Lecture 7 Notes, 95.657 Electromagnetic Theory I, Fall 2011
Dr. Christopher S. Baird, UMass Lowell
1. Expansion of Green Functions in Spherical Coordinates
 Consider the problem of a spherical boundary with radius
a
, the potential is known on the
boundary, there is charge present, and we wish to find the the potential anywhere external to the
sphere.
 We have already solved this problem using Green's functions:
x
=
1
4
0
∫
x
'
G
D
d
3
x
'
−
1
4
∮
d G
D
d n
'
da
'
External to a spherical boundary we have already found the Green's function, it being the potential
created by a unit charge and its image:
G
x
,
x
'
=
1
∣
x
−
x
'
∣
−
1
∣
x '
a
x
−
a
x '
x
'
∣
 When solving problems with both charges and boundary surfaces, the mathematics is simplified if
the Green's function is expanded in spherical harmonics.
 The addition theorem was used to find the spherical harmonics expansion of a unit charge:
1
∣
r
−
r
0
∣
=
4
∑
l
=
0
∞
∑
m
=−
l
l
1
2
l
1
r
l
r
0
l
1
Y
l m
*
'
,
'
Y
lm
,
if
r
r
0
and
1
∣
r
−
r
0
∣
=
4
∑
l
=
0
∞
∑
m
=−
l
l
1
2
l
1
r
0
l
r
l
1
Y
lm
*
'
,
'
Y
lm
,
if
r
r
0
 Both the point charge and its image can be expanded in this way to give the spherical Green's
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 Fall '11
 Staff
 Charge, Mass, Boundary value problem, LM, Green's function, boundary condition r=b

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