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Uniqueness Theorem: With Normal Components Specified on External Spherical Surface

Rajavardhan Talashila

TL;DR

This paper addresses the uniqueness of time-harmonic electromagnetic fields inside a spherical volume that encloses sources, under the condition of an exterior homogeneous medium supporting only outgoing waves. It develops a multipole-expansion framework showing that the radial components of $E$ and $B$ on the outer sphere, together with tangential data on interior surfaces, uniquely determine the interior field, and it demonstrates the equivalence between radial boundary data and tangential boundary data for outside fields. The main contribution is a new uniqueness theorem for spherical volumes with lossless, homogeneous exteriors, backed by multipole theory and the classical uniqueness theorem, with practical implications for antenna simulations and boundary-data formulations. The half-wave dipole example validates that radial-boundary data suffice to reproduce correct far-fields, reinforcing the method's physical relevance and potential for broader applicability.

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.

Uniqueness Theorem: With Normal Components Specified on External Spherical Surface

TL;DR

This paper addresses the uniqueness of time-harmonic electromagnetic fields inside a spherical volume that encloses sources, under the condition of an exterior homogeneous medium supporting only outgoing waves. It develops a multipole-expansion framework showing that the radial components of and on the outer sphere, together with tangential data on interior surfaces, uniquely determine the interior field, and it demonstrates the equivalence between radial boundary data and tangential boundary data for outside fields. The main contribution is a new uniqueness theorem for spherical volumes with lossless, homogeneous exteriors, backed by multipole theory and the classical uniqueness theorem, with practical implications for antenna simulations and boundary-data formulations. The half-wave dipole example validates that radial-boundary data suffice to reproduce correct far-fields, reinforcing the method's physical relevance and potential for broader applicability.

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 that contains only perfect conductors and homogeneous lossless materials and for which the impressed currents 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 and over the exterior spherical surface and the tangential components of either or 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 having loss-less materials is considered to be the case of lossy materials in the limit the dissipation approaching zero.

Paper Structure

This paper contains 6 sections, 15 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Uniqueness Theorem
  3. Multipole Theory
  4. Proposed Theorem
  5. Conclusion
  6. Half-wave Dipole

Figures (3)

  • Figure 1: The representation of the volume $V$ with external spherical surface $S_{e}$ and internal surface $S_{i}$. It has the source currents $\mathbf{J}$ specified.
  • Figure 2: This is the comparision of the far-field radiation for the dipole calculated directly from the coefficients and from the radial electric field expansion. They agree well.
  • Figure 3: The real and imaginary parts of the "source" $\mathbf{r \centerdot E}$ on the $\lambda / 4$ sphere surrounding the halfwave dipole antenna.