The Mathematical Model

Abstract

Electromagnetic phenomena in a space–time domain $\mathbb {R}^4_>:=\mathbb {R}^3\times (0,\infty )$ can be governed by the system of macroscopic Maxwell’s differential equations

$$\begin{aligned} \begin{array}{r@{\ }c@{\ }l@{\quad }r@{\ }c@{\ }l} \dfrac{1}{c}\dfrac{\partial {\mathbf B }}{\partial t} + \nabla \times {\mathbf E }&{}=&{} 0, &{}\dfrac{1}{c}\dfrac{\partial {\mathbf D }}{\partial t} - \nabla \times {\mathbf H }&{}=&{}-\dfrac{4\pi }{c}{\mathbf J }\,,\\ \nabla \cdot {\mathbf D }&{}=&{} 4\pi \rho , &{} \nabla \cdot {\mathbf B }&{}=&{} 0, \end{array} \end{aligned}$$

where the Gaussian unit system is used (see, for example, Born and Wolf in Principles of Optic, Pergamon Press, Oxford, 1970, [1, Sect. 1.1.1], Landau et al. in Electrodynamics of Continuous Media, Elsevier Butterworth-Heinemann, Oxford, 1984, [2, Chap. IX]). Here, ${\mathbf E },$ ${\mathbf H },$ ${\mathbf D },$ ${\mathbf B }:\,\mathbb {R}^4_>\rightarrow \mathbb {R}^3$ denote the unknown vector fields of electric and magnetic field intensity, electric and magnetic induction, respectively, c is a positive constant—the velocity of light. The function $\rho :\,\mathbb {R}^4_>\rightarrow \mathbb {R}$ and the vector field ${\mathbf J }:\,\mathbb {R}^4_>\rightarrow \mathbb {R}^3$ are called the electric charge density and the electric current density, respectively. These macroscopic quantities are obtained by averaging rapidly varying microscopic quantities over spatial scales that are much larger than the typical material microstructure scales. Details of the averaging procedure can be found in standard electrodynamic textbooks, for instance, in Jackson, Classical Electrodynamics, Wiley, New York, 1999, [3].