Physics: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 [math]\displaystyle{ \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]\displaystyle{ (\rho^{\prime}, \theta^{\prime}) }[/math] refer to the position of the line charge(s), whereas the unprimed coordinates such as [math]\displaystyle{ (\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]\displaystyle{ \mathbf{r} }[/math] has coordinates [math]\displaystyle{ ( \rho, \theta, z) }[/math] where [math]\displaystyle{ \rho }[/math] is the radius from the [math]\displaystyle{ z }[/math] axis, [math]\displaystyle{ \theta }[/math] is the azimuthal angle and [math]\displaystyle{ z }[/math] is the normal Cartesian coordinate. By assumption, the line charges are infinitely long and aligned with the [math]\displaystyle{ z }[/math] axis.
Cylindrical multipole moments of a line charge
The electric potential of a line charge [math]\displaystyle{ \lambda }[/math] located at [math]\displaystyle{ (\rho', \theta') }[/math] is given by [math]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ z }[/math]-dependence. The line charge [math]\displaystyle{ \lambda }[/math] is the charge per unit length in the [math]\displaystyle{ z }[/math]-direction, and has units of (charge/length). If the radius [math]\displaystyle{ \rho }[/math] of the observation point is greater than the radius [math]\displaystyle{ \rho' }[/math] of the line charge, we may factor out [math]\displaystyle{ \rho^{2} }[/math] [math]\displaystyle{ \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]\displaystyle{ (\rho'/\rho)\lt 1 }[/math] [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \rho }[/math] of the observation point is less than the radius [math]\displaystyle{ \rho' }[/math] of the line charge, we may factor out [math]\displaystyle{ \left( \rho' \right)^{2} }[/math] and expand the logarithms in powers of [math]\displaystyle{ (\rho/\rho')\lt 1 }[/math] [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \lambda(\rho', \theta') }[/math] is straightforward. The functional form is the same [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \lambda(\rho', \theta') }[/math] represents the line charge per unit area in the [math]\displaystyle{ (\rho-\theta) }[/math] plane.
Interior cylindrical multipole moments
Similarly, the interior cylindrical multipole expansion has the functional form [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ f(\mathbf{r}^{\prime}) }[/math] be the second charge density, and define [math]\displaystyle{ \lambda(\rho, \theta) }[/math] as its integral over z [math]\displaystyle{ \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]\displaystyle{ U = \int d\theta \, d\rho \, \rho \, \lambda(\rho, \theta) \Phi(\rho, \theta) }[/math]
If the cylindrical multipoles are exterior, this equation becomes [math]\displaystyle{ 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]\displaystyle{ Q_{1} }[/math], [math]\displaystyle{ C_{1k} }[/math] and [math]\displaystyle{ S_{1k} }[/math] are the cylindrical multipole moments of charge distribution 1. This energy formula can be reduced to a remarkably simple form [math]\displaystyle{ 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]\displaystyle{ I_{2k} }[/math] and [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ I_{1k} }[/math] and [math]\displaystyle{ J_{1k} }[/math] are the interior cylindrical multipole moments of charge distribution 1, and [math]\displaystyle{ C_{2k} }[/math] and [math]\displaystyle{ 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.
See also
- Axial multipole moments
- Potential theory
- Quantum cylindrical quadrupole
- Multipole expansion
- Spherical multipole moments
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