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 and in the Bopp–Podolsky theory. The theory of second gradient electromagnetostatics delivers a singularity-free electromagnetic field theory with weak spatial nonlocality.
Highlights
In recent years, there has been a continuous interest in the so-called Bopp–Podolsky theory [1,2,3,4,5,6,7], a first-order linear gradient theory of electrodynamics
We will study the textbook examples of electric point charge, electrostatic dipole and magnetostatic dipole in the framework of generalized electrodynamics, and show that second gradient electromagnetostatics yields nonsingular dipole fields and gives a straightforward regularization of the dipole singularities based on higher-order partial differential equations
We study the static version of it called second gradient electromagnetostatics
Summary
There has been a continuous interest in the so-called Bopp–Podolsky theory [1,2,3,4,5,6,7], a first-order linear gradient theory of electrodynamics. In order to obtain electromagnetic fields of a non-uniformly moving point charge with no directional discontinuity on the light cone, the theory of second gradient electrodynamics has recently been proposed and used in [18]. We will study the textbook examples of electric point charge, electrostatic dipole and magnetostatic dipole in the framework of generalized electrodynamics, and show that second gradient electromagnetostatics yields nonsingular dipole fields and gives a straightforward regularization of the dipole singularities based on higher-order partial differential equations. The limits of those electromagnetic fields to the Bopp–Podolsky theory and to the classical Maxwell theory are given in Sections 5 and 6, respectively.
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