EAD 234A: E&M
Homework #4
Due Thursday, February 4, 2010
1.
A thin insulating rod, running from z = a to z = +a, carries the following line charges:
(a)
0
cos
2
z
a
π
λλ
⎛⎞
=
⎜⎟
⎝⎠
(b)
0
sin
2
z
a
=
(c)
0
cos
z
a
=
In each case, find the leading term in the multipole expansion of the potential.
2.
A charge Q is distributed uniformly along the z axis from z = a to z =a. Find the first
three nonvanishing terms in the multipole expansion of the electric potential V(r ,
θ
) for r
> a.
3.
(
a) Evaluate the exterior spherical multipole moments for a shell of radius R which carries a
surface charge density
σ
(
θ
,
φ
) =
σ
0
sin
θ
cos
φ
.
(b) Write
ϕ
(r > R,
θ
,
φ
) in the form
ϕ
(x, y, z, r).
(c) Evaluate the interior spherical multipole moments for the shell of part (a).
(d) Write
ϕ
(r < R,
θ
,
φ
) in the form
ϕ
(x, y, z, r)
(e) Check the matching conditions for
ϕ
and
E
at r = R.
(f) Extract the dipole moment
p
of the shell from your answer to part (b).
4.
(a). Calculate the multipole moments q
lm
of each of the two charge distributions shown in the figure
below. Try to obtain results for the nonvanishing moments valid for all l, but in each case find the first
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 Spring '10
 NCL
 Electrostatics, Electric charge, Fundamental physics concepts, Multipole expansion

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