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

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

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

By symmetry, the electric potential of an infinite linecharge 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^{\prime} 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^{\prime}/\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^{\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

\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
Q = \lambda ,
C_{k} = \frac{\lambda}{k} \left( \rho^{\prime} \right)^{k} \cos k\theta^{\prime} ,
and
S_{k} = \frac{\lambda}{k} \left( \rho^{\prime} \right)^{k} \sin k\theta^{\prime} .

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

\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

\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
Q = \lambda ,
I_{k} = \frac{\lambda}{k} \frac{\cos k\theta^{\prime}}{\left( \rho^{\prime} \right)^{k}},
and
J_{k} = \frac{\lambda}{k} \frac{\sin k\theta^{\prime}}{\left( \rho^{\prime} \right)^{k}}.

General cylindrical multipole moments

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

\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

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

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

If the cylindrical multipoles are exterior, this equation becomes

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} C_{1k} \int d\theta \int d\rho \left[ \frac{\cos k\theta}{\rho^{k-1}} \right] \lambda(\rho, \theta)
\ \ \ \ \ \ \ \ + \ \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 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 \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 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^{\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 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