Chapter 3
MAXWELL'S EQUATIONS IN MAGNETIC MEDIA

The most striking feature of a magnetic material is the fact that it can give rise by itself to a static magnetic field in outer space. According to Maxwell's equations, a static magnetic field is produced by stationary electric currents, and the problem arises of the nature of the currents responsible for this field. The question is non trivial and poses deep conceptual difficulties, as confirmed by the fact that the existence of permanent magnets has been known to mankind since ancient times, but the discovery of the relationship between this kind of magnetism and electric currents is less than two centuries old, and Ampère's intuition of the existence of molecular currents inside matter could be given a scientific basis only with the advent of relativistic quantum mechanics.
    The basic quantum nature of the magnetism of magnetic materials raises the point of how these quantum effects should be treated in the classical setting of macroscopic Maxwell's equations. An acceptable solution is to postulate the existence in magnetized matter of elementary pointlike and permanent magnetic moments, and to describe a magnetic material as a collection of such moments. All quantum effects are lumped in the properties of the individual moments. By accepting their existence as an additional fact, to be added to the existence of electric charges, one is able to describe the behavior of magnetic materials by purely classical means, in terms of solutions of magnetostatic Maxwell's equations.
    A second relevant point is that the motion of elementary charge and moment carriers in magnetic bodies is extremely intricate and irregular. A detailed treatment would be hopeless. However, if we are interested in effects taking place on a sufficiently coarse scale, we can get rid of these complications by taking convenient space averages over elementary volumes, small enough with respect to the characteristic scale of interest, but still large enough to contain at any time a substantial number of particles. [...] By working with these local averages, one looses the fine details of the processes occurring inside each elementary volume, but one obtains a description in terms of smooth quantities, perfectly suited to the study of phenomena taking place over a scale much larger than that of the elementary volumes.
    In the following sections, we shall concentrate on those aspects of Maxwell's equations that are of direct relevance to magnetic materials. No attempt is made to give a comprehensive presentation of the general properties of Maxwell's equations, for which many texts can be found in the literature, nor to give detailed derivations of all the relations that will be stated. In Section 3.1, we briefly summarize Maxwell's equations, we discuss magnetostatic equations and we introduce the concept of elementary magnetic moment. Section 3.2 discusses how the magnetic state of a body can be characterized in terms of the magnetization vector. Finally, Section 3.3 discusses energy conservation and energy dissipation caused by eddy currents.
Giorgio Bertotti
Materials Department, IEN Galileo Ferraris Corso Massimo d'Azeglio 42, I-10125 Torino, Italy