Reduction of the classical electromagnetism to a two-dimensional curved surface

TL;DR

This paper explores two methods of reducing three-dimensional classical electromagnetism to a two-dimensional surface, resulting in different field and source configurations relevant for thin-layer and interface electrodynamics.

Contribution

It introduces two distinct reduced models of electromagnetism on a surface, detailing their field structures and source properties, with potential applications in thin-layer physics.

Findings

Two models of 2D electromagnetism derived from 3D theory.

One model features a tangent electric field with conserved currents.

The other involves a scalar electric field with non-conserved sources.

Abstract

The reduction of the three-dimensional classical electromagnetism is performed in a twofold way. In the first case the ordinary two-dimensional electromagnetism is obtained with sources in the form of conserved electric currents flowing along the surface. The electric field is a two-vector tangent to the surface and magnetic field is a scalar quantity. In the second approach the reduced theory is that of the two-vector magnetic field and a scalar electric one. The only source coupled to the fields is now a scalar subject to no conservation law. In the redefined theory this scalar source is may be converted into an eddy magnetic current flowing in the surface. No magnetic monopoles appear. Our results can find some applications in the electrodynamics of thin layers and of metal-dielectric interfaces.