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 ${\displaystyle \ln \ R}$. 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 ${\displaystyle (\rho ^{\prime },\theta ^{\prime })}$ refer to the position of the line charge(s), whereas the unprimed coordinates such as ${\displaystyle (\rho ,\theta )}$ refer to the point at which the potential is being observed. We use cylindrical coordinates throughout, e.g., an arbitrary vector ${\displaystyle \mathbf {r} }$ has coordinates ${\displaystyle (\rho ,\theta ,z)}$ where ${\displaystyle \rho }$ is the radius from the ${\displaystyle z}$ axis, ${\displaystyle \theta }$ is the azimuthal angle and ${\displaystyle z}$ is the normal Cartesian coordinate. By assumption, the line charges are infinitely long and aligned with the ${\displaystyle z}$ axis.

Cylindrical multipole moments of a line chargeEdit

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

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

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

where ${\displaystyle R}$ is the shortest distance between the line charge and the observation point.

By symmetry, the electric potential of an infinite linecharge has no ${\displaystyle z}$-dependence. The line charge ${\displaystyle \lambda }$ is the charge per unit length in the ${\displaystyle z}$-direction, and has units of (charge/length). If the radius ${\displaystyle \rho }$ of the observation point is greater than the radius ${\displaystyle \rho ^{\prime }}$ of the line charge, we may factor out ${\displaystyle \rho ^{2}}$

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

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

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

which may be written as

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

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

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

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

which may be written as

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

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

General cylindrical multipole momentsEdit

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

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

and the moments can be written

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

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

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

Note that the ${\displaystyle \lambda (\rho ^{\prime },\theta ^{\prime })}$ represents the line charge per unit area in the ${\displaystyle (\rho -\theta )}$ plane.

Interior cylindrical multipole momentsEdit

Similarly, the interior cylindrical multipole expansion has the functional form

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

where the moments are defined

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

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

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

Interaction energies of cylindrical multipolesEdit

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

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

If the cylindrical multipoles are exterior, this equation becomes

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

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

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

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

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

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

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

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

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

This article uses material from the Wikipedia article  Metasyntactic variable, which is released under the  Creative Commons Attribution-ShareAlike 3.0 Unported License.