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

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

By assumption, the line charges are infinitely long and aligned with the

The electric potential of a line charge

is the shortest distance between the line charge and the observation point.

By symmetry, the electric potential of an infinite line charge has no

is the charge per unit length in the

of the observation point is greater than the radius

of the line charge, we may factor out

and expand the logarithms in powers of

where the multipole moments are defined as

of the line charge, we may factor out

where the interior multipole moments are defined as

The generalization to an arbitrary distribution of line charges

represents the line charge per unit area in the

Similarly, the interior cylindrical multipole expansion has the functional form

A simple formula for the interaction energy of cylindrical multipoles (charge density 1) with a second charge density can be derived.

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

are the cylindrical multipole moments of charge distribution 1.

This energy formula can be reduced to a remarkably simple form

are the interior cylindrical multipoles of the second charge density.

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

are the interior cylindrical multipole moments of charge distribution 1, 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.