Download Part I (50 points)

Part I (50 points)
1. Prove the mean value theorems:
(a) For charge-free space, the value of the electrostatic potential  (x ) at
any point is equal to the average of the potential over the surface of
any sphere centered on that point. (10 points)
[Hint: (x)  V
 (x ')
R
d 3 x '
1
4

  1 
 1 

  da ' ]

S R n '
n '  R  

(b) For a function f (r ) which satisfies the Laplace equation 2 f (r)  0 ,
the value of f (r ) at any point is equal to the average of the function
over the surface of any sphere centered on that point. (10 points)
[Hint: Green’s theorem
     d
2
V
2. We know the expansion:
2
3
x

 
 



da ]
S
n 
 n

^ ^
1
rl
  l1 Pl ( r r ') .
| r  r ' | l 0 r
(a) Show that this implies Pl ( x  1)  ( 1)l . (10 points)
(b) Three point charges are located on the z axis as follows, 2q at z=0
and –q at z=±a. Find the electric potential in the limit a→0, but keep
qa2 finite. (20 points)
[Hint: Legendre polynomials:
P0 ( x)  1 ; P1 ( x )  x ; P2 ( x )  (3x 2  1) / 2 ; P3 ( x )  (5 x 3  3x ) / 2 ; …]
Electrodynamics (Part II 50 points).
March 2005
1. A dc current I0 flows through the edge of a rotatable metal disk of radius R0 and its
axle. If the disk is perpendicularly placed in an external uniform magnetic

induction B , find the magnetic torque on the disk. (10%)
R0

B
2.
I0

(a) Find the B fields inside and outside a sphere with uniform magnetization

M and radius R. (10%) (b) If a small sphere cavity of radius r0 is located inside
the uniformly magnetized sphere (r0<R), and the distance between the centers of

the small cavity and the magnetized sphere is c, find the B field inside the cavity.
(5%)
r0
c
R
3. An ideal solenoid (length L>>radius R) with n closely wound coil-turns per unit
length is applied by a fixed current I.

(a) (10 points) Find the vector potential A inside and outside the solenoid in
cylindrical coordinate.
(b) (15 points) Find the magnetic force per unit area on the surface of the
solenoid.
Electrodynamics (Part III 50 points)
1. Use Cauchy integral formula f ( ) 
March 2005
1
f ( )
d with f ( )   ( )  1 ,

C
2i   
where  ( ) is the dielectric function, to derive the Kramers-Kronig relations:
Re  ( )  1 
Im  ( )  
2

2


P
0

P
0
  Im  ( )
d  ,
2   2
Re  ( ) - 1
d  .
2   2
Where Re  ( ) represents the real part of dielectric function, and Im  ( ) the
imaginary part. The symbol P denotes the principal value of the integral. (25%)
2. Write down Maxwell’s equations assuming that no dielectrics or magnetic
material are present. (State your system of units.) (10%)
In all of the following questions, you must justify your answer.
(a) If the signs of all source charges are reversed, what happens to the electric


and magnetic fields, E and B ? (5%)


(b) If the system is space inverted, i.e. x   x , what happens to the charge



density and current density,  and j , and to E and B ? (5%)
 


(c) If the system is time reversed, i.e. t  t , what happens to  , j , E

and B ? (5%)