An Alternative to Maxwell's Displacement Current

TL;DR

This paper proposes an alternative to Maxwell's displacement current that enforces local charge conservation and describes an instantaneous action theory, potentially useful where the finite speed of light is negligible.

Contribution

It introduces a new formulation replacing Maxwell's displacement current with an instantaneous action approach based on a revision of Ampere's law, independent of current constitution.

Findings

Provides explicit electric and magnetic fields in terms of charge and current densities.

Enforces local charge conservation without relying on displacement current.

Approximates Maxwell's equations when the finite speed of light can be ignored.

Abstract

Though sufficient for local conservation of charge, we show that Maxwells displacement current is not necessary. An alternative to the Ampere Maxwell equation is exhibited and the alternative s electric and magnetic fields and scalar and vector potentials are expressed in terms of the charge and current densities. The alternative describes a theory in which action is instantaneous and so may provide a good approximation to Maxwells equations where and when the finite speed of light can be neglected. The result is reminiscent of the Darwin approximation which arose from the study classical charged point particles to order (v/c)2 in the Lagrangian. Unlike Darwin, this approach does not depend on the constitution of the electric current. Instead, this approach grows from a straightforward revision of the Ampere Equation which revision enforces the local conservation of charge.