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Classical formulation

Let us denote the density of electric current by $J$, the electric field by $E$, the electric displacements by $D$, the magnetic field by $H$ and the magnetic induction by $B$. Then are valid in a domain $G$ the following laws:
\begin{eqnarray*}c\nabla \times H&=&4\pi J+\partial _t D\quad (\mbox{Ampère's l......\\ [2ex]\nabla\cdot B&=&0\quad(\mbox{no free magnetic charges})\end{eqnarray*}
where $c$ is the velocitiy of the light in vacuum and $\rho$ the electric charge density. A coupling with constitutive laws leads to
\begin{eqnarray*}D=\varepsilon E,\quad J=\sigma (E+g)\quad\mbox{and}\quad B=f(H)\end{eqnarray*}
where $f:\R^3\to \R^3$ is a monotonic function and has for instance to represent the behaviour of soft iron for high field saturation. The function $g$ is a given electric source. In so-called chiral materials is assumed the Drude-Born-Fedorov constitutive law. Let be $\varepsilon $ the electric permittivity, $\mu$ the magnetic permeability, $\beta$ the chirality measure then the constitutive relations read as follows:
\begin{eqnarray*}D=\varepsilon (E+\beta{\rm rot} E),\quad B==\mu(H+\beta{\rm rot} H).\end{eqnarray*}


After a simple calculation one can find in case of an achiral material with constants $\mu$ and $\varepsilon $:

\begin{eqnarray*}\varepsilon \partial _{tt} B+4\pi\sigma \partial _t B+ c^2(\na......nabla\times B)=4\pi c\sigma \nabla\times g\quad\mbox{in}\quad G.\end{eqnarray*}
In an isotropic chiral medium an has the socalled Tellegen representation :
\begin{eqnarray*}D=\varepsilon E+\alpha H\quad B=\mu H+\beta E.\end{eqnarray*}
In a source-free region we have
\begin{eqnarray*}&&{\rm rot}\; E=i\nu\; B,\\ [2ex]&&{\rm rot}\; H=-i\nu\; D.\end{eqnarray*}


Sometimes one has $f(H)=\mu H$. Often the displacement current term $\partial _t D$ can be neglected. Then the equation simplifies.
 

A late reference: Augusto Visintin, (1996) Models of Phase transitions, Progress in Nonlinear Differential Equations and their Applications, Volume 28, Birkhäuser Basel.
 


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collected and worked out by Wolfgang Sprößig, TU-Bergakademie Freiberg