3 電位
What is the danger if your hair suddenly stands up?
Lightning bolt and ground current
How can you reduce your risk from ground current?
3-1 電位能
Fe and Fg are
mathematically identical
Fe is a conservative force
r d F
W U
U
U f i e
3-2 電位
Ue depends on q, but Ve does not
q W q
U
q U q
V U V
V
q V U
f i i
f
3-3 等位面
Four Equipotential Surfaces
FIGURE 23.22 CONTOUR LINES, CURVES OF CONSTANT ELEVATION
等位面3例
E.S. for a uniform field, a point charge, and an electric dipole
3-4 由電場算電位
The differential work done by F
f
i
E d s
q W
s d E
q dW
s d F
dW
0
,
0Ex. 1 均勻電場
(a) Along path (a)
Ex. 1 均勻電場︰路徑(a)
Ed ds
E ds
E
ds E
s d E
V V
f i f
i
f i f
i i f
(cos 0 )
Ex. 1 均勻電場︰路徑(b)
Ed E d
E ds
ds E
s d E
V V
f c
f i f
i i f
2 2 2
) 45 (cos
3-5 點電荷的電位
cos180
f f
i
E ds E ds E dr
V E ds E dr
r V q
r r q
r d V q
r E q
r d E s
d E
V
f f
f f
i
0
0 2
0 2 0
4 1
1 4
1 4
4 1
點電荷的電位
點 電 荷 的 電 位 圖
3-6 一組點電荷的電位
• The principle of superposition
ni i
i n
i
i
r
V q V
0 1
1
4
1
(a) on a circle
(b) on an arc
0
4 , 12 1
0
E
R V e
Ex.2 Vc and Ec for 12 electrons
0
4 , 12 1
0
E
R V e
3-7 電雙極的電位
)
, cos
(
cos 4
1 cos
4 4
) 4 (
1
2
2 0
2 0
0
0 2
1
r r
r d
r r
r p r
d q
r r
r r
q
r q r
V q V
V V
i
i
雙極矩 p = qd
p 由負電荷指向正電荷
Induced Dipole Moment
感應雙極矩
極化 (polarization)
水分子與微波爐
3-8 Polarization (偏振)
Polarized Light
The Plane of Polarization
Linear, Circular and Elliptical polarization
偏振片
Polarization of Light
偏 振 與 太 陽 眼 鏡
3-9 連續電荷分佈的電位
r
dV dq r
V q V
n
i i
i n
i
i
1 0
1 0 4
1 4
1
Line of Charge
2 / 1 2
2 0
0 4 ( )
1 4
1 ,
d x
dx r
dV dq
dx dq
d
d L
V L
d d
L L
d x
x
d x
dx
d x
dV dx V
L L
L
] )
ln ( 4
ln ]
) (
4 ln[
) (
4 ln(
) (
4
) (
4 1
2 / 1 2 2
0
2 / 1 2 2
0
0 2 / 1 2 2
0
0 2 2 1/ 2
0
0 2 2 1/ 2
0
計
算
) 1
2 0 ( z2 R2 E z
Charged Disk
) 2 (
) (
2
) (
) )(
2 ( 4
1 4
1
) )(
2 (
2 2
0
0 2 2 1/2
0
2 / 1 2 2
0 0
z R
z
R z
R d dV R
V
R z
R d R
r dV dq
R d R
dq
R
計算
3-10 由電位算電場
z E V
y E V
x E V
s E V
ds E dV
ds E
q dV
q dW
z y
x
s
, ,
cos
)
0 (cos
0
• The work done by the electric field
3-11 The gradient
operator 梯度運算子
, ,
ˆ ˆ ˆ
ˆ ˆ ˆ
=
ˆ ˆ ˆ
( )
x y z
x y z
V V V
E E E
x y z
E E i E j E k
V V V
i j k
x y z
i j k V V
x y z
The Laplacian operator
0 2
0 2
2 2 2
2 2 2
2 2 2
2 2 2
0
( )
ˆ ˆ
ˆ ˆ ˆ ˆ
( ) ( )
( )
E V E
V V
i j k i j k
x y z x y z
x y z
x y z V
The Laplace Equation
2 2 2
2 2 2
2 2
2 2
2 2
2 2
2 2
2 2
( ) 0
( ) 0, If V(x,y)=X(x)Y(y) ( ) ( ) ( ) ( )
0
( ) ( )
( ) ( ) 0
x y z V
x y V
X x Y y X x Y y
x y
X x Y y
Y y X x
x y
For 2 dimensional problems
x sin
e y
Solving Laplace Equation by separation of variables
2 2
2 2
2 2
2 2
2 2
2 2
2 2
2 2
( ) ( )
( ) ( ) 0
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) 0
1 ( ) 1 ( )
( ) ( ) 0
1 ( ) 1 ( )
( ) , ( )
X x Y y
Y y X x
x y
Y y X x X x Y y
X x Y y x X x Y y y
X x Y y
X x x Y y y
X x Y y
X x x Y y y
The Solution
2 2
2 2
2 2
2
2 2
2
2 2
1 ( ) 1 ( )
( ) , ( )
( ) ( ) 0 ( )
( ) ( ) 0 ( ) sin
x
X x Y y
X x x Y y y
X x X x X x e
x
Y y Y y X x y
y
A boundary value problem
Ex.3
The charged disk
) 1
2 (
) 2 (
) 2 (
2 2
0
2 2
0 2 2
0
R z
z
z R
dz z d z
E V
z R
z V
z
3-12一組點電荷的電位能
r q W q
r U
V q 1 2
0 1
0 4
1 4
1
• For charges q1 and q2
Ex.4 Three charges
Ex.4 An alpha particle and a gold nucleus
fm Mev e K e
U
K 24.6
23 .
9
) 79 )(
2 ( 4
1
0
3-13 Potential of a charged isolated conductor
) 0 (
0
V E d s E
V
fi i f
• 導體內部及表面為一等位面
(equipotentail surface)
Discharge
Plots of E (r) and V(r)
Calculating electric potential
Vsurface = EsurfaceR
Vmax = EmaxR
e.g. Emax = 3×106 V/m
If R=1 cm Vmax = 30,000V If R=2 m Vmax = 6MV
Example 23.8 A charged conducting sphere