Electrostatic charges in v × B fields and the phenomenon of induction

The appearance of electrostatic charges in a moving conductor subjected to a static magnetic field is reviewed, and the ensuing electric field is shown to obey Faraday’s law of induction. The charge density and the electric field are determined analytically in detail in the case of a circular loop rotating in a uniform magnetic field. The case of a non-conductor moving in a magnetic field is also dealt with. Non-relativistic reasoning and calculations are used throughout. 1. Statement of the issue It is well known that, if an electric conductor is moved in a static magnetic field B, electric currents may be induced in the bulk. Lorrain (1990) argued that the conductor is also the seat of free charge. The free-charge density gives rise to an electric field E which subsequently transports the carriers together with the electromotive (that is, v × B) field, where v is the velocity of a point of the conductor. Redžić (2001) corrected two relatively minor points in Lorrain’s article, putting the reflection in the framework of the electrodynamics of moving media, and further developed the relativistic aspect of the problem in a particular geometry (2002). Lorrain (2001) remarked that not only text-books, but also many papers on the magnetohydrodynamics of the Earth and the Sun, were fraught with misconceptions due to the disregard of the free charge generated by the electromotive field. The issue therefore concerns not only undergraduate-level electromagnetism, but also the leading edge of research. It is the purpose of the present paper to address that issue again from a different perspective which, it is hoped, will fully clarify the most basic points. The salient features of our treatment are fourfold: (i) we keep within the non-relativistic realm, in the Galilean limit appropriate to the description of usual electromechanical generators and motors, but we carefully specify the reference frames involved; (ii) a rigid ohmic conductor is considered throughout; (iii) the link-up with Faraday’s law of induction is dealt with and (iv) the response of the conductor to the electromotive field is compared with that of a perfect dielectric. 0143-0807/03/010021+09$30.00 © 2003 IOP Publishing Ltd Printed in the UK 21