A consistency condition for the vector potential in multiply-connected domains

TL;DR

This paper introduces a gauge-invariant consistency condition for the vector potential in multiply-connected domains, addressing non-uniqueness issues in static electromagnetics and improving the inversion of magnetic field integral equations.

Contribution

It proposes a novel consistency condition that ensures uniqueness of the vector potential in multiply-connected domains, resolving a longstanding problem in electromagnetics.

Findings

The condition guarantees a unique vector potential representation.

It enables accurate inversion of magnetic field integral equations.

The approach is gauge-invariant and applicable to static and low-frequency regimes.

Abstract

A classical problem in electromagnetics concerns the representation of the electric and magnetic fields in the low-frequency or static regime, where topology plays a fundamental role. For multiply connected conductors, at zero frequency the standard boundary conditions on the tangential components of the magnetic field do not uniquely determine the vector potential. We describe a (gauge-invariant) consistency condition that overcomes this non-uniqueness and resolves a longstanding difficulty in inverting the magnetic field integral equation.