Electric potential energy of a system of point charges

'

,

1

2 iji j

U U

0

1

4i j

ijij

q qU

r

1( )

2 i j iji j i

U q V r

Potential of a charged isolated conductor

Surface charge density of a conductor

1 1 11

0 1 0

1

4

Q RV

R

2 2 22

0 2 0

1

4

Q RV

R

1 1 2 2R R

Electric fields near a conductor

1 1 2 2R R

1 1 0/E 2 2 0/E

1 2

2 1

E R

E R

The field strength is strongest at the point on a conductor where its local curvature of radius is the smallest.

Image charge

Home work

Question ( 問題 ): 8, 15, 21

Exercise ( 練習題 ): 5, 12, 19

Problem ( 習題 ): 12, 28, 29, 37