3 電位

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(1)

3 電位

What is the danger if your hair suddenly stands up?

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Lightning bolt and ground current

How can you reduce your risk from ground current?

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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

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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

(5)

3-3 等位面

Four Equipotential Surfaces

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FIGURE 23.22 CONTOUR LINES, CURVES OF CONSTANT ELEVATION

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等位面3例

E.S. for a uniform field, a point charge, and an electric dipole

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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

,

0

(9)

Ex. 1 均勻電場

(a) Along path (a)

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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 )

(11)

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

 

(12)

3-5 點電荷的電位

cos180

f f

i

E ds E ds E dr

V E ds E dr

   

       

(13)

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

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點 電 荷 的 電 位 圖

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3-6 一組點電荷的電位

• The principle of superposition

n

i i

i n

i

i

r

V q V

0 1

1

4

1



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(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



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3-7 電雙極的電位

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)

, 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

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Induced Dipole Moment

感應雙極矩

極化 (polarization)

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水分子與微波爐

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3-8 Polarization (偏振)

Polarized Light

The Plane of Polarization

Linear, Circular and Elliptical polarization

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偏振片

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Polarization of Light

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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





(26)

Line of Charge

2 / 1 2

2 0

0 4 ( )

1 4

1 ,

d x

dx r

dV dq

dx dq

 





(27)

 

 

 

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











(28)

) 1

2 0 ( z2 R2 E z

 

Charged Disk

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) 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

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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

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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

     

   

   

(32)

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

   

    

 

 

(33)

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

(34)

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

 

(35)

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

 

 

(36)

A boundary value problem

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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

(38)

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

(39)

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



(40)

3-13 Potential of a charged isolated conductor

) 0 (

0 

VE d s E

V

f

i i f

 

• 導體內部及表面為一等位面

(equipotentail surface)

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Discharge

(42)

Plots of E (r) and V(r)

(43)

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

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Figure

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