Electrodynamics of Rotating Systems

A phenomenological and electron-theoretical study of the electrodynamics of rotating systems is developed in the threedimensional form. In the former part of the present paper, the electromagnetic field equations in an isotropic and homogeneous medium performing arbitrary accelerated motion relative to an inertial reference frame are derived from the phenomenological viewpoint, under the assumptions that the observer is at rest in the inertial or rotating reference frame and that the macroscopic properties of the material medium are not changed by the acceleration acting upon it. With the aid of the classical electron theory, the latter part of the present paper discusses the electromagnetic field equations in rotating media as viewed from the inertial or rotating frame, presenting an electrontheoretical basis for the electrodynamics of rotating media. In addition, the electromagnetic effects caused by the inertial forces accompanying the rotational motion, i.e., the centrifugal and Coriolis forces, are investigated in detail. I . INTRODUCTION T HE ELECTRODYNAMICS of moving media was founded in 1908 by Minkowski {l], on the basis of the special theory of relativity. Although new formulations [4]-[6] to replace Minkowski’s theory [2], [3] have been presented by several authors after its appearance, it has been recently clarified by Tai [7] that they are not independent of the Minkowski formulation but rather can be deduced directly from i t by the appropriate linear transformations.1 Thus we may now consider Minkowski’s theory as giving the basis for the macroscopic electromagnetism. However, Minkowski’s theory seems to be unsatisfactory in that it can be applied only to the electromagnetic phenomena in inertial reference Manuscript receivedMay 7, 1973; revised August 6, 1973. Engineering, Osaka University, Suita, Osaka 565, Japan. The author is with the Department of Electrical Communication 1 Recently, a comparative study of various formulations of electrodynamics of moving media has atso been presented by Penfield and Haus [SI, but they have not discussed the constitutive relations between field vectors. As pointed out by Tai [7 1, the constitutive relations play an important role in the electrcdynamiar of moving media. frames: i t is applicable only to the case where both observers and material media are at rest in inertial frames. Most recently, several attempts [9]-[12] have been made to extend Minkowski’s theory to the case that observers or material media are performing accelerated motion with respect to inertial frames. In these attempts, the electromagnetic field equations in accelerated systems have been obtained from those in inertial reference frames by means of the mathematical tool in the general theory of relativity, Le., the tensor calculus in the four-dimensional Riemannian space [2], [IS][IS] . Although the tensor calculus is an elegant and useful means for describing the physical phenomena in accelerated systems, it is also possible, as in the case of the MaxwellMinkowski equations in inertial frames [16], [22], to develop the electrodynamics of accelerated systems in the threedimensional form, without using the tensor calculus. The purpose of this paper is to derive, by the latter method, the electromagnetic field equations for the typical cases of accelerated motion that observers or material media are uniformly rotating with respect to inertial reference frames. In this article, let us discuss the aforementioned problem from two distinct points of view, namely, from the phenomenological and electron-theoretical viewpoints. By the phenomenological viewpoint is meant a viewpoint in which one describes the electromagnetic effects of matter in terms of given parameters such as permittivity E, permeability p, and conductivity u, without inquiring into its internal structures. In this sense, the works presented so far [9]-[ 121 stand all on the phenomenological viewpoint. In the electron-theoretical point of view, on the other hand, material media are regarded as consisting of positively charged nuclei and negatively charged electrons, and their electromagnetic effects are taken into account by considering the interaction f electromagnetic fields with the aggregate of these charged particles. For the case of material media performing accelerated motion, in SHIOZAWA: ELECTRODYNAMICS OF ROTATING SYSTEMS general, their macroscopic properties may be considered to be modified by the acceleration acting upon them. To take the effects caused by acceleration into account, one must rely upon the electron theory, and thus the phenomenological treatment cannot be justified as long as the change in the macroscopic properties of matter due to the acceleration or the inertial forces cannot be neglected. In the subsequent sections, we first consider, from the phenomenological point of view, the electromagnetic field equations in an isotropic and homogeneous medium with permittivity c, permeability p, and conductivity u which is performing arbitrary accelerated motion, with the observer assumed to be a t rest in an inertial or rotating reference frame. With the aid of the classical electron theory [19], [20], we next discuss the electromagnetic field equations in rotating media as viewed from an inertial or rotating reference frame, giving an electron-theoretical basis for the electrodynamics of rotating media. In addition, we investigate in detail the electromagnetic effects produced by the inertial forces appearing in the rotational motion, Le., the centrifugal and Coriolis forces. 11. PHENOMENOLOGICAL CONSIDERATION A . Electromagnetic Field Equations in Inertial Reference Frame2 Let us first consider the electromagnetic field equations for arbitrarily moving media as viewed from an inertial reference frame. Let a reference frame I be an inertial frame, and let an isotropic and homogeneous medium with permittivity e, permeability p, and conductivity u be performing an accelerated motion relative to the inertial frame I with velocity urn(%, y, e, t ) which is generally nonuniform in space and time. We also assume that the change in the macroscopic properties of the medium due to the acceleration acting upon it may be neglected: the constitutive parameters of the medium, E , p, and u are constants independent of its acceleration. Then we can show that Minkowski's theory still holds good in the inertial reference frame I under these conditions [23], [24]. Namely, in the inertial frame I , we have the basic field equations aB V X E = , V * B = 0 at together with the constitutive relations