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 $z'$ axis

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

$${\begin{aligned}Q&=\lambda ,\\C_{k}&={\frac {\lambda }{k}}\left(\rho '\right)^{k}\cos k\theta ',\\S_{k}&={\frac {\lambda }{k}}\left(\rho '\right)^{k}\sin k\theta '.\end{aligned}}$$

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

$${\begin{aligned}Q&=\lambda ,\\I_{k}&={\frac {\lambda }{k}}{\frac {\cos k\theta '}{\left(\rho '\right)^{k}}},\\J_{k}&={\frac {\lambda }{k}}{\frac {\sin k\theta '}{\left(\rho '\right)^{k}}}.\end{aligned}}$$

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

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

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

$${\begin{aligned}Q&=\int d\theta '\,d\rho '\,\rho '\lambda (\rho ',\theta ')\\I_{k}&={\frac {1}{k}}\int d\theta '\,d\rho '{\frac {\cos k\theta '}{\left(\rho '\right)^{k-1}}}\lambda (\rho ',\theta ')\\J_{k}&={\frac {1}{k}}\int d\theta '\,d\rho '{\frac {\sin k\theta '}{\left(\rho '\right)^{k-1}}}\lambda (\rho ',\theta ')\end{aligned}}$$

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.

See also

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