Electromagnetic Theory I
Problem Set 10
Due: 7 November 2011
37. A point magnetic dipole with moment
m
=
m
ˆ
z
is placed at the center of a spherical
shell with uniform magnetic permeability
μ
, and with inner and outer radii
a, b
respectively.
a) Set up the boundary equations that determine the magnetic field in the three regions
0
< r < a
,
a < r < b
,
r > b
.
b) Solve the equations of part a) to find the
B
and
H
fields in all three regions.
c) Discuss the two extreme limits
μ
→ ∞
and
μ
→
0. Determine the limiting
B
and
H
fields in each region, and discuss the qualitative differences of the two limits.
d) Calculate the difference Δ
U
=
U
μ
=0

U
μ
=
∞
of the total magnetic field energy stored in
the two limiting situations. Handle the divergence when
r
→
0 by excluding the region
0
< r
≤
δ
a
in the energy integral. The divergence cancels in Δ
U
, and you can
then take
δ
→
0. Can you understand the sign in terms of the qualitative behavior of
the fields?
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 Spring '08
 Staff
 field lines, 5 m, ∆U

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