Physics 505
Fall 2007
Homework Assignment #6 — Due Thursday, October 25
Textbook problems: Ch. 4: 4.1, 4.7, 4.8, 4.9
4.1 Calculate the multipole moments
q
lm
of the charge distributions shown as parts
a
)
and
b
). Try to obtain results for the nonvanishing moments valid for all
l
, but in each
case ﬁnd the ﬁrst
two
sets of nonvanishing moments at the very least.
a
)
z
a
a
q
q
q
q


a
a
x
y
b
)

a
x
y
z
q
q
a
q
2
c
) For the charge distribution of the second set
b
) write down the multipole expansion
for the potential. Keeping only the lowestorder term in the expansion, plot the
potential in the
x

y
plane as a function of distance from the origin for distances
greater than
a
.
d
) Calculate directly from Coulomb’s law the exact potential for
b
) in the
x

y
plane.
Plot it as a function of distance and compare with the result found in part
c
).
Divide out the asymptotic form in parts
c
) and
d
) to see the behavior at large distances
more clearly.
4.7 A localized distribution of charge has a charge density
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 Fall '08
 Stephens,P
 Charge, Magnetism, Work, Electric charge, Multipole moments, Axial multipole moments, quadrupole, Multipole expansion, multipole moments qlm

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