Convection displacement current and alternative form of Maxwell-Lorentz equations

TL;DR

This paper introduces a novel convection displacement current to address inconsistencies in Maxwell-Lorentz equations, providing a relativistically invariant formulation and new gauge conditions that offer a fresh interpretation of electromagnetic fields.

Contribution

It proposes a new convection displacement current and a relativistically invariant form of Maxwell-Lorentz equations, along with a novel gauge condition, enhancing the theoretical framework of classical electrodynamics.

Findings

The new form of equations is relativistically invariant.

A new gauge condition replaces the Coulomb gauge.

Provides a physical interpretation of the four-vector potentials.

Abstract

Some mathematical inconsistencies in the conventional form of Maxwell's equations extended by Lorentz for a single charge system are discussed. To surmount these in framework of Maxwellian theory, a novel convection displacement current is considered as additional and complementary to the famous Maxwell displacement current. It is shown that this form of the Maxwell-Lorentz equations is similar to that proposed by Hertz for electrodynamics of bodies in motion. Original Maxwell's equations can be considered as a valid approximation for a continuous and closed (or going to infinity) conduction current. It is also proved that our novel form of the Maxwell-Lorentz equations is relativistically invariant. In particular, a relativistically invariant gauge for quasistatic fields has been found to replace the non-invariant Coulomb gauge. The new gauge condition contains the famous relationship between electric and magnetic potentials for one uniformly moving charge that is usually attributed to the Lorentz transformations. Thus, for the first time, using the convection displacement current, a physical interpretation is given to the relationship between the components of the four-vector of quasistatic potentials. A rigorous application of the new gauge transformation with the Lorentz gauge transforms the basic field equations into an independent pair of differential equations responsible for longitudinal and transverse fields, respectively. The longitudinal components can be interpreted exclusively from the standpoint of the instantaneous "action at a distance" concept and leads to necessary conceptual revision of the conventional Faraday-Maxwell field. The concept of electrodynamic dualism is proposed for self-consistent classical electrodynamics. It implies simultaneous coexistence