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


The electric potential of a line charge $$\lambda$$ located at $$(\rho', \theta')$$ is given by $$ \Phi(\rho, \theta) = \frac{-\lambda}{2\pi\epsilon} \ln R = \frac{-\lambda}{4\pi\epsilon} \ln \left| \rho^{2} + \left( \rho' \right)^{2} - 2\rho\rho'\cos (\theta - \theta' ) \right| $$ where $$R$$ 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 $$\rho^{2}$$ $$ \Phi(\rho, \theta) = \frac{-\lambda}{4\pi\epsilon} \left\{ 2\ln \rho + \ln \left( 1 - \frac{\rho^{\prime}}{\rho} e^{i \left(\theta - \theta^{\prime}\right)} \right) \left( 1 - \frac{\rho^{\prime}}{\rho} e^{-i \left(\theta - \theta^{\prime} \right)} \right) \right\} $$ and expand the logarithms in powers of $$(\rho'/\rho)<1$$ $$\Phi(\rho, \theta) = \frac{-\lambda}{2\pi\epsilon} \left\{\ln \rho - \sum_{k=1}^{\infty} \frac{1}{k} \left( \frac{\rho'}{\rho} \right)^k \left[ \cos k\theta \cos k\theta' + \sin k\theta \sin k\theta' \right] \right\} $$ which may be written as $$\Phi(\rho, \theta) = \frac{-Q}{2\pi\epsilon} \ln \rho + \frac{1}{2\pi\epsilon} \sum_{k=1}^{\infty} \frac{C_{k} \cos k\theta + S_{k} \sin k\theta}{\rho^{k}} $$ where the multipole moments are defined as $$\begin{align} Q &= \lambda ,\\ C_k &= \frac{\lambda}{k} \left( \rho' \right)^k \cos k\theta', \\ S_{k} &= \frac{\lambda}{k} \left( \rho' \right)^k \sin k\theta'. \end{align}$$

Conversely, if the radius $$\rho$$ of the observation point is less than the radius $$\rho'$$ of the line charge, we may factor out $$\left( \rho' \right)^{2}$$ and expand the logarithms in powers of $$(\rho/\rho')<1$$ $$\Phi(\rho, \theta) = \frac{-\lambda}{2\pi\epsilon} \left\{\ln \rho' - \sum_{k=1}^{\infty} \left( \frac{1}{k} \right) \left( \frac{\rho}{\rho'} \right)^k \left[ \cos k\theta \cos k\theta' + \sin k\theta \sin k\theta' \right] \right\}$$ which may be written as $$\Phi(\rho, \theta) = \frac{-Q}{2\pi\epsilon} \ln \rho' + \frac{1}{2\pi\epsilon} \sum_{k=1}^{\infty} \rho^{k} \left[ I_{k} \cos k\theta + J_{k} \sin k\theta \right] $$ where the interior multipole moments are defined as $$\begin{align} Q &= \lambda, \\ I_k &= \frac{\lambda}{k} \frac{\cos k\theta'}{\left( \rho' \right)^k}, \\ J_k &= \frac{\lambda}{k} \frac{\sin k\theta'}{\left( \rho' \right)^k}.\end{align}$$

General cylindrical multipole moments
The generalization to an arbitrary distribution of line charges $$\lambda(\rho', \theta')$$ is straightforward. The functional form is the same $$\Phi(\mathbf{r}) = \frac{-Q}{2\pi\epsilon} \ln \rho + \frac{1}{2\pi\epsilon} \sum_{k=1}^{\infty} \frac{C_{k} \cos k\theta + S_{k} \sin k\theta}{\rho^k}$$ and the moments can be written $$\begin{align} Q &= \int d\theta' \, d\rho' \, \rho' \lambda(\rho', \theta') \\ C_k &= \frac{1}{k} \int d\theta' \, d\rho' \left(\rho'\right)^{k+1} \lambda(\rho', \theta') \cos k\theta' \\ S_k &= \frac{1}{k} \int d\theta' \, d\rho' \left(\rho'\right)^{k+1} \lambda(\rho', \theta') \sin k\theta' \end{align}$$ 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 $$ \Phi(\rho, \theta) = \frac{-Q}{2\pi\epsilon} \ln \rho' + \frac{1}{2\pi\epsilon} \sum_{k=1}^{\infty} \rho^{k} \left[ I_{k} \cos k\theta + J_{k} \sin k\theta \right] $$ where the moments are defined $$\begin{align} Q &= \int d\theta' \, d\rho' \, \rho' \lambda(\rho', \theta') \\ I_{k} &= \frac{1}{k} \int d\theta' \, d\rho' \frac{\cos k\theta'}{\left(\rho'\right)^{k-1}} \lambda(\rho', \theta') \\ J_{k} &= \frac{1}{k} \int d\theta' \, d\rho' \frac{\sin k\theta'}{\left(\rho'\right)^{k-1}} \lambda(\rho', \theta') \end{align}$$

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

If the cylindrical multipoles are exterior, this equation becomes $$U = \frac{-Q_1}{2\pi\epsilon} \int d\rho \, \rho \, \lambda(\rho, \theta) \ln \rho + \frac{1}{2\pi\epsilon} \sum_{k=1}^{\infty} \int d\theta \, d\rho \left[ C_{1k} \frac{\cos k\theta}{\rho^{k-1}} + S_{1k} \frac{\sin k\theta}{\rho^{k-1}}\right] \lambda(\rho, \theta)$$ where $$Q_{1}$$, $$C_{1k}$$ and $$S_{1k}$$ are the cylindrical multipole moments of charge distribution 1. This energy formula can be reduced to a remarkably simple form $$U = \frac{-Q_{1}}{2\pi\epsilon} \int d\rho \, \rho \, \lambda(\rho, \theta) \ln \rho + \frac{1}{2\pi\epsilon} \sum_{k=1}^{\infty} k \left( C_{1k} I_{2k} + S_{1k} J_{2k} \right)$$ where $$I_{2k}$$ 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 $$ U = \frac{-Q_1\ln \rho'}{2\pi\epsilon} \int d\rho \, \rho \, \lambda(\rho, \theta) + \frac{1}{2\pi\epsilon} \sum_{k=1}^{\infty} k \left( C_{2k} I_{1k} + S_{2k} J_{1k} \right)$$ where $$I_{1k}$$ 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.