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Maxwell's Equations

  Maxwell's equations (in macroscopic form and MKSA units) are

(Other system units are discussed e.g. in [Jackson75].)

is the electric field, is the displacement, the permittivity of free space and the polarization. is the magnetic induction (magnetic flux density), the magnetic field, the permeability of free space and the magnetization. is the density of electric charge and is the current density. The relations between and , and between and , are called constitutive equations; they describe the medium. In a linear, isotropic medium and , where and are constants. In general (or ) is not even a unique function of (or ), but depends upon the earlier time evolution (hysteresis).

where c is the speed of light in vacuum and by definition

The continuity equation

follows from Maxwell's equations and expresses the conservation of electric charge.

Defining the scalar potential and vector potential by

explicitly solves half of Maxwell's equations. The potentials are not unique, since any gauge transformation

leaves the physical fields and unchanged, being an arbitrary function.

Other types of potentials may be useful in certain cases (e.g. a scalar potential for ).

Rudolf K. Bock, 9 April 1998