Multipole expansions for time-dependent charge and current distributions in quasistatic approximation

TL;DR

This paper develops a comprehensive multipole expansion framework for time-dependent charge and current distributions within the quasistatic approximation, including electric, magnetic, and toroidal moments, with explicit formulas for arbitrary orders.

Contribution

It introduces a consistent method to define and expand electric, magnetic, and toroidal multipole moments, including closed-form expressions for arbitrary expansion orders.

Findings

Derived closed-form multipole expressions for arbitrary order.

Unified framework for electric, magnetic, and toroidal multipoles.

Clarified the quasistatic approximation's role in multipole expansions.

Abstract

We propose a consistent approach to the definition of electric, magnetic, and toroidal multipole moments. Electric and magnetic fields are split into potential, vortex, and radiative terms, with the latter ones dropped off in the quasistatic approximation. The potential part of the electric field, the vortex parts of the magnetic field and vector potential contain gradients of scalar functions. Formally introducing magnetic and toroidal analogs of the electric charge, we apply multipole expansions for those scalars. Closed-form expressions are derived in an arbitrary order for electric, magnetic, and toroidal multipoles, which constitute a full system for expansions of the electromagnetic field.