Coulomb Law in the Non-Uniform Euler-Heisenberg Theory

TL;DR

This paper explores how non-uniform, higher-order corrections to classical electrodynamics modify Coulomb's law, deriving a new differential equation and analyzing the electric field of point charges within this extended framework.

Contribution

It introduces a non-uniform Euler-Heisenberg extension to Maxwell's theory, deriving a modified Gauss law with higher-order derivatives for electrostatics.

Findings

Derived a higher-order differential equation for electrostatics.

Calculated corrections to Coulomb's law for point charges.

Analyzed the impact of non-uniform terms on electric fields.

Abstract

We consider the non-linear classical field theory which results from adding to the Maxwell's Lagrangian the contributions from the weak-field Euler-Heisenberg Lagrangian and a non-uniform part which involves derivatives of the electric and magnetic fields. We focus on the electrostatic case where the magnetic field is set to zero, and we derive the modified Gauss law, resulting in a higher order differential equation. This equation gives the electric field produced by stationary charges in the higher order non-linear electrodynamics. Specializing for the case of a point charge, we investigate the solutions of the modified Gauss law and calculate the correction to the Coulomb law.