Download Chapter 26 Electric Potential

Chapter 26 Electric Potential
第二十六章 電位
Lightning
Just before lightning
Electric potential energy
Electric potential energy
U  U f  U i  W
where W is the work done by the static electric field.
For convenience we often define
U  U f  W
Where W is the work done by the field to move the charge
particle from infinity to its current position and U   0.
Electric potential
U  U f  U i  qE  s
V  U / q   E  s
In general we have
sf
V    E  ds
si
Work done by an applied force
K  K f  Ki  Wapp  W
If ΔK = 0, then Wapp  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
sf
V  V f  Vi    E  ds
si
or
sf
V f  Vi   E  ds
si
Vi can be assigned to any convenient value such as 0.
Potential due to a point charge
sf
V f  Vi   E  ds
si
 Vi  
rf
 Vi  
rf
ri
ri
1
q
rˆ  dr
2
4 0 r
1
q
dr
2
4 0 r
1
q rf
 Vi 
( ) ri
4 0 r
1
q
1 q

 (Vi 
)
4 0 rf
4 0 ri
V (r ) 
1
q
4 0 r
If
1
q
Vi 
4 0 ri
Potential due to a point charge
1
q
V (r ) 
4 0 r
Potential due to an electric dipole
V  V  V 
P
r

p
1
4 0
(
q q
 )
r r
r  r
V
(
)
4 0 r r
q
r
r̂
If the point of interest P is
far away from the dipole,
then
d cos 
V
(
)
2
4 0
r
1 p cos 
V
(
)
2
4 0
r
1 p  rˆ
V
( 2 )
4 0 r
q
Potential due to an electric dipole
Induced dipole moment
Potential due to a group of point
charges
n
n
qi
V  Vi 

4

i 1
0 i ri
Example
1
Potential due to continuous charge
distribution
dV 
1 dq
4 0 r
V   dV 
1
dq
4 0  r
Line of charge
V 
l
0
1
 dx
4 0 ( x 2  a 2 )1/ 2
l

2
2 1/ 2
ln( x  ( x  a ) ) 

0
4 0
Charged disk
V 
a
0
1
 (2 r )dr
4 0 (r 2  x 2 )1/ 2

2
2 1/ 2 a
(r  x ) 

0
2 0
Calculating the field from the
potential
q0 dV  q0 E cos  ds
dV
E cos   
ds
V
Ex  
x
E  V
V
V
V
ˆ
ˆ
 (
x
y
zˆ )
x
y
z
Electric potential energy of a
system of point charges
1
U   'U ij
2 i, j
U ij 
1
qi q j
4 0 rij
1
U   qi  V j (rij )
2 i
j i
Potential of a charged isolated
conductor
Surface charge density of a
conductor
1 Q1  1 R1
V1 

4 0 R1
0
1 Q2  2 R2
V2 

4 0 R2
0
1R1   2 R2
Electric fields near a conductor
1R1   2 R2
E1  1 /  0
E2   2 /  0
E1 R2

E2 R1
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