Physics 3323, Electromagnetism I, Spring 2002
EXAM #2
Wednesday, March 20, 2002
Remember to work each problem as fully as possible. Partial credit will be given so
SHOW ALL OF YOUR WORK!!
Useful information:
∫ u sin u du
&
E dipole =
= sin u − u cosu
[
1 p
2 cosθ rˆ + sinθ θˆ
3
4πε 0 r
]
1. A point charge q is placed a distance d above the surface a semi-infinite slab of
homogenous dielectric material (with relative permittivity εr). The surface of the slab is
located at z=0.
In this problem, we will find the potential above and below the surface using the method
of images. Before you begin, here are a few hints:
i) You have to use different image charge configurations for the regions z > 0 and z < 0.
ii) For the region z > 0, the location of the image charge below the surface is where we
would expect, but the presence of the dielectric will modify its magnitude (q’).
iii) For the region z < 0, there is only one image charge (located above surface where we
would expect), but it’s magnitude will also be different, too (q’’).
iv) The usual boundary conditions for potentials at dielectric surfaces apply.
a) For z > 0, write down an expression for Vabove, the potential above the surface in terms
of the charge q and unknown image charge q’. (6 points)
b) For z < 0, write down an expression for Vbelow, the potential below the surface in terms
of the unknown image charge q’’. (6 points)
c) Using the appropriate boundary conditions, find q’ and q’’ in terms of q. (14 points)
d) Find the force on the charge q exerted by the dielectric. (7 points)
2. A thin rod located along the x-axis from x = -a to x = a has a charge density given by
λ = k sin(πx/a).
a) Write down a general expression for the potential in terms of a multipole expansion,
keeping in mind that the integrals over linear charge density are one dimensional over x.
(7 points)
b) Draw a graph of λ vs. x. From your graph, what can you conclude about the monopole
term in the potential? (6 points)
c) Prove yourself correct by computing the monopole term in the potential. (10 points)
d) Compute the dipole term in the potential. (10 points)
3. A point charge q is located a distance r from a neutral molecule having polarizability
α.
a) Compute the induced dipole moment on the molecule due to the point charge. (8
points)
b) Compute the field felt by the point charge due to the induced dipole moment. (15
points)
c) Compute the force exerted on the point charge by the induced dipole. Is it attractive or
repulsive? (10 points)