Point charge representations of multipoles

Abstract

We generalize the notion of a dipole made of two point charges placed infinitesimally close together to multipoles of all degrees. The charge distributions are derived first by differentiating the delta function for the monopole in the same way that differentiating the potential $1/r$ generates the spherical multipoles $r^{-\ell-1}P_\ell^m(\cos\theta)e^{im\phi}$. This method surprisingly generates messy distributions for spherical multipoles with $m\geq3$. So instead we use a ``bracelet" of $2m$ charges for the planar spherical multipoles of order $m=\ell$, and stack these to construct multipoles for all $m,\ell$. We show that the spherical multipoles of degree $\ell$ can be made of $\ell+1$ charges for $m=0$ and $2m(\ell-m+1)$ charges for $1\leq m\leq \ell$. A similar construction is found for the Cartesian multipoles.

Current loop and point dipole configurations for magnetic multipoles

“…The field of this configuration can be evaluated by substituting the dipole field equation (44) in for each dipole, and has been checked to evaluate exactly to C m m using . Maybe it could be proved for all m using a similar approach to that in [9] for point charges.…”

“…The multipoles with m > 0 have m-fold symmetry around f. It was found for the electrostatic multipoles in [9] that simply differentiating the charge distributions in the x and y directions gets quite messy. Instead, the best representation was to use a 'bracelet' of 2m point charges placed symmetrically around the z-axis with an alternating sign.…”

“…The most straightforward approach is to layer copies of the m = n configurations in the z-direction. Mathematically we apply the differential relation (9) n − m times to the current J m m in (29), which gives…”

“…So this paper is dedicated to exploring the currents that produce all multipole fields. It is an extension of [9] which treated the point charge distributions for electrostatic multipoles. Despite the mathematical similarity between the electrostatic and magnetostatic multipole fields, we will find that more interesting distributions are possible when working with currents.…”

“…Nevertheless, the configurations are useful as idealizations to understand the nature of the multipolar fields. Now we can generalize these configurations to all n. For the layered loops, we have n loops of radius a spaced along the z-axis, and the currents can be determined by differentiating those for the previous n, giving a Pascal's triangle pattern similar to the electrostatic case [9]. Here there are two dimensions-the loop radii and their spacing in the z-direction, and we can be more general by allowing a loop spacing of βa for some β > 0.…”