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 <math>\ln \ R</math>. 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 <math>(\rho^{\prime}, \theta^{\prime})</math> refer to the position of the line charge(s), whereas the unprimed coordinates such as <math>(\rho, \theta)</math> refer to the point at which the potential is being observed. We use cylindrical coordinates throughout, e.g., an arbitrary vector <math>\mathbf{r}</math> has coordinates <math>( \rho, \theta, z)</math> where <math>\rho</math> is the radius from the <math>z</math> axis, <math>\theta</math> is the azimuthal angle and <math>z</math> is the normal Cartesian coordinate. By assumption, the line charges are infinitely long and aligned with the <math>z</math> axis.
Cylindrical multipole moments of a line charge
The electric potential of a line charge <math>\lambda</math> located at <math>(\rho', \theta')</math> is given by <math display="block"> \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| </math> where <math>R</math> is the shortest distance between the line charge and the observation point.
By symmetry, the electric potential of an infinite line charge has no <math>z</math>-dependence. The line charge <math>\lambda</math> is the charge per unit length in the <math>z</math>-direction, and has units of (charge/length). If the radius <math>\rho</math> of the observation point is greater than the radius <math>\rho'</math> of the line charge, we may factor out <math>\rho^{2}</math> <math display="block"> \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\} </math> and expand the logarithms in powers of <math>(\rho'/\rho)<1</math> <math display="block">\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\} </math> which may be written as <math display="block">\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}} </math> where the multipole moments are defined as <math display="block">\begin{align} 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{align}</math>
Conversely, if the radius <math>\rho</math> of the observation point is less than the radius <math>\rho'</math> of the line charge, we may factor out <math>\left( \rho' \right)^{2}</math> and expand the logarithms in powers of <math>(\rho/\rho')<1</math> <math display="block">\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\}</math> which may be written as <math display="block">\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] </math> where the interior multipole moments are defined as <math display="block">\begin{align} 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{align}</math>
General cylindrical multipole moments
The generalization to an arbitrary distribution of line charges <math>\lambda(\rho', \theta')</math> is straightforward. The functional form is the same <math display="block">\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}</math> and the moments can be written <math display="block">\begin{align} 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{align}</math> Note that the <math>\lambda(\rho', \theta')</math> represents the line charge per unit area in the <math>(\rho-\theta)</math> plane.
Interior cylindrical multipole moments
Similarly, the interior cylindrical multipole expansion has the functional form <math display="block"> \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] </math> where the moments are defined <math display="block">\begin{align} 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{align}</math>
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 <math>f(\mathbf{r}^{\prime})</math> be the second charge density, and define <math>\lambda(\rho, \theta)</math> as its integral over z <math display="block">\lambda(\rho, \theta) = \int dz \, f(\rho, \theta, z)</math>
The electrostatic energy is given by the integral of the charge multiplied by the potential due to the cylindrical multipoles <math display="block">U = \int d\theta \, d\rho \, \rho \, \lambda(\rho, \theta) \Phi(\rho, \theta)</math>
If the cylindrical multipoles are exterior, this equation becomes <math display="block">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)</math> where <math>Q_{1}</math>, <math>C_{1k}</math> and <math>S_{1k}</math> are the cylindrical multipole moments of charge distribution 1. This energy formula can be reduced to a remarkably simple form <math display="block">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)</math> where <math>I_{2k}</math> and <math>J_{2k}</math> are the interior cylindrical multipoles of the second charge density.
The analogous formula holds if charge density 1 is composed of interior cylindrical multipoles <math display="block"> 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)</math> where <math>I_{1k}</math> and <math>J_{1k}</math> are the interior cylindrical multipole moments of charge distribution 1, and <math>C_{2k}</math> and <math>S_{2k}</math> 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.
