Cylindrical multipole moments are the coefficients in a series expansion of a potential that varies logarithmically with the distance to a source, i.e., as
ln
R
. Such potentials arise in the electric potential of long line charges, and the analogous sources for the magnetic potential and gravitational potential.
For clarity, we illustrate the expansion for a single line charge, then generalize to an arbitrary distribution of line charges. Through this article, the primed coordinates such as
(
ρ
′
,
θ
′
)
refer to the position of the line charge(s), whereas the unprimed coordinates such as
(
ρ
,
θ
)
refer to the point at which the potential is being observed. We use cylindrical coordinates throughout, e.g., an arbitrary vector
r
has coordinates
(
ρ
,
θ
,
z
)
where
ρ
is the radius from the
z
axis,
θ
is the azimuthal angle and
z
is the normal Cartesian coordinate. By assumption, the line charges are infinitely long and aligned with the
z
axis.
The electric potential of a line charge
λ
located at
(
ρ
′
,
θ
′
)
is given by
Φ
(
ρ
,
θ
)
=
−
λ
2
π
ϵ
ln
R
=
−
λ
4
π
ϵ
ln
|
ρ
2
+
(
ρ
′
)
2
−
2
ρ
ρ
′
cos
(
θ
−
θ
′
)
|
where
R
is the shortest distance between the line charge and the observation point.
By symmetry, the electric potential of an infinite linecharge has no
z
-dependence. The line charge
λ
is the charge per unit length in the
z
-direction, and has units of (charge/length). If the radius
ρ
of the observation point is greater than the radius
ρ
′
of the line charge, we may factor out
ρ
2
Φ
(
ρ
,
θ
)
=
−
λ
4
π
ϵ
{
2
ln
ρ
+
ln
(
1
−
ρ
′
ρ
e
i
(
θ
−
θ
′
)
)
(
1
−
ρ
′
ρ
e
−
i
(
θ
−
θ
′
)
)
}
and expand the logarithms in powers of
(
ρ
′
/
ρ
)
<
1
Φ
(
ρ
,
θ
)
=
−
λ
2
π
ϵ
{
ln
ρ
−
∑
k
=
1
∞
(
1
k
)
(
ρ
′
ρ
)
k
[
cos
k
θ
cos
k
θ
′
+
sin
k
θ
sin
k
θ
′
]
}
which may be written as
Φ
(
ρ
,
θ
)
=
−
Q
2
π
ϵ
ln
ρ
+
(
1
2
π
ϵ
)
∑
k
=
1
∞
C
k
cos
k
θ
+
S
k
sin
k
θ
ρ
k
where the multipole moments are defined as
Q
=
λ
,
C
k
=
λ
k
(
ρ
′
)
k
cos
k
θ
′
,
and
S
k
=
λ
k
(
ρ
′
)
k
sin
k
θ
′
.
Conversely, if the radius
ρ
of the observation point is less than the radius
ρ
′
of the line charge, we may factor out
(
ρ
′
)
2
and expand the logarithms in powers of
(
ρ
/
ρ
′
)
<
1
Φ
(
ρ
,
θ
)
=
−
λ
2
π
ϵ
{
ln
ρ
′
−
∑
k
=
1
∞
(
1
k
)
(
ρ
ρ
′
)
k
[
cos
k
θ
cos
k
θ
′
+
sin
k
θ
sin
k
θ
′
]
}
which may be written as
Φ
(
ρ
,
θ
)
=
−
Q
2
π
ϵ
ln
ρ
′
+
(
1
2
π
ϵ
)
∑
k
=
1
∞
ρ
k
[
I
k
cos
k
θ
+
J
k
sin
k
θ
]
where the interior multipole moments are defined as
Q
=
λ
,
I
k
=
λ
k
cos
k
θ
′
(
ρ
′
)
k
,
and
J
k
=
λ
k
sin
k
θ
′
(
ρ
′
)
k
.
The generalization to an arbitrary distribution of line charges
λ
(
ρ
′
,
θ
′
)
is straightforward. The functional form is the same
Φ
(
r
)
=
−
Q
2
π
ϵ
ln
ρ
+
(
1
2
π
ϵ
)
∑
k
=
1
∞
C
k
cos
k
θ
+
S
k
sin
k
θ
ρ
k
and the moments can be written
Q
=
∫
d
θ
′
∫
ρ
′
d
ρ
′
λ
(
ρ
′
,
θ
′
)
C
k
=
(
1
k
)
∫
d
θ
′
∫
d
ρ
′
(
ρ
′
)
k
+
1
λ
(
ρ
′
,
θ
′
)
cos
k
θ
′
S
k
=
(
1
k
)
∫
d
θ
′
∫
d
ρ
′
(
ρ
′
)
k
+
1
λ
(
ρ
′
,
θ
′
)
sin
k
θ
′
Note that the
λ
(
ρ
′
,
θ
′
)
represents the line charge per unit area in the
(
ρ
−
θ
)
plane.
Similarly, the interior cylindrical multipole expansion has the functional form
Φ
(
ρ
,
θ
)
=
−
Q
2
π
ϵ
ln
ρ
′
+
(
1
2
π
ϵ
)
∑
k
=
1
∞
ρ
k
[
I
k
cos
k
θ
+
J
k
sin
k
θ
]
where the moments are defined
Q
=
∫
d
θ
′
∫
ρ
′
d
ρ
′
λ
(
ρ
′
,
θ
′
)
I
k
=
(
1
k
)
∫
d
θ
′
∫
d
ρ
′
[
cos
k
θ
′
(
ρ
′
)
k
−
1
]
λ
(
ρ
′
,
θ
′
)
J
k
=
(
1
k
)
∫
d
θ
′
∫
d
ρ
′
[
sin
k
θ
′
(
ρ
′
)
k
−
1
]
λ
(
ρ
′
,
θ
′
)
A simple formula for the interaction energy of cylindrical multipoles (charge density 1) with a second charge density can be derived. Let
f
(
r
′
)
be the second charge density, and define
λ
(
ρ
,
θ
)
as its integral over z
λ
(
ρ
,
θ
)
=
∫
d
z
f
(
ρ
,
θ
,
z
)
The electrostatic energy is given by the integral of the charge multiplied by the potential due to the cylindrical multipoles
U
=
∫
d
θ
∫
ρ
d
ρ
λ
(
ρ
,
θ
)
Φ
(
ρ
,
θ
)
If the cylindrical multipoles are exterior, this equation becomes
U
=
−
Q
1
2
π
ϵ
∫
ρ
d
ρ
λ
(
ρ
,
θ
)
ln
ρ
+
(
1
2
π
ϵ
)
∑
k
=
1
∞
C
1
k
∫
d
θ
∫
d
ρ
[
cos
k
θ
ρ
k
−
1
]
λ
(
ρ
,
θ
)
+
(
1
2
π
ϵ
)
∑
k
=
1
∞
S
1
k
∫
d
θ
∫
d
ρ
[
sin
k
θ
ρ
k
−
1
]
λ
(
ρ
,
θ
)
where
Q
1
,
C
1
k
and
S
1
k
are the cylindrical multipole moments of charge distribution 1. This energy formula can be reduced to a remarkably simple form
U
=
−
Q
1
2
π
ϵ
∫
ρ
d
ρ
λ
(
ρ
,
θ
)
ln
ρ
+
(
1
2
π
ϵ
)
∑
k
=
1
∞
k
(
C
1
k
I
2
k
+
S
1
k
J
2
k
)
where
I
2
k
and
J
2
k
are the interior cylindrical multipoles of the second charge density.
The analogous formula holds if charge density 1 is composed of interior cylindrical multipoles
U
=
−
Q
1
ln
ρ
′
2
π
ϵ
∫
ρ
d
ρ
λ
(
ρ
,
θ
)
+
(
1
2
π
ϵ
)
∑
k
=
1
∞
k
(
C
2
k
I
1
k
+
S
2
k
J
1
k
)
where
I
1
k
and
J
1
k
are the interior cylindrical multipole moments of charge distribution 1, and
C
2
k
and
S
2
k
are the exterior cylindrical multipoles of the second charge density.
As an example, these formulae could be used to determine the interaction energy of a small protein in the electrostatic field of a double-stranded DNA molecule; the latter is relatively straight and bears a constant linear charge density due to the phosphate groups of its backbone.
