On the $\delta$-singularities of the electromagnetic field

TL;DR

This paper derives comprehensive delta-singularities for electromagnetic fields, including all multipole moments, and introduces an efficient symmetric, trace-free representation for higher-order multipoles, applicable in static and dynamic cases.

Contribution

It provides a unified method to derive delta-singularities for all multipole orders using symmetric trace-free moments, extending to higher orders with an adaptable algorithm.

Findings

Derived delta-singularities for all multipole orders in static fields

Presented an efficient symmetric, trace-free multipole representation

Extended the method to dynamic cases for lower orders

Abstract

The singularities of the electromagnetic field are derived to include all the point-like multipoles representing an electric charge and current distribution. We show that for higher orders, it is more efficient to have fields represented in terms of symmetric and trace free moments. In the static case, the delta-singularities are expressed for arbitrary multipole orders, while in the dynamic case we restrict ourselves to the lower orders. The algorithm we give can be easily extended to the next orders.