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 $(\rho^{\prime}, \theta^{\prime})$ 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 $\mathbf{r}$ 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

Figure 1: Definitions for cylindrical multipoles; looking down the $z^{\prime}$ axis

The electric potential of a line charge $\lambda$ located at $(\rho^{\prime}, \theta^{\prime})$ is given by

$\Phi(\rho, \theta) = \frac{-\lambda}{2\pi\epsilon} \ln R = \frac{-\lambda}{4\pi\epsilon} \ln \left| \rho^{2} + \left( \rho^{\prime} \right)^{2} - 2\rho\rho^{\prime}\cos (\theta-\theta^{\prime} ) \right|$

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 $\lambda$ 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 $\rho^{\prime}$ of the line charge, we may factor out $\rho^{2}$

$\Phi(\rho, \theta) = \frac{-\lambda}{4\pi\epsilon} \left\{ 2\ln \rho + \ln \left( 1 - \frac{\rho^{\prime}}{\rho} e^{i \left(\theta - \theta^{\prime}\right)} \right) \left( 1 - \frac{\rho^{\prime}}{\rho} e^{-i \left(\theta - \theta^{\prime} \right)} \right) \right\}$

and expand the logarithms in powers of $(\rho^{\prime}/\rho)<1$

$\Phi(\rho, \theta) = \frac{-\lambda}{2\pi\epsilon} \left\{\ln \rho - \sum_{k=1}^{\infty} \left( \frac{1}{k} \right) \left( \frac{\rho^{\prime}}{\rho} \right)^{k} \left[ \cos k\theta \cos k\theta^{\prime} + \sin k\theta \sin k\theta^{\prime} \right] \right\}$

which may be written as

$\Phi(\rho, \theta) = \frac{-Q}{2\pi\epsilon} \ln \rho + \left( \frac{1}{2\pi\epsilon} \right) \sum_{k=1}^{\infty} \frac{C_{k} \cos k\theta + S_{k} \sin k\theta}{\rho^{k}}$

where the multipole moments are defined as
$Q = \lambda ,$
$C_{k} = \frac{\lambda}{k} \left( \rho^{\prime} \right)^{k} \cos k\theta^{\prime} ,$
and
$S_{k} = \frac{\lambda}{k} \left( \rho^{\prime} \right)^{k} \sin k\theta^{\prime} .$

Conversely, if the radius $\rho$ of the observation point is less than the radius $\rho^{\prime}$ of the line charge, we may factor out $\left( \rho^{\prime} \right)^{2}$ and expand the logarithms in powers of $(\rho/\rho^{\prime})<1$

$\Phi(\rho, \theta) = \frac{-\lambda}{2\pi\epsilon} \left\{\ln \rho^{\prime} - \sum_{k=1}^{\infty} \left( \frac{1}{k} \right) \left( \frac{\rho}{\rho^{\prime}} \right)^{k} \left[ \cos k\theta \cos k\theta^{\prime} + \sin k\theta \sin k\theta^{\prime} \right] \right\}$

which may be written as

$\Phi(\rho, \theta) = \frac{-Q}{2\pi\epsilon} \ln \rho^{\prime} + \left( \frac{1}{2\pi\epsilon} \right) \sum_{k=1}^{\infty} \rho^{k} \left[ I_{k} \cos k\theta + J_{k} \sin k\theta \right]$

where the interior multipole moments are defined as
$Q = \lambda ,$
$I_{k} = \frac{\lambda}{k} \frac{\cos k\theta^{\prime}}{\left( \rho^{\prime} \right)^{k}},$
and
$J_{k} = \frac{\lambda}{k} \frac{\sin k\theta^{\prime}}{\left( \rho^{\prime} \right)^{k}}.$

General cylindrical multipole moments

The generalization to an arbitrary distribution of line charges $\lambda(\rho^{\prime}, \theta^{\prime})$ is straightforward. The functional form is the same

$\Phi(\mathbf{r}) = \frac{-Q}{2\pi\epsilon} \ln \rho + \left( \frac{1}{2\pi\epsilon} \right) \sum_{k=1}^{\infty} \frac{C_{k} \cos k\theta + S_{k} \sin k\theta}{\rho^{k}}$

and the moments can be written

$Q = \int d\theta^{\prime} \int \rho^{\prime} d\rho^{\prime} \lambda(\rho^{\prime}, \theta^{\prime})$
$C_{k} = \left( \frac{1}{k} \right) \int d\theta^{\prime} \int d\rho^{\prime} \left(\rho^{\prime}\right)^{k+1} \lambda(\rho^{\prime}, \theta^{\prime}) \cos k\theta^{\prime}$
$S_{k} = \left( \frac{1}{k} \right) \int d\theta^{\prime} \int d\rho^{\prime} \left(\rho^{\prime}\right)^{k+1} \lambda(\rho^{\prime}, \theta^{\prime}) \sin k\theta^{\prime}$

Note that the $\lambda(\rho^{\prime}, \theta^{\prime})$ 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

$\Phi(\rho, \theta) = \frac{-Q}{2\pi\epsilon} \ln \rho^{\prime} + \left( \frac{1}{2\pi\epsilon} \right) \sum_{k=1}^{\infty} \rho^{k} \left[ I_{k} \cos k\theta + J_{k} \sin k\theta \right]$

where the moments are defined

$Q = \int d\theta^{\prime} \int \rho^{\prime} d\rho^{\prime} \lambda(\rho^{\prime}, \theta^{\prime})$
$I_{k} = \left( \frac{1}{k} \right) \int d\theta^{\prime} \int d\rho^{\prime} \left[ \frac{\cos k\theta^{\prime}}{\left(\rho^{\prime}\right)^{k-1}} \right] \lambda(\rho^{\prime}, \theta^{\prime})$
$J_{k} = \left( \frac{1}{k} \right) \int d\theta^{\prime} \int d\rho^{\prime} \left[ \frac{\sin k\theta^{\prime}}{\left(\rho^{\prime}\right)^{k-1}} \right] \lambda(\rho^{\prime}, \theta^{\prime})$

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 $f(\mathbf{r}^{\prime})$ 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

$U = \int d\theta \int \rho d\rho \ \lambda(\rho, \theta) \Phi(\rho, \theta)$

If the cylindrical multipoles are exterior, this equation becomes

$U = \frac{-Q_{1}}{2\pi\epsilon} \int \rho d\rho \ \lambda(\rho, \theta) \ln \rho$
$\ \ \ \ \ \ \ \ \ \ + \ \left( \frac{1}{2\pi\epsilon} \right) \sum_{k=1}^{\infty} C_{1k} \int d\theta \int d\rho \left[ \frac{\cos k\theta}{\rho^{k-1}} \right] \lambda(\rho, \theta)$
$\ \ \ \ \ \ \ \ + \ \left( \frac{1}{2\pi\epsilon} \right) \sum_{k=1}^{\infty} S_{1k} \int d\theta \int d\rho \left[ \frac{\sin k\theta}{\rho^{k-1}} \right] \lambda(\rho, \theta)$

where $Q_{1}$, $C_{1k}$ and $S_{1k}$ are the cylindrical multipole moments of charge distribution 1. This energy formula can be reduced to a remarkably simple form

$U = \frac{-Q_{1}}{2\pi\epsilon} \int \rho d\rho \ \lambda(\rho, \theta) \ln \rho + \left( \frac{1}{2\pi\epsilon} \right) \sum_{k=1}^{\infty} k \left( C_{1k} I_{2k} + S_{1k} J_{2k} \right)$

where $I_{2k}$ and $J_{2k}$ are the interior cylindrical multipoles of the second charge density.

The analogous formula holds if charge density 1 is composed of interior cylindrical multipoles

$U = \frac{-Q_{1}\ln \rho^{\prime}}{2\pi\epsilon} \int \rho d\rho \ \lambda(\rho, \theta) + \left( \frac{1}{2\pi\epsilon} \right) \sum_{k=1}^{\infty} k \left( C_{2k} I_{1k} + S_{2k} J_{1k} \right)$

where $I_{1k}$ and $J_{1k}$ are the interior cylindrical multipole moments of charge distribution 1, and $C_{2k}$ and $S_{2k}$ 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.