Physics 505
Fall 2007
Practice Final
The final will be a three hour open book, open notes exam. Do all four problems.
1. Two point charges
q
and

q
are located on the
z
axis at
z
= +
d/
2 and
z
=

d/
2,
respectively.
a
) If the charges are isolated in space, show that the potential admits a Legendre
expansion
Φ(
r, θ
) =
2
q
4
π
0
l
odd
r
l
<
r
l
+1
>
P
l
(cos
θ
)
where
r
<
= min(
r, d/
2) and
r
>
= max(
r, d/
2).
b
) Now consider the charges to be contained inside a linear dielectric sphere of
permittivity
and radius
a
(where
a > d/
2).

q
q
Find the electric potential everywhere as an expansion in Legendre polynomials.
2. A point magnetic dipole
m
is located in vacuum pointing away from and a distance
d
away from a semiinfinite slab of material with relative permeability
μ
r
.
a
) Find the magnetic induction everywhere.
b
) What is the force on the dipole (magnitude and direction)?
3. An infinitely long solenoid of radius
a
has
N
tightly wound turns per unit length. For
a constant current
I
, elementary considerations tells us that the magnetic induction
is uniform inside the solenoid. In cylindrical coordinates,
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 Fall '08
 Stephens,P
 Charge, Magnetism, Magnetic Field, 1 L, three hour

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