[Paper Review] On the Solvability of Maxwell's Equations

W. Engelhardt|arXiv (Cornell University)|Sep 13, 2012

Advanced Scientific and Engineering Studies6 references3 citations

TL;DR

This paper investigates the solvability of Maxwell's equations in Lorenz and Coulomb gauges using an oscillating electric dipole source, revealing that retarded integral solutions fail to satisfy the inhomogeneous wave equations they are meant to solve. The key finding is an internal inconsistency in the standard formulation: retarded potentials derived from wave equations do not actually solve them, undermining the foundation of classical electrodynamics as conventionally applied.

ABSTRACT

Complementing a study which was published in this journal in 2005, we present explicit calculations of fields predicted by Maxwell's equations both in Lorenz and in Coulomb gauge. Analytic expressions are obtainable, when the source of the fields is an oscillating electric dipole. As before it is found that the fields calculated by different methods are at variance. In addition, the reason for the discrepancies is revealed: The retarded integrals turn out not to satisfy the inhomogeneous wave equations which they are supposed to solve.

Motivation & Objective

• To investigate the consistency of solutions to Maxwell’s equations in different gauges, particularly Lorenz and Coulomb gauge.
• To determine whether retarded integral solutions of the inhomogeneous wave equation actually satisfy the wave equation.
• To resolve discrepancies between fields calculated via different gauge methods, especially for an oscillating dipole source.
• To examine the validity of Duhamel’s principle in constructing solutions to inhomogeneous wave equations.
• To clarify the theoretical basis for the separation of near-field (instantaneous) and far-field (radiative) solutions in electrodynamics.

Proposed method

• Explicit calculation of electromagnetic potentials and fields using the Lorenz gauge for an oscillating electric dipole, yielding analytic expressions.
• Application of the Coulomb gauge to the same dipole source, computing improper integrals with well-defined limiting values.
• Derivation and analysis of the inhomogeneous wave equation for a spatially extended source, testing whether standard retarded solutions satisfy it.
• Comparison of solutions from different gauges to identify contradictions in field predictions.
• Use of Duhamel’s principle to construct solutions from homogeneous solutions, testing its validity for wave equations.
• Separation of fields into instantaneous (Coulomb-like) and wave (radiative) components via decomposition of the electric and magnetic fields.

Experimental results

Research questions

• RQ1 Do retarded integral solutions of the inhomogeneous wave equation actually satisfy the wave equation they are supposed to solve?
• RQ2 Why do solutions of Maxwell’s equations in Lorenz and Coulomb gauges yield different electromagnetic fields for the same dipole source?
• RQ3 Is Duhamel’s principle valid for constructing solutions of the inhomogeneous wave equation from solutions of the homogeneous equation?
• RQ4 Can the standard retarded potential formalism be consistently applied to wave equations with finite-speed propagation?
• RQ5 What is the theoretical basis for separating near-field and far-field solutions in classical electrodynamics?

Key findings

• The retarded integral solutions of the inhomogeneous wave equation do not satisfy the wave equation they are meant to solve, revealing a fundamental inconsistency in the standard approach.
• Solutions in Lorenz and Coulomb gauges yield different electromagnetic fields for the same oscillating dipole source, confirming prior findings of gauge-dependent results.
• The use of Duhamel’s principle for wave equations is invalid because the wave equation's finite propagation speed contradicts the principle’s assumption of instantaneous response.
• The standard retarded potential formalism fails because the time variable in the wave equation’s source term is not properly retarded, leading to logical inconsistencies.
• The near-field (instantaneous) and far-field (radiative) solutions must be treated separately, as they stem from different types of equations: elliptic vs. hyperbolic.
• The paper concludes that a unified theory of both instantaneous and radiative fields within Maxwell’s equations is not currently possible without modifying the system.