Direct Derivation of Liénard–Wiechert Potentials, Maxwell’s Equations and Lorentz Force from Coulomb’s Law

Hrvoje Dodig

doi:10.3390/math9030237

Abstract

In this paper, the solution to long standing problem of deriving Maxwell’s equations and Lorentz force from first principles, i.e., from Coulomb’s law, is presented. This problem was studied by many authors throughout history but it was never satisfactorily solved, and it was never solved for charges in arbitrary motion. In this paper, relativistically correct Liénard–Wiechert potentials for charges in arbitrary motion and Maxwell equations are both derived directly from Coulomb’s law by careful mathematical analysis of the moment just before the charge in motion stops. In the second part of this paper, the electrodynamic energy conservation principle is derived directly from Coulomb’s law by using similar approach. From this energy conservation principle the Lorentz force is derived. To make these derivations possible, the generalized Helmholtz theorem was derived along with two novel vector identities. The special relativity was not used in our derivations, and the results show that electromagnetism as a whole is not the consequence of special relativity, but it is rather the consequence of time retardation.

Highlights

In his famous Treatise [1,2], Maxwell derived his equations of electrodynamics based on the knowledge about the three experimental laws known at the time: Coulomb’s law describing the electric force between charges at rest; Ampere’s law describing the force between current carrying wires, and Faraday’s law of induction
The special relativity was not used in our derivations, and the results show that electromagnetism as a whole is not the consequence of special relativity, but it is rather the consequence of time retardation
Maxwell equations and the Lorentz force were derived in this paper directly from Coulomb’s law

Summary

Introduction

In his famous Treatise [1,2], Maxwell derived his equations of electrodynamics based on the knowledge about the three experimental laws known at the time: Coulomb’s law describing the electric force between charges at rest; Ampere’s law describing the force between current carrying wires, and Faraday’s law of induction. In addition to the criticism above, it should be emphasized that the derivations of Maxwell’s equations from Coulomb’s law using Lorentz transformation should only be considered valid for the special case of the charge moving along the straight line with constant velocity. To discover the mathematical form of this “unknown” electrodynamic force acting in the past from the knowledge of known electrostatic force (Coulomb’s law) acting at the present time, the generalized Helmholtz decomposition theorem was applied. This was again achieved by the careful application of generalized Helmholtz decomposition theorem which allowed us to transform electrostatic energy conservation law valid at present to dynamic energy conservation law valid in the past This dynamic energy conservation law states that the work of non-conservative force along closed contour is equal to the time derivative of the flux of certain vector field through the surface bounded by this closed contour. It was derived from time retardation, electromagnetism should be considered to be the consequence of time retardation and not as the consequence of special relativity

Generalized Helmholtz Decomposition Theorem

Derivation of Maxwell Equations from Coulomb’s Law

Derivation of Electrodynamic Energy Conservation Law and Lorentz Force

Conclusions