Multipole Distributions and Hyper-Flux Fields

TL;DR

This paper introduces a unified mathematical framework for multipole expansions and hyper-flux fields applicable beyond electrostatics, enabling simplified calculations of bound charges, multipoles, and forces in continuum mechanics.

Contribution

It presents a novel, general approach to multipole and hyper-flux concepts, extending classical electrostatics to tensorial flux fields in continuum mechanics.

Findings

Provides formulas for bound multipoles and charges

Derives a general expression for mechanical force functional

Shows moving multipoles generate hyper-fluxes

Abstract

We outline here a simple mathematical introduction to the notions of multipoles for a general extensive property $\Pi$ from the point of view of continuum mechanics. Classically, $\Pi$ is the electric charge, but the theory is not limited to electrostatics. The proposed framework allows a simple computation of the bound "charges" and bound multipoles of lower orders. In addition, if the property $\Pi$ has a potential function in the sense described below, a general expression for the mechanical force (power) functional acting on bodies containing the property is presented. Finally, using a similar viewpoint, we consider hyper-fluxes -- flux fields of tensorial order greater than one -- and show that moving multipoles (in particular, a moving dielectric) give rise to hyper-fluxes.