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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