Particles in an Electromagnetic Field

Abstract

It is well known [JA 75] that in classical electrodynamics, an electromagnetic field is described by the electric and magnetic fields E(r; t),B(r; t). These fields satisfy Maxwell’s equations, which in unrationalized Gaussian units become

$$\begin{gathered}\nabla \times E\left( {r;t} \right) + \frac{1}{c}\frac{{\partial B\left( {r;t} \right)}}{{\partial t}} = 0, \hfill \\\nabla \cdot E\left( {r;t} \right) = 4\pi {\ell _e}\left( {r;t} \right), \hfill \\\nabla \times B\left( {r;t} \right) - \frac{1}{c}\frac{{\partial E\left( {r;t} \right)}}{{\partial t}} = \frac{{4\pi }}{c}{J_{em}}\left( {r;t} \right), \hfill \\\end{gathered} $$

(12.1)

where ϱ_e(r; t) and J _em (r; t) are respectively the charge and current densities, related by the continuity equation

$$\frac{{{\partial _{{\varrho _e}}}\left( {r;t} \right)}}{{\partial t}} + \nabla \cdot{J_{em}}\left( {r;t} \right) = 0$$

(12.2)

.