The Equation of an Electromagnetic Field in a Moving Magnetically Anisotropic Conducting Medium

In most problems concerning calculation of an electromagnetic field, it is reasonable to use vector magnetic potential which is an auxiliary function that meets the condition = rot In vector algebra, div rot = 0; therefore, in the calculation of a field, one of the main conditions, div = 0, is always met. Since the rotor of the gradient of any scalar quantity is zero, the gradient of an arbitrary scalar is introduced into the left-hand part of the second Maxwell equation. The time derivative of this scalar is not a factor that can change magnetic and electric fields. Thus, using the basic equations of an electromagnetic field, its potentials can be determined with an accuracy equivalent to the scalar gradient. Such invariance is called the “gauge” or “gradient” invariance. To eliminate this uncertainty when finding the field potentials, they are subjected to an additional condition set by a special gauge. For an electromagnetic field propagating in vacuum, the Lorentz gauge is known; for a conducting medium, this gauge loses its meaning. When choosing a gauge for conducting media, the condition div = 0 should be met; therefore, it is reasonable to use it as a gauge one. It is shown that, in solving a one-dimensional problem, div = 0 can be taken to be zero, while a special gauge of the potential field is required in solving two- or three-dimensional problems. It is proposed to use the condition div = 0 as a gauge, since it is satisfied in all conducting media.