Uniqueness Theorem: With Normal Components Specified on External Spherical Surface

TL;DR

This paper presents and proves a new uniqueness theorem for time-harmonic electromagnetic fields, showing that specifying normal components on a spherical surface uniquely determines the field inside, even with lossy materials.

Contribution

It introduces a novel uniqueness theorem for electromagnetic fields with specified normal components on a spherical surface, extending previous results to lossy materials.

Findings

The theorem guarantees unique solutions given boundary data.

Proof relies on multipole expansion uniqueness.

Applicable to lossy and lossless materials.

Abstract

A uniqueness theorem for time-harmonic electromagnetic fields which requires the normal components of electromagnetic fields specified on a spherical surface is proposed and proved. The statement of the theorem is : "For a spherical volume $V$ that contains only perfect conductors and homogeneous lossless materials and for which the impressed currents $\mathbf{J}$ are specified, a time-harmonic solution to the Maxwell's equations within the volume, having outgoing waves alone, is uniquely specified by the values of the radial components of both $\mathbf{E}$ and $\mathbf{B}$ over the exterior spherical surface $V$ and the tangential components of either $\mathbf{E}$ or $\mathbf{B}$ on the interior surfaces." The proof of this theorem relies on the uniqueness of multipole expansion of electromagnetic fields outside the enclosing sphere. The conventional uniqueness theorem for the volume $V$ having loss-less materials is considered to be the case of lossy materials in the limit the dissipation approaching zero.