Download Physics PHYS 354 Electricity and Magnetism II Problem Set #2

Physics PHYS 354
Electricity and Magnetism II
Problem Set #2
1.
Method of Images: Conducting Sphere in a Uniform Electric Field
A conducting sphere in a uniform electric field can be thought of as a conducting
sphere placed between extremely distant positive and negative point charges as
shown below.
a)
Show that near the origin there is an approximately constant electric field,
+Q
-Q
O
z=-R
z=+R
E0 
Q
2 0 R 2
.
In the limit as R and Q approach infinity, but with Q R 2 a constant, this
approximation becomes exact.
b)
Using the method of images, it was shown that the potential due to a point
charge near a grounded conducting sphere is




q  1
1

V x  
  

y
4 0  x  y
x anˆ  


a
where nˆ  is a unit vector in the direction of the original source charge.
r
P

+Q
z=-R
a
-Q
z=+R
By using this relationship twice (once for each charge on either side of the
sphere) write down the total potential, expand, then let R go to infinity
while Q R 2 remains a constant. Show that to first order,


1  2Q
2Q a 3

r
cos


cos    

2
2
2
4 0  R
R r

where the omitted terms vanish in the limit R   .
c)
Show then that
 a3 
  E0  r  2  cos  .
 r 
where the first term is just due to the original electric field, and the second
term is the dipole strength of the charge distribution induced on the
surface.
2
2.
Laplace's Equation in Rectangular Coordinates
Consider an infinitely long rectangular tube with potentials on each of its four
sides as specified in the figure.
y
V=V0
(0,b)
V(x,y)
V=0
(0,0)
V=0
(a,b)
V=0
(a,0)
x
Show that the potential inside the rectangular tube is given by the following
expression:
V ( x, y ) 
1  nx  sinhny a 
sin
.

 n1,3,5 n  a  sinhnb a 
4V0


3