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NOTE 12 11/06/2007 3.2 The Method of Images

Suppose a point charge q is held a distance d above an infinite grounded conducting plane.

What is the potential in the region above the plane?

Boundary Conditions: (The first uniqueness theorem guarantees one solution only.) (p. 121-122) (1) V 0 at z 0 (since the conducting plane is grounded.)

(2) V 0 far from the charge (i.e. r(x2y2z2)1/2 >> d )

,( ), { 2 2 2 2/1 2 2 2 2/1 }

]) ( [ ]) ( 4 [

1

dz y x

q dz

y x

q

zy o

xV       

(3.9)

Above solution satisfies both boundary conditions.

3.2.2 Induced Surface Charge

(2.35) VaboveVbelow o nˆ Vbelow 0 (for conductor)

  ( )ˆ ( )ˆ z0

z V o above

o above

o V nV zabove

0

]) ( [

) ( ])

( [

) ( 4

1 { }

),

( 2 2 2 2/3 2 2 2 2/3





 

 z

dz y x

dz q dz

y x

dz o q

yx o

( , ) 21 [ 2 2 2]3/2 d y x y qd

x

(3.10)

 

2

0 0 0

) ( )

(

2 2 2 3/2 rdrd 2 2 1/2 q da

Q r d

qd d

r induced qd

total (3.11)

3.2.3 Force and Energy z

F d

q

o 22 ˆ

) 2 41 (



 

One could determine the energy by calculating the work required to

bring q in from infinity. The force required (to oppose the electric force) is z

d q

o 22 ˆ

) 2 41 (

 ,

d qz d qd

z q

o o

o z dz

W 4 4

1 4

4 1 )

2 4 (

1 2 2

2

2 ˆ ( )







3.2.4 Other Image Problems:

Example 3.2 A point charge q is situated a distance a from the center of a grounded conducting sphere of radius R. Find the potential outside the sphere.

HM#7 Problems: 3.7, 3.10, 3.12 (Due: 11/22)

期中考:11 月 15 日(四),10:10-12:00 am, LH108, 範圍: p.58-126

q

V=0 z

d

V (x,y,z)

-q d

(2)

q

q' Ra , bRa2

( , ) [ ]

cos ) / ( 2 ) / (

) / ( cos

41 2

2 2 2 2 2

2



a R r a R r

q a R ra

a r

q

r

o

V

0 ]

[

( / ) 2 cos

) / ( cos

41 2

2 2 2

2

 

 ra R R ra q

a R ra

a r

q

o2 2 2 2 2

) 4 (

1 )

( ' 4

1

R a

Ra q b

a qq

o

F o

 

(3.18)

3.3 Separation of Variations

Example 3.3: Boundary conditions:

(i) V 0 at y 0 (ii) V 0 at ya (iii) V Vo( y) at x 0 (iv) V 0 as

x  

2 0

2 2

2  

y V x

V , and V(x,y) X(x)Y(y) ( ) 2 2( ) ( ) 2 2( ) 0

dy y Y d dx

x X

d X x

y Y

2 0

2 2

2 ( )

) (1 ) ( ) (

1

dy y Y d y dx Y

x X d x

X 1( ) 2 2( ) k2

dx x X d x

X and Y(1y)d2dyY(2y) k2

0 ) 2 ( ) (

2

2 k X x

dx x X

dX(x) Aekx Bekx, Y(y)CsinkyDcosky

V(x,y)(Aekx Bekx)(CsinkyDcosky)

(iv) condition  V(x,y)ekx(CsinkyDcosky) (i) condition  V(x,y)Cekxsinky

(ii) condition  sinka0 kan, i.e. k n/a (n=1,2,3,….)

( , ) / sin( / )

1C e n y a

y x

V n x a

n n

(iii) condition (0, ) sin( / ) ( )

1

y V a y n C

y

V o

n

n

a a

n Cn n y a m y a dy Vo y m y a dy

0 0

1 sin(/ )sin(/ ) ( )sin(/ )

a n m

m a n

dy a y m a y n 0

) (

) ( 0/2 )

/ sin(

) /

sin(  

Cn aaVo y n y a dy

0

2 ( )sin(/ ) , if Vo(y) is a constant potential, then

( )

) ( 0 / 4 2

0

2 sin( / ) (1 cos ) even

n odd n V

a V a

n Vo n y a dy o n o

C

( , ) / sin( / )

...

5 , 3 , 1 4 1

a y n e

y x

V n x a

n n

Vo

(3)

(3.36)

Example 3.4: Boundary conditions (i) V 0 at y 0

(ii) V 0 at ya (iii) V Vo at x b (iv) V Vo at x b

2 0

2 2

2  

y V x

V

V(x,y)(Aekx Bekx)(CsinkyDcosky) )

, ( ) ,

( x y V x y

V

AB

V(x,y)coshkx(CsinkyDcosky)

(i) and (ii) conditions  D0 and k n/a

V(x,y)Ccosh(nx/a)sin(ny/a)

1 cosh( / )sin( / ) )

, (

n Cn n x a n y a

y x

V

(iii) condition  o

n Cn n b a n y a V

y b

V

1 cosh( / )sin( / ) )

,

(

(( ))

0 /

) 4 /

cosh( even

n odd

n n b a Vo

C

( , ) cosh(cosh( // ))sin( / )

...

5 , 3 , 1 4 1

a y n y

x

V nn bx aa

n n

Vo

3.3.2 Spherical Coordinates

0 )

(sin )

(

2

2 2 2 2

2 sin

1 sin

2 1

2

1

  

V

r r

r

Vr r

V r V (3.53) azimuthal symmetry: r(r2Vr)sin1 (sin V)0 (3.54)

) ( ) ( ) ,

(r R r

V (3.55)

[See Example 3.5 and do it by yourself.]

(4)

(3.56)  (3.65)

0

) (cos ) (

) ,

( 1

l

r l B l

lr P

A r

V ll (3.65)

1 )

0(cos

P , P1(cos)cos , 2

/ ) 1 cos 3 ( )

(cos 2

2   

P , P3(cos)(5cos33cos)/2

Example 3.6: The potential Vo()is specified on the surface of a hollow sphere of radius R.

Find the potential inside the sphere.

0

) (cos )

, (

l

l l lr P A r

V , ( , ) (cos ) ( )

0

o

l

l l

lR P V

A R

V

1 2 2

0 '

{ 0 sin

) (cos ) (cos

 

l l

l P d

P    

, ll''ll or ll' 2l21

AR P Pl d Vo Pl d

l

l l

l (cos ) (cos )sin ( ) '(cos )sin

' 0

0 0

 

1 ' 22 ' ' 0

1 22 '

0 0 (cos ) '(cos )sin

 

l l l

l

l ll l l l

l l

l

lR P P d AR A R

A

d P

V

Al lRl o( ) l(cos )sin

2 0 1 2

Let Vo()ksin2(/2), sin2(/2) 12(1cos) 21[Po(cos)P1(cos)], then

d P

P P

Al lRl k [ o(cos ) (cos )] l(cos )sin

0 1

2 2 1

2

] sin ) (cos ) (cos sin

) (cos ) (cos

[ 0 0 1

2 2 1

2lRl k

PoPl  d

PPl  d

] sin ) (cos ) (cos sin

) (cos ) (cos

[ 0 0 1

2 2 1

2lRl k

PoPl  d

PPl  d

2 2 2 1 k2 k

Ao   ,A123R k2(32)2kR, Al1 0

) 1

( cos ) ( )

(cos )

,

( 0 1 1 1 2 2 2 cos

0 k r R

R k o k

o l

l l

lr P A r P Ar P r

A r

V

(3.71)

Strongly suggested examples: Examples: 3.7, 3.8, 3.9

參考文獻

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