Which electric fields are realizable in conducting materials?

TL;DR

This paper investigates when a smooth periodic gradient field can be realized as a divergence-free current in conducting materials, establishing conditions for local, global, and isotropic realizability in various dimensions.

Contribution

It provides new criteria for the realizability of gradient fields as conductivity-induced currents, including conditions for isotropic and periodic cases, using a dynamical systems approach.

Findings

Realizability is always possible locally if the gradient is non-vanishing.

In 2D, non-vanishing gradient fields are necessary for realizability.

Isotropic realizability holds globally in space if the gradient never vanishes.

Abstract

In this paper we study the realizability of a given smooth periodic gradient field $\nabla u$ defined in $\RR^d$, in the sense of finding when one can obtain a matrix conductivity $\si$ such that $\si\nabla u$ is a divergence free current field. The construction is shown to be always possible locally in $\RR^d$ provided that $\nabla u$ is non-vanishing. This condition is also necessary in dimension two but not in dimension three. In fact the realizability may fail for non-regular gradient fields, and in general the conductivity cannot be both periodic and isotropic. However, using a dynamical systems approach the isotropic realizability is proved to hold in the whole space (without periodicity) under the assumption that the gradient does not vanish anywhere. Moreover, a sharp condition is obtained to ensure the isotropic realizability in the torus. The realizability of a matrix field is also investigated both in the periodic case and in the laminate case. In this context the sign of the matrix field determinant plays an essential role according to the space dimension.