Cylindrical Multipole Moments

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 $(\rho^, \theta^)$ refer to the position of the line charge(s), whereas the unprimed coordinates such as $(\rho, \theta)$ refer to the point at which the potential is being observed. We use cylindrical coordinates throughout, e.g., an arbitrary vector $\mathbf$ has coordinates $( \rho, \theta, z)$
where $\rho$ is the radius from the $z$ axis, $\theta$ 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.

Cylindrical multipole moments of a line charge

The electric potential of a line charge $\lambda$ located at $(\rho', \theta')$ is given by
$\Phi(\rho, \theta) = \frac \ln R = \frac \ln \left, \rho^ + \left( \rho' \right)^ - 2\rho\rho'\cos (\theta - \theta' ) \$
where $R$ is the shortest distance between the line charge and the observation point.

By symmetry, the electric potential of an infinite line charge has no $z$-dependence. The line charge $\lambda$ is the charge per unit length in the $z$-direction, and has units of (charge/length). If the radius $\rho$ of the observation point is greater than the radius $\rho'$ of the line charge, we may factor out $\rho^$
$\Phi(\rho, \theta) = \frac \left\$
and expand the logarithms in powers of $(\rho'/\rho)<1$
$\Phi(\rho, \theta) = \frac \left\$
which may be written as
$\Phi(\rho, \theta) = \frac \ln \rho + \frac \sum_^ \frac$
where the multipole moments are defined as
$\begin Q &= \lambda ,\\ C_k &= \frac \left( \rho' \right)^k \cos k\theta' , \\ S_ &= \frac \left( \rho' \right)^k \sin k\theta'. \end$

Conversely, if the radius $\rho$ of the observation point is less than the radius $\rho'$ of the line charge, we may factor out $\left( \rho' \right)^$ and expand the logarithms in powers of $(\rho/\rho')<1$
$\Phi(\rho, \theta) = \frac \left\$
which may be written as
\Phi(\rho, \theta) =
\frac \ln \rho' +
\frac \sum_^
\rho^ \left I_ \cos k\theta + J_ \sin k\theta \right
where the interior multipole moments are defined as
$\begin Q &= \lambda \ln \rho', \\ I_k &= \frac \frac, \\ J_k &= \frac \frac.\end$

General cylindrical multipole moments

The generalization to an arbitrary distribution of line charges $\lambda(\rho', \theta')$ is straightforward. The functional form is the same
$\Phi(\mathbf) = \frac \ln \rho + \frac \sum_^ \frac$
and the moments can be written
$\begin Q &= \int d\theta' \, d\rho' \, \rho' \lambda(\rho', \theta') \\ C_k &= \frac \int d\theta' \, d\rho' \left(\rho'\right)^ \lambda(\rho', \theta') \cos k\theta' \\ S_k &= \frac \int d\theta' \, d\rho' \left(\rho'\right)^ \lambda(\rho', \theta') \sin k\theta' \end$
Note that the $\lambda(\rho', \theta')$ represents the line charge per unit area in the $(\rho-\theta)$ plane.

Interior cylindrical multipole moments

Similarly, the interior cylindrical multipole expansion has the functional form

\Phi(\rho, \theta) =
\frac \ln \rho' +
\frac \sum_^
\rho^ \left I_ \cos k\theta + J_ \sin k\theta \right

where the moments are defined
$\begin Q &= \int d\theta' \, d\rho' \, \rho' \lambda(\rho', \theta') \\ I_ &= \frac \int d\theta' \, d\rho' \frac \lambda(\rho', \theta') \\ J_ &= \frac \int d\theta' \, d\rho' \frac \lambda(\rho', \theta') \end$

Interaction energies of cylindrical multipoles

A simple formula for the interaction energy of cylindrical multipoles (charge density 1) with a second charge density can be derived. Let $f(\mathbf^)$ be the second charge density, and define $\lambda(\rho, \theta)$ as its integral over z
$\lambda(\rho, \theta) = \int dz \, f(\rho, \theta, z)$

The electrostatic energy is given by the integral of the charge multiplied by the potential due to the cylindrical multipoles
$U = \int d\theta \, d\rho \, \rho \, \lambda(\rho, \theta) \Phi(\rho, \theta)$

If the cylindrical multipoles are exterior, this equation becomes
U = \frac \int d\rho \, \rho \, \lambda(\rho, \theta) \ln \rho
+ \frac \sum_^ \int d\theta \, d\rho
\left C_ \frac + S_ \frac\right\lambda(\rho, \theta)
where $Q_$, $C_$ and $S_$ are the cylindrical multipole moments of charge distribution 1. This energy formula can be reduced to a remarkably simple form
$U = \frac \int d\rho \, \rho \, \lambda(\rho, \theta) \ln \rho + \frac \sum_^ k \left( C_ I_ + S_ J_ \right)$
where $I_$ and $J_$ are the interior cylindrical multipoles of the second charge density.

The analogous formula holds if charge density 1 is composed of interior cylindrical multipoles
$U = \frac \int d\rho \, \rho \, \lambda(\rho, \theta) + \frac \sum_^ k \left( C_ I_ + S_ J_ \right)$
where $I_$ and $J_$ are the interior cylindrical multipole moments of charge distribution 1, and $C_$ and $S_$ 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

* Axial multipole moments
* Potential theory
* Quantum cylindrical quadrupole
* Multipole expansion
* Spherical multipole moments

{{DEFAULTSORT:Cylindrical Multipole Moments
Electromagnetism
Potential theory
Moment (physics)