Mean-Field Formulation of Maxwell Equations to Model Electrically Inhomogeneous and Isotropic Media

Maxwell equations were originally designed to describe classic electromagnetic phenomena in any type of medium. In particular, to describe electromagnetic phenomena under the quasistatic electric approximation in media that are electrically inhomogeneous and isotropic, such as for example when there are strong spatial variations of conductivity, the formalism must be adapted according to the problem considered. We review here two approaches to this problem, first a “microscopic” model, where the spatial variations of conductivity and permittivity are explicitly taken into account. In a second “macroscopic” model, these spatial variations are taken on average by using a mean-field formulation of Maxwell equations. Both of these models can describe the electromagnetic behavior of inhomogeneous media. We illustrate this formalism to describe the electric behavior of biological media, such as brain tissue, which is typically very inhomogeneous. We show that the theory predicts that for the typical frequency range of biological phenomena (lower than about 1000 Hz), the inhomogeneous nature of the medium has a determinant influence.