Physics 505
Fall 2005
Homework Assignment #4 — Solutions
Textbook problems: Ch. 3: 3.4, 3.6, 3.9, 3.10
3.4 The surface of a hollow conducting sphere of inner radius
a
is divided into an
even
number
of equal segments by a set of planes; their common line of intersection is the
z
axis and they are distributed uniformly in the angle
φ
. (The segments are like the
skin on wedges of an apple, or the earth’s surface between successive meridians of
longitude.) The segments are kept at fixed potentials
±
V
, alternately.
a
) Set up a series representation for the potential inside the sphere for the general
case of 2
n
segments, and carry the calculation of the coefficients in the series far
enough to determine exactly which coefficients are different from zero. For the
nonvanishing terms, exhibit the coefficients as an integral over cos
θ
.
The general spherical harmonic expansion for the potential inside a sphere of
radius
a
is
Φ(
r, θ, φ
) =
l,m
α
lm
r
a
l
Y
lm
(
θ, φ
)
where
α
lm
=
V
(
θ, φ
)
Y
*
lm
(
θ, φ
)
d
Ω
In this problem,
V
(
θ, φ
) =
±
V
is independent of
θ
, but depends on the azimuthal
angle
φ
. It can in fact be thought of as a square wave in
φ
2
π
V
V

ϕ
n
=4
This has a familiar Fourier expansion
V
(
φ
) =
4
V
π
∞
k
=0
1
2
k
+ 1
sin[(2
k
+ 1)
nφ
]
This is already enough to demonstrate that the
m
values in the spherical harmonic
expansion can only take on the values
±
(2
k
+1)
n
. In terms of associated Legendre
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polynomials, the expansion coefficients are
α
lm
=
2
l
+ 1
4
π
(
l

m
)!
(
l
+
m
)!
2
±
0
V
(
φ
)
e

imφ
dφ
1

1
P
m
l
(
x
)
dx
=
4
V
π
2
l
+ 1
4
π
(
l

m
)!
(
l
+
m
)!
∞
k
=0
1
2
k
+ 1
2
π
0
sin[(2
k
+ 1)
nφ
]
e

imφ
dφ
×
1

1
P
m
l
(
x
)
dx
=

4
iV
2
l
+ 1
4
π
(
l

m
)!
(
l
+
m
)!
∞
k
=0
δ
m,
(2
k
+1)
n

δ
m,

(2
k
+1)
n
2
k
+ 1
1

1
P
m
l
(
x
)
dx
Using
P

m
l
(
x
) = (

)
m
[(
l

m
)!
/
(
l
+
m
)!]
P
m
l
(
x
), we may write the nonvanishing
coefficients as
α
l,

(2
k
+1)
n
= (

)
n
+1
α
l,
(2
k
+1)
n
=

4
iV
2
k
+ 1
2
l
+ 1
4
π
(
l

(2
k
+ 1)
n
)!
(
l
+ (2
k
+ 1)
n
)!
1

1
P
(2
k
+1)
n
l
(
x
)
dx
(1)
for
k
= 0
,
1
,
2
, . . .
. Since
l
≥
(2
k
+ 1)
n
, we see that the first nonvanishing term
enters at order
l
=
n
. Making note of the parity of associated Legendre polyno
mials,
P

l
m
(

x
) = (

)
l
+
m
P
m
l
(
x
), we see that the nonvanishing coefficients
are given by the sequence
α
n,n
,
α
n
+2
,n
,
α
n
+4
,n
,
α
n
+6
,n
, . . .
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 Spring '09
 Fourier Series, Magnetism, Work, Spherical Harmonics, Cos, Associated Legendre function

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