Cylindrical multipole moments

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

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

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

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

By symmetry, the electric potential of an infinite line charge has no ${\displaystyle z}$-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 ${\displaystyle \rho '}$ 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 ${\displaystyle (\rho '/\rho )<1}$

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

which may be written as

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

where the multipole moments are defined as

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

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

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

which may be written as

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

where the interior multipole moments are defined as

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

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

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

and the moments can be written

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

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

Similarly, the interior cylindrical multipole expansion has the functional form

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

where the moments are defined

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

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

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

If the cylindrical multipoles are exterior, this equation becomes

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

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

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

where ${\displaystyle I_{2k}}$ 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

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

where ${\displaystyle I_{1k}}$ 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.

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