4Partial differential equations
IB Methods
4.5 Multipole expansions for Laplace’s equation
One can quickly check that
φ(r) =
1
r − r
0

solves Laplace’s equation
∇
2
φ
= 0 for all
r ∈ R
3
\ r
0
. For example, if
r
0
=
ˆ
k
,
where
ˆ
k is the unit vector in the z direction, then
1
r −
ˆ
k
=
1
√
r
2
+ 1 − 2r cos θ
=
∞
X
`=0
c
`
r
`
P
`
(cos θ).
To find these coefficients, we can employ a little trick. Since
P
`
(1) = 0, at
θ
= 0,
we have
∞
X
`=0
c
`
r
`
=
1
1 − r
=
∞
X
`=0
r
`
.
So all the coefficients must be 1. This is valid for r < 1.
More generally, we have
1
r − r
0

=
1
r
0
∞
X
`=0
r
r
0
`
P
`
(
ˆ
r ·
ˆ
r
0
).
This is called the multiple expansion, and is valid when r < r
0
. Thus
1
r − r
0

=
1
r
0
+
r
r
02
ˆ
r ·
ˆ
r
0
+ ··· .
The first term
1
r
0
is known as the monopole, and the second term is known as
the dipole, in analogy to charges in electromagnetism. The monopole term is
what we get due to a single charge, and the second term is the result of the
interaction of two charges.