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 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 $\rho ^{\prime }$ 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 $Q\equiv \lambda$, $C_{k}\equiv {\frac {\lambda }{k}}\left(\rho ^{\prime }\right)^{k}\cos k\theta ^{\prime }$, and $S_{k}\equiv {\frac {\lambda }{k}}\left(\rho ^{\prime }\right)^{k}\sin k\theta ^{\prime }$.

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 $Q\equiv \lambda$, $I_{k}\equiv {\frac {\lambda }{k}}{\frac {\cos k\theta ^{\prime }}{\left(\rho ^{\prime }\right)^{k}}}$, and $J_{k}\equiv {\frac {\lambda }{k}}{\frac {\sin k\theta ^{\prime }}{\left(\rho ^{\prime }\right)^{k}}}$.

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

$Q\equiv \int d\theta ^{\prime }\int \rho ^{\prime }d\rho ^{\prime }\lambda (\rho ^{\prime },\theta ^{\prime })$

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

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

Note that the $\lambda (\rho ^{\prime },\theta ^{\prime })$ represents the line charge per unit area in the $\rho -\theta$ plane, and has units of $\mathrm {(charge/length)/length} ^{2}$.

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

$Q\equiv \int d\theta ^{\prime }\int \rho ^{\prime }d\rho ^{\prime }\lambda (\rho ^{\prime },\theta ^{\prime })$

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

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

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

$\lambda (\rho ,\theta )\equiv \int dz\ f(\rho ,\theta ,z)$

The electrostatic energy is given by the integral of the charge multiplied by the potential due to the cylindrical multipoles

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

If the cylindrical multipoles are exterior, this equation becomes

$U\equiv {\frac {-Q_{1}}{2\pi \epsilon }}\int \rho d\rho \ \lambda (\rho ,\theta )\ln \rho$

$\ \ \ \ \ \ \ \ \ \ +\ \left({\frac {1}{2\pi \epsilon }}\right)\sum _{k=1}^{\infty }C_{1k}\int d\theta \int d\rho \left[{\frac {\cos k\theta }{\rho ^{k-1}}}\right]\lambda (\rho ,\theta )$

$\ \ \ \ \ \ \ \ +\ \left({\frac {1}{2\pi \epsilon }}\right)\sum _{k=1}^{\infty }S_{1k}\int d\theta \int d\rho \left[{\frac {\sin k\theta }{\rho ^{k-1}}}\right]\lambda (\rho ,\theta )$

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

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

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

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

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.

See also

• Potential theory
• Multipole moments
• Multipole expansion
• Axial multipole moments
• Spherical multipole moments