current - 國立臺灣大學 · 與 太 陽 眼 鏡. 4-9 連續電荷分佈的電位 ......
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4 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?
4-1 電位能
• Fe and Fg are
mathematically identical
• Fe is a conservative force
rdFWUUU eif
4-2 電位
• Ue depends on q, but Ve does not
f if i
f
i
UV
q
U UV V V
q q
U WE ds
q q
4-3 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
4-4 由電場算電位
• The differential
work done by F
f
isdEqW
sdEqdWsdFdW
0
0 ,
Ex. 1 均勻電場
(a) Along path (a)
Ex. 1 均勻電場︰路徑(a)
EddsEdsE
dsEsdEVV
f
i
f
i
f
i
f
iif
)0(cos
Ex. 1 均勻電場︰路徑(b)
EddE
dsE
dsEsdEVV
f
c
f
i
f
iif
222
)45(cos
4-5 點電荷的電位
cos180
f f
i
E ds E ds E dr
V E ds E dr
r
qV
r
qrd
r
qV
r
qE
rdEsdEV
ff
ff
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
4-6 一組點電荷的電位
• The principle of superposition
n
i i
in
i
ir
qVV
101 4
1
(a) on a circle
(b) on an arc
0 ,4
112
0
ER
eV
Ex.2 Vc and Ec for 12 electrons
0 ,4
112
0
ER
eV
4-7 電雙極的電位
) ,cos(
cos
4
1cos
44
)(4
1
2
2
0
2
00
0
2
1
rrrdrr
r
p
r
dq
rr
rrq
r
q
r
qVVVV
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
4-8 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
4-9 連續電荷分佈的電位
r
dqdV
r
qVV
n
i i
in
i
i
0101 4
1
4
1
Line of Charge
2/122
00 )(4
1
4
1
,
dx
dx
r
dqdV
dxdq
d
dLLV
ddLL
dxx
dx
dx
dx
dxdVV
L
L
L
])(ln
4
ln])(ln[4
)(ln(4
)(4
)(4
1
2/122
0
2/122
0
0
2/122
0
0 2/122
0
0 2/122
0
計算
)1(2 22
0 Rz
zE
Charged Disk
)(2
)(
2
)(
))(2(
4
1
4
1
))(2(
22
0
0 2/122
0
2/122
00
zRz
Rz
RdRdVV
Rz
RdR
r
dqdV
RdRdq
R
計算
4-10 由電位算電場
z
VE
y
VE
x
VE
s
VE
ds
dVE
dsEqdVqdW
zyx
s
,,
cos
)(cos00
• The work done by the
electric field
4-11 The gradient operator 梯度運算子
, ,
ˆˆ ˆ
ˆˆ ˆ =
ˆˆ ˆ ( )
x y z
x y z
V V VE E E
x y z
E E i E j E k
V V Vi j k
x y z
i j k V Vx y z
The Laplacian operator
0
2
0
2
2 2 2
2 2 2
2 2 2
2 2 2
0
( )
ˆ ˆˆ ˆ ˆ ˆ( ) ( )
( )
E V E
V V
i j k i j kx y z x y z
x y z
Vx y z
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
Vx y z
Vx y
X x Y y X x Y y
x y
X x Y yY y X x
x y
For 2 dimensional
problems
sinxe 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 yY 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 22 2
2 2
22
2
22
2
1 ( ) 1 ( ),
( ) ( )
( )( ) 0 ( )
( )( ) 0 ( ) sin
x
X x Y y
X x x Y y y
X xX x X x e
x
Y yY y Y y y
y
A boundary value problem
Ex.3 The charged disk
)1(2
)(2
)(2
220
22
0
22
0
Rz
z
zRzdz
d
z
VE
zRzV
z
4-12一組點電荷的電位能
r
qqWU
r
qV 21
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
Mevfm
eeKUK 6.24
23.9
)79)(2(
4
1
0
4-13 Potential of a charged isolated
conductor
)0( 0 EsdEVVf
iif
• 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