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Homework 7 Answers, 95.657 Electromagnetic Theory I
Dr. Christopher S. Baird, UMass Lowell
Jackson 3.14
A line charge of length 2
d
with a total charge
Q
has a linear charge density varying as (
d
2
–
z
2
), where
z
is the distance from the midpoint. A grounded, conducting, spherical shell of inner radius
b
>
d
is
centered at the midpoint of the line charge.
(a) Find the potential everywhere inside the spherical shell as an expansion in Legendre polynomials.
SOLUTION:
This problem includes both a charge distribution and a boundary condition. We must use the Green
function method to include both. The Green function method solution for Dirichlet boundary conditions
is:
x
=
1
4
0
∫
x
'
G
D
d
3
x
'
−
1
4
∮
d G
D
d n
'
da
'
In this particular case, the boundary condition states that the potential is zero on the spherical surface.
This causes the entire surface integral in the Green function solution to equate to zero. The solution is
then:
x
=
1
4
0
∫
x
'
G
D
d
3
x
'
The charge density that satisfies the description above is:
r ,
,
=
3
Q
8
d
3
d
2
−
r
2
r
2
[
cos
−
1
cos
1
]
if
r
<
d
and
r ,
,
=
0
r
>
d
The
r
2
in the denominator was required to convert the linear charge density into spherical coordinates.
The constants out front were found by integrating over all space and setting the result equal to the total
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 Charge, Magnetism, Mass, Work

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