Cylindrical multipole moments

Summary

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 . 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 refer to the position of the line charge(s), whereas the unprimed coordinates such as refer to the point at which the potential is being observed. We use cylindrical coordinates throughout, e.g., an arbitrary vector has coordinates where is the radius from the axis, is the azimuthal angle and is the normal Cartesian coordinate. By assumption, the line charges are infinitely long and aligned with the axis.

Cylindrical multipole moments of a line charge

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Figure 1: Definitions for cylindrical multipoles; looking down the   axis

The electric potential of a line charge   located at   is given by   where   is the shortest distance between the line charge and the observation point.

By symmetry, the electric potential of an infinite line charge has no  -dependence. The line charge   is the charge per unit length in the  -direction, and has units of (charge/length). If the radius   of the observation point is greater than the radius   of the line charge, we may factor out     and expand the logarithms in powers of     which may be written as   where the multipole moments are defined as  

Conversely, if the radius   of the observation point is less than the radius   of the line charge, we may factor out   and expand the logarithms in powers of     which may be written as   where the interior multipole moments are defined as  

General cylindrical multipole moments

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The generalization to an arbitrary distribution of line charges   is straightforward. The functional form is the same   and the moments can be written   Note that the   represents the line charge per unit area in the   plane.

Interior cylindrical multipole moments

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Similarly, the interior cylindrical multipole expansion has the functional form   where the moments are defined  

Interaction energies of cylindrical multipoles

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A simple formula for the interaction energy of cylindrical multipoles (charge density 1) with a second charge density can be derived. Let   be the second charge density, and define   as its integral over z  

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

If the cylindrical multipoles are exterior, this equation becomes   where  ,   and   are the cylindrical multipole moments of charge distribution 1. This energy formula can be reduced to a remarkably simple form   where   and   are the interior cylindrical multipoles of the second charge density.

The analogous formula holds if charge density 1 is composed of interior cylindrical multipoles   where   and   are the interior cylindrical multipole moments of charge distribution 1, and   and   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

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