Equations of electrodynamics in a rotating solid dielectric

Abstract

The equations of electrodynamics in a rotating isotropic homogeneous dielectric are obtained in a covariant form in coordinates of a reference frame that accompanies the rotation of the dielectric. It is found from these equations, which have variable coefficients, that the medium of the rotating dielectric is anisotropic and inhomogeneous. To derive tensors of the electromagnetic field in a rotating reference frame (RRF), the fields and inductions of a virtual inertial reference frame (IRF) that instantaneously accompanies the motion of one of the points of the dielectric are used twice. Initially, using instantaneous local relations, they are expressed in terms of real fields and inductions of the rotating medium, and then they are transformed into fields and inductions of a stationary IRF, in which they are used as components of the tensors of the electromagnetic field. Thus, the electromagnetic field tensors in the IRF are determined taking into account a priori unknown real inhomogeneous permittivity $$\bar \varepsilon $$ and permeability $$\bar \varepsilon $$ of the rotating medium. At the final stage, the tensors in the RRF are obtained by transformation rules for covariant and contravariant tensor components in accordance with known analytical relationships of fixed and rotating coordinates. The properties of modes of a rotating ring resonator in the form of homogeneous TE waves that travel along and against the direction of rotation and, in particular, their normal frequencies are examined. The contribution of inhomogeneous properties of the medium of a rotating dielectric to the difference between the normal frequencies of the counterpropagating waves (to the Sagnac effect) is determined. In a solid material with known elastic and striction characteristics, its density and dielectric permittivity depend on the radial coordinate. These dependences are caused by the action of the centrifugal force and changes in the polarization and magnetization of the medium because of the rotational motion of charged particles. The coordinate dependences of permittivity $$\bar \varepsilon $$ and permeability $$\bar \varepsilon $$ make additional contributions to the inhomogeneous properties of the medium of the rotating dielectric and to the Sagnac effect.