Cylindrical multipole moments

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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 (ρ,θ) 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 (ρ,θ,z) where ρ is the radius from the z axis, θ 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.

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[edit] Cylindrical multipole moments of a line charge

Figure 1: Definitions for cylindrical multipoles; looking down the axis
Figure 1: Definitions for cylindrical multipoles; looking down the z^{\prime} axis

The electric potential of a line charge λ 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 line charge has no z-dependence. The line charge λ is the charge per unit length in the z-direction, and has units of (charge/length).

If the radius ρ of the observation point is greater than the radius \rho^{\prime} of the line charge, we may factor out ρ2

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

Conversely, if the radius ρ 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 Q \ \stackrel{\mathrm{def}}{=}\ \lambda , I_{k} \ \stackrel{\mathrm{def}}{=}\ \frac{\lambda}{k} \frac{\cos k\theta^{\prime}}{\left( \rho^{\prime} \right)^{k}}, and J_{k} \ \stackrel{\mathrm{def}}{=}\ \frac{\lambda}{k} \frac{\sin k\theta^{\prime}}{\left( \rho^{\prime} \right)^{k}}.

[edit] 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 \ \stackrel{\mathrm{def}}{=}\ \int d\theta^{\prime} \int \rho^{\prime} d\rho^{\prime} \lambda(\rho^{\prime}, \theta^{\prime})
C_{k} \ \stackrel{\mathrm{def}}{=}\ \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} \ \stackrel{\mathrm{def}}{=}\ \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 ρ − θ plane, and has units of (charge / length) / length2.

[edit] 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 \ \stackrel{\mathrm{def}}{=}\ \int d\theta^{\prime} \int \rho^{\prime} d\rho^{\prime} \lambda(\rho^{\prime}, \theta^{\prime})
I_{k} \ \stackrel{\mathrm{def}}{=}\ \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} \ \stackrel{\mathrm{def}}{=}\ \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})

[edit] 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 λ(ρ,θ) as its integral over z

\lambda(\rho, \theta) \ \stackrel{\mathrm{def}}{=}\ \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 \ \stackrel{\mathrm{def}}{=}\ \int d\theta \int \rho d\rho \ \lambda(\rho, \theta) \Phi(\rho, \theta)

If the cylindrical multipoles are exterior, this equation becomes

U \ \stackrel{\mathrm{def}}{=}\ \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 Q1, C1k and S1k are the cylindrical multipole moments of charge distribution 1. This energy formula can be reduced to a remarkably simple form

U \ \stackrel{\mathrm{def}}{=}\ \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 I2k and J2k 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 \ \stackrel{\mathrm{def}}{=}\ \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 I1k and J1k are the interior cylindrical multipole moments of charge distribution 1, and C2k and S2k 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.

[edit] See also