Solutions to Homework Set 10
Dielectric Sphere
: The solution outside will be of the form
Φ
out
(
r, θ
)
=

E
0
rP
1
(cos
θ
) +
C
1
r
2
P
1
(cos
θ
)
=

E
0
r
+
C
r
2
cos
θ.
while, inside
Φ
in
(
r, θ
) =
A
1
rP
1
(cos
θ
)
.
These terms have been selected to match the asymptotically uniform electric field at infinity,
to be finite at the centre of the sphere, and to all have the same angular dependence.
The continuity of the tangential electric field will be ensured if Φ is continuous at the
surface of the sphere. The continuity of the radial component of
D
requires
∂
Φ
in
∂r
=
0
∂
Φ
out
∂r
at
r
=
a
. These two conditions are
A
1
=

E
0
+
C
1
a
3
,
and
A
1
=

0
E
0

2
0
C
1
a
3
,
respectively. This pair of equations may be solved to give
A
1
=

3
0
+ 2
0
E
0
C
1
=

0
+ 2
0
a
3
E
0
.
Therefore
E
in
=
3
0
+ 2
0
E
0
.
1
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Hollow Sphere
: The generating function for Legendre polynomials is
1
(1 +
r
2

2
r
cos
θ
)
1
/
2
=
∞
X
l
=0
r
l
P
l
(cos
θ
)
,
r <
1
.
If we differentiate this relation with respect to
r
, and then multiply the result by
r
, we have

r
2
+
r
cos
θ
(1 +
r
2

2
r
cos
θ
)
3
/
2
=
∞
X
l
=0
lr
l
P
l
(cos
θ
)
.
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 Fall '07
 Stone
 Work, Cos, Nuclear weapon, Heisenberg, Rcrit, Rudolph Peierls

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