[Paper Review] Potential Theory in Classical Electrodynamics

W. Engelhardt|arXiv (Cornell University)|Sep 8, 2012

Geophysics and Sensor Technology7 references3 citations

TL;DR

This paper challenges the foundational consistency of classical electrodynamics by demonstrating that Maxwell's first-order equations contain an internal contradiction between Faraday's law and Maxwell's flux law, which undermines gauge invariance and unique solutions. It shows that retarded potentials—commonly used to solve inhomogeneous wave equations—fail to satisfy the equations when sources move at constant velocity, and that the potential method obscures this inconsistency, especially in systems like a plate capacitor where field solutions depend on arbitrary choices of vector potential divergence.

ABSTRACT

In Maxwell's classical theory of electrodynamics the fields are frequently expressed by potentials in order to facilitate the solution of the first order system of equations. This method obscures, however, that there exists an inconsistency between Faraday's law of induction and Maxwell's flux law. As a consequence of this internal contradiction there is neither gauge invariance, nor exist unique solutions in general. It is also demonstrated that inhomogeneous wave equations cannot be solved by retarded integrals.

Motivation & Objective

• To identify and analyze an internal inconsistency in Maxwell's first-order equations of classical electrodynamics.
• To investigate whether the standard potential method—using scalar and vector potentials—yields unique and consistent solutions.
• To challenge the validity of retarded integrals as solutions to inhomogeneous wave equations in electrodynamics.
• To demonstrate that the divergence of the vector potential cannot be arbitrarily chosen without violating field consistency.
• To show that the potential method conceals a fundamental contradiction between Faraday’s law and Maxwell’s flux law.

Proposed method

• Analyzes the potential ansatz $\vec{B} = \nabla \times \vec{A}$, $\vec{E} = -\nabla\phi - \frac{1}{c}\frac{\partial\vec{A}}{\partial t}$ and its gauge invariance under $\vec{A} \to \vec{A} + \nabla\psi$, $\phi \to \phi - \frac{1}{c}\frac{\partial\psi}{\partial t}$.
• Splits the scalar and vector potentials into source-dependent and gauge-dependent parts: $\phi = \phi_1 + \phi_2$, $\vec{A} = \vec{A}_1 + \vec{A}_2$, with $\chi = \nabla \cdot \vec{A}$ as the gauge parameter.
• Applies Helmholtz’s theorem to show that $\phi_1$ and $\vec{A}_1$ uniquely determine fields when $\chi = 0$, corresponding to Coulomb gauge.
• Derives necessary conditions $\nabla \times \vec{A}_2 = 0$ and $\nabla\phi_2 + \frac{1}{c}\frac{\partial\vec{A}_2}{\partial t} = 0$ for gauge independence, leading to $\vec{A}_2 = \nabla U$ and $\phi_2 = -\frac{1}{c}\frac{\partial U}{\partial t}$.
• Evaluates the Liénard-Wiechert potential for a uniformly moving charge and shows it does not satisfy the inhomogeneous wave equation.
• Uses volume integrals and spherical coordinates to compute field contributions in a plate capacitor, revealing inconsistency between two derived expressions for $\vec{B}$.

Experimental results

Research questions

• RQ1 Does the standard potential method in classical electrodynamics yield unique and gauge-invariant solutions for the electromagnetic fields?
• RQ2 Can the inhomogeneous wave equation for the vector potential be consistently solved using retarded potentials, especially for moving charges?
• RQ3 Is there an internal inconsistency between Faraday’s law of induction and Maxwell’s flux law in the first-order Maxwell equations?
• RQ4 Why do solutions in Lorenz gauge differ from those in Coulomb gauge, and what does this imply about the validity of the potential method?
• RQ5 Can the Liénard-Wiechert potential be considered a valid solution to the inhomogeneous wave equation for a uniformly moving point charge?

Key findings

• The potential method in classical electrodynamics conceals an internal inconsistency between Faraday’s law and Maxwell’s flux law, which invalidates gauge invariance and prevents unique solutions.
• The divergence of the vector potential $\chi = \nabla \cdot \vec{A}$ cannot be chosen arbitrarily; it is constrained by the requirement that field expressions remain independent of gauge choice.
• The Liénard-Wiechert scalar potential for a uniformly moving charge fails to satisfy the inhomogeneous wave equation, invalidating the standard retarded solution approach.
• In a plate capacitor, two derivations of the magnetic field $\vec{B}$—one from the current and one from the time-varying electric field—yield inconsistent results, exposing the system’s inconsistency.
• The inhomogeneous wave equation cannot be solved by retarded integrals in general, as they assume source and field are evaluated at different times, contradicting the simultaneous dependence in the equations.
• Only the homogeneous wave equation, as considered by Maxwell in his theory of light, consistently describes electromagnetic waves disconnected from their sources; the inhomogeneous case is fundamentally unsolvable in the standard framework.