current - 國立臺灣大學 · 與 太 陽 眼 鏡. 4-9 連續電荷分佈的電位 ......

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