Electrodynamics 2nd midterm exam
95.12.23
Total: 110%
z
1. As shown in Fig. 1, the surface charge density on a thin spherical shell with radius b is
f’
’
f’fcos2’. Find the electric potential inside and outside the spherical shell by two
different methods. (20%)
b

2. A point electric dipole p is placed at the center of a thin-grounded conducting spherical
Fig. 1
z
shell with radius a (Fig. 2). Find the electric potential inside the spherical shell by the
following methods: (a) Considering the general solutions of Laplace equation. (10%); (b)
Using the Green function method. (10%)

p
3. Consider a Dirichlet boundary value problem in the space z 0. As shown in Fig.3, a thin
flat disk of radius a fits in an infinite conducting sheet at z=0. The center of disk is at the
a
origin. The disk is maintained at a fixed potential (s=V), while outside the disk, the
infinite sheet is kept at zero potential s=0).
(a) Find the Green function in cylindrical coordinates. (5%)
Fig. 2
z
(b) Find an integral expression for the potential (z) at any point above the plane. (5%)
(c) Find the potential at the point with a perpendicular distance z above the center of the disc. (5%)
a
s0
4. In Fig. 4, a hollow conducting cylindrical box is with inner radius a and outer radius b.
The potentials of the caps at two ends and the outer shell (=b) are kept at zero, while the
potential of the inner shell (=a) is kept at a constant V0. Find the potential at any point
inside the box. (b   a; L  z  0). (20%)
a
b
s0
L
sV0
5. As shown in Fig. 5, a grounded rectangular box is bounded by the surfaces x=0, x=a, y=-b/2, y=b/2,
z=0, z=c. If a point charge q is located at the point (x’, y’, z’) inside the box, show that the potential
s0
inside the box can be expressed in the following alternative forms:
  lx   lx'   my   my '   nz   nz '  
 sin 
 cos
 cos
 sin 
 sin 

 sin 
8q
 
a
a
b
b
c
c 












 (r , r ' ) 

2
2
2

 0 2 abc l 1, m  0, n 1
 l  m n


     


a  b  c


Fig. 4
z

4q    lx   lx'   my   my '  sinh(  lm z  ) sinh[  lm (c  z  )]
 
 (r , r ' ) 

 sin   sin   cos  cos
 0 ab l 1,m 0   a   a   b   b 
sinh(  lm c)
c
b
a
terms in the potential expansion. (10%)
y
x
Fig.5
where  lm   (l 2 / a 2  m 2 / b 2 ) 1 / 2 ; z  , z  are from the comparison between z and z’. (20%)
6. In Fig. 6, a uniformly charged wire with total charge Q, and total length 2L. (a) Find the
multipole moment qm for  =0,  =1 and  =2. (10%) (b) For r>L, find the first three
Fig.3
z
z=L
z=－L
Fig. 6