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
{\displaystyle \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
′
)
{\displaystyle (\rho ^{\prime },\theta ^{\prime })}
refer to the position of the line charge(s), whereas the unprimed coordinates such as
(
ρ
,
θ
)
{\displaystyle (\rho ,\theta )}
refer to the point at which the potential is being observed. We use cylindrical coordinates throughout, e.g., an arbitrary vector
r
{\displaystyle \mathbf {r} }
has coordinates
(
ρ
,
θ
,
z
)
{\displaystyle (\rho ,\theta ,z)}
where
ρ
{\displaystyle \rho }
is the radius from the
z
{\displaystyle z}
axis,
θ
{\displaystyle \theta }
is the azimuthal angle and
z
{\displaystyle z}
is the normal Cartesian coordinate . By assumption, the line charges are infinitely long and aligned with the
z
{\displaystyle z}
axis.
Cylindrical multipole moments of a line charge
edit
Figure 1: Definitions for cylindrical multipoles; looking down the
z
′
{\displaystyle z'}
axis
The electric potential of a line charge
λ
{\displaystyle \lambda }
located at
(
ρ
′
,
θ
′
)
{\displaystyle (\rho ',\theta ')}
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
′
)
|
{\displaystyle \Phi (\rho ,\theta )={\frac {-\lambda }{2\pi \epsilon }}\ln R={\frac {-\lambda }{4\pi \epsilon }}\ln \left|\rho ^{2}+\left(\rho '\right)^{2}-2\rho \rho '\cos(\theta -\theta ')\right|}
where
R
{\displaystyle R}
is the shortest distance between the line charge and the observation point.
By symmetry, the electric potential of an infinite line charge has no
z
{\displaystyle z}
-dependence. The line charge
λ
{\displaystyle \lambda }
is the charge per unit length in the
z
{\displaystyle z}
-direction, and has units of (charge/length). If the radius
ρ
{\displaystyle \rho }
of the observation point is greater than the radius
ρ
′
{\displaystyle \rho '}
of the line charge, we may factor out
ρ
2
{\displaystyle \rho ^{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
′
)
)
}
{\displaystyle \Phi (\rho ,\theta )={\frac {-\lambda }{4\pi \epsilon }}\left\{2\ln \rho +\ln \left(1-{\frac {\rho ^{\prime }}{\rho }}e^{i\left(\theta -\theta ^{\prime }\right)}\right)\left(1-{\frac {\rho ^{\prime }}{\rho }}e^{-i\left(\theta -\theta ^{\prime }\right)}\right)\right\}}
and expand the logarithms in powers of
(
ρ
′
/
ρ
)
<
1
{\displaystyle (\rho '/\rho )<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
′
]
}
{\displaystyle \Phi (\rho ,\theta )={\frac {-\lambda }{2\pi \epsilon }}\left\{\ln \rho -\sum _{k=1}^{\infty }{\frac {1}{k}}\left({\frac {\rho '}{\rho }}\right)^{k}\left[\cos k\theta \cos k\theta '+\sin k\theta \sin k\theta '\right]\right\}}
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
{\displaystyle \Phi (\rho ,\theta )={\frac {-Q}{2\pi \epsilon }}\ln \rho +{\frac {1}{2\pi \epsilon }}\sum _{k=1}^{\infty }{\frac {C_{k}\cos k\theta +S_{k}\sin k\theta }{\rho ^{k}}}}
where the multipole moments are defined as
Q
=
λ
,
C
k
=
λ
k
(
ρ
′
)
k
cos
k
θ
′
,
S
k
=
λ
k
(
ρ
′
)
k
sin
k
θ
′
.
{\displaystyle {\begin{aligned}Q&=\lambda ,\\C_{k}&={\frac {\lambda }{k}}\left(\rho '\right)^{k}\cos k\theta ',\\S_{k}&={\frac {\lambda }{k}}\left(\rho '\right)^{k}\sin k\theta '.\end{aligned}}}
Conversely, if the radius
ρ
{\displaystyle \rho }
of the observation point is less than the radius
ρ
′
{\displaystyle \rho '}
of the line charge, we may factor out
(
ρ
′
)
2
{\displaystyle \left(\rho '\right)^{2}}
and expand the logarithms in powers of
(
ρ
/
ρ
′
)
<
1
{\displaystyle (\rho /\rho ')<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
′
]
}
{\displaystyle \Phi (\rho ,\theta )={\frac {-\lambda }{2\pi \epsilon }}\left\{\ln \rho '-\sum _{k=1}^{\infty }\left({\frac {1}{k}}\right)\left({\frac {\rho }{\rho '}}\right)^{k}\left[\cos k\theta \cos k\theta '+\sin k\theta \sin k\theta '\right]\right\}}
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
]
{\displaystyle \Phi (\rho ,\theta )={\frac {-Q}{2\pi \epsilon }}\ln \rho '+{\frac {1}{2\pi \epsilon }}\sum _{k=1}^{\infty }\rho ^{k}\left[I_{k}\cos k\theta +J_{k}\sin k\theta \right]}
where the interior multipole moments are defined as
Q
=
λ
,
I
k
=
λ
k
cos
k
θ
′
(
ρ
′
)
k
,
J
k
=
λ
k
sin
k
θ
′
(
ρ
′
)
k
.
{\displaystyle {\begin{aligned}Q&=\lambda ,\\I_{k}&={\frac {\lambda }{k}}{\frac {\cos k\theta '}{\left(\rho '\right)^{k}}},\\J_{k}&={\frac {\lambda }{k}}{\frac {\sin k\theta '}{\left(\rho '\right)^{k}}}.\end{aligned}}}
General cylindrical multipole moments
edit
The generalization to an arbitrary distribution of line charges
λ
(
ρ
′
,
θ
′
)
{\displaystyle \lambda (\rho ',\theta ')}
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
{\displaystyle \Phi (\mathbf {r} )={\frac {-Q}{2\pi \epsilon }}\ln \rho +{\frac {1}{2\pi \epsilon }}\sum _{k=1}^{\infty }{\frac {C_{k}\cos k\theta +S_{k}\sin k\theta }{\rho ^{k}}}}
and the moments can be written
Q
=
∫
d
θ
′
d
ρ
′
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
′
{\displaystyle {\begin{aligned}Q&=\int d\theta '\,d\rho '\,\rho '\lambda (\rho ',\theta ')\\C_{k}&={\frac {1}{k}}\int d\theta '\,d\rho '\left(\rho '\right)^{k+1}\lambda (\rho ',\theta ')\cos k\theta '\\S_{k}&={\frac {1}{k}}\int d\theta '\,d\rho '\left(\rho '\right)^{k+1}\lambda (\rho ',\theta ')\sin k\theta '\end{aligned}}}
Note that the
λ
(
ρ
′
,
θ
′
)
{\displaystyle \lambda (\rho ',\theta ')}
represents the line charge per unit area in the
(
ρ
−
θ
)
{\displaystyle (\rho -\theta )}
plane.
Interior cylindrical multipole moments
edit
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
]
{\displaystyle \Phi (\rho ,\theta )={\frac {-Q}{2\pi \epsilon }}\ln \rho '+{\frac {1}{2\pi \epsilon }}\sum _{k=1}^{\infty }\rho ^{k}\left[I_{k}\cos k\theta +J_{k}\sin k\theta \right]}
where the moments are defined
Q
=
∫
d
θ
′
d
ρ
′
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
′
)
{\displaystyle {\begin{aligned}Q&=\int d\theta '\,d\rho '\,\rho '\lambda (\rho ',\theta ')\\I_{k}&={\frac {1}{k}}\int d\theta '\,d\rho '{\frac {\cos k\theta '}{\left(\rho '\right)^{k-1}}}\lambda (\rho ',\theta ')\\J_{k}&={\frac {1}{k}}\int d\theta '\,d\rho '{\frac {\sin k\theta '}{\left(\rho '\right)^{k-1}}}\lambda (\rho ',\theta ')\end{aligned}}}
Interaction energies of cylindrical multipoles
edit
A simple formula for the interaction energy of cylindrical multipoles (charge density 1) with a second charge density can be derived. Let
f
(
r
′
)
{\displaystyle f(\mathbf {r} ^{\prime })}
be the second charge density, and define
λ
(
ρ
,
θ
)
{\displaystyle \lambda (\rho ,\theta )}
as its integral over z
λ
(
ρ
,
θ
)
=
∫
d
z
f
(
ρ
,
θ
,
z
)
{\displaystyle \lambda (\rho ,\theta )=\int dz\,f(\rho ,\theta ,z)}
The electrostatic energy is given by the integral of the charge multiplied by the potential due to the cylindrical multipoles
U
=
∫
d
θ
d
ρ
ρ
λ
(
ρ
,
θ
)
Φ
(
ρ
,
θ
)
{\displaystyle U=\int d\theta \,d\rho \,\rho \,\lambda (\rho ,\theta )\Phi (\rho ,\theta )}
If the cylindrical multipoles are exterior , this equation becomes
U
=
−
Q
1
2
π
ϵ
∫
d
ρ
ρ
λ
(
ρ
,
θ
)
ln
ρ
+
1
2
π
ϵ
∑
k
=
1
∞
∫
d
θ
d
ρ
[
C
1
k
cos
k
θ
ρ
k
−
1
+
S
1
k
sin
k
θ
ρ
k
−
1
]
λ
(
ρ
,
θ
)
{\displaystyle U={\frac {-Q_{1}}{2\pi \epsilon }}\int d\rho \,\rho \,\lambda (\rho ,\theta )\ln \rho +{\frac {1}{2\pi \epsilon }}\sum _{k=1}^{\infty }\int d\theta \,d\rho \left[C_{1k}{\frac {\cos k\theta }{\rho ^{k-1}}}+S_{1k}{\frac {\sin k\theta }{\rho ^{k-1}}}\right]\lambda (\rho ,\theta )}
where
Q
1
{\displaystyle Q_{1}}
,
C
1
k
{\displaystyle C_{1k}}
and
S
1
k
{\displaystyle S_{1k}}
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
∫
d
ρ
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
)
{\displaystyle U={\frac {-Q_{1}}{2\pi \epsilon }}\int d\rho \,\rho \,\lambda (\rho ,\theta )\ln \rho +{\frac {1}{2\pi \epsilon }}\sum _{k=1}^{\infty }k\left(C_{1k}I_{2k}+S_{1k}J_{2k}\right)}
where
I
2
k
{\displaystyle I_{2k}}
and
J
2
k
{\displaystyle J_{2k}}
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
∫
d
ρ
ρ
λ
(
ρ
,
θ
)
+
1
2
π
ϵ
∑
k
=
1
∞
k
(
C
2
k
I
1
k
+
S
2
k
J
1
k
)
{\displaystyle U={\frac {-Q_{1}\ln \rho '}{2\pi \epsilon }}\int d\rho \,\rho \,\lambda (\rho ,\theta )+{\frac {1}{2\pi \epsilon }}\sum _{k=1}^{\infty }k\left(C_{2k}I_{1k}+S_{2k}J_{1k}\right)}
where
I
1
k
{\displaystyle I_{1k}}
and
J
1
k
{\displaystyle J_{1k}}
are the interior cylindrical multipole moments of charge distribution 1, and
C
2
k
{\displaystyle C_{2k}}
and
S
2
k
{\displaystyle S_{2k}}
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.
See also
edit