Second gradient electromagnetostatics: electric point charge, electrostatic and magnetostatic dipoles

TL;DR

This paper develops a second gradient electromagnetostatics theory that regularizes classical electromagnetic fields, eliminating singularities for point charges and dipoles by incorporating higher-order derivatives and weak nonlocality.

Contribution

It introduces a novel second gradient electromagnetostatics framework with fourth-order operators, providing singularity-free solutions for static electromagnetic sources.

Findings

Electromagnetic fields are free of singularities.

The theory incorporates higher-order derivatives for regularization.

Fields exhibit weak spatial nonlocality.

Abstract

In this paper, we study the theory of second gradient electromagnetostatics as the static version of second gradient electrodynamics. The theory of second gradient electrodynamics is a linear generalization of higher order of classical Maxwell electrodynamics whose Lagrangian is both Lorentz and U(1)-gauge invariant. Second gradient electromagnetostatics is a gradient field theory with up to second-order derivatives of the electromagnetic field strengths in the Lagrangian. Moreover, it possesses a weak nonlocality in space and gives a regularization based on higher-order partial differential equations. From the group theoretical point of view, in second gradient electromagnetostatics the (isotropic) constitutive relations involve an invariant scalar differential operator of fourth order in addition to scalar constitutive parameters. We investigate the classical static problems of an electric point charge, and electric and magnetic dipoles in the framework of second gradient electromagnetostatics, and we show that all the electromagnetic fields (potential, field strength, interaction energy, interaction force) are singularity-free unlike the corresponding solutions in the classical Maxwell electromagnetism as well as in the Bopp-Podolsky theory. The theory of second gradient electromagnetostatics delivers a singularity-free electromagnetic field theory with weak spatial nonlocality.