Introduction to Electrodynamics

MAGNETIZATION

Diamagnets, Paramagnets, Ferromagnets

If you ask the average person what “magnetism” is, you will probably be told about refrigerator decorations, compass needles, and the North Pole—none of which has any obvious connection with moving charges or current-carrying wires. Yet all magnetic phenomena are due to electric charges in motion, and in fact, if you could examine a piece of magnetic material on an atomic scale you would find tiny currents: electrons orbiting around nuclei and spinning about their axes. For macroscopic purposes, these current loops are so small that we may treat them as magnetic dipoles. Ordinarily, they cancel each other out because of the random orientation of the atoms. But when a magnetic field is applied, a net alignment of these magnetic dipoles occurs, and the medium becomes magnetically polarized, or magnetized.

Unlike electric polarization, which is almost always in the same direction as E, some materials acquire a magnetization parallel to B (paramagnets) and some opposite to B (diamagnets). A few substances (called ferromagnets, in deference to the most common example, iron) retain their magnetization even after the external field has been removed—for these, the magnetization is not determined by the present field but by the whole magnetic “history” of the object. Permanent magnets made of iron are the most familiar examples of magnetism, but from a theoretical point of view they are the most complicated; I'll save ferromagnetism for the end of the chapter, and begin with qualitative models of paramagnetism and diamagnetism.

Torques and Forces on Magnetic Dipoles

A magnetic dipole experiences a torque in a magnetic field, just as an electric dipole does in an electric field. Let's calculate the torque on a rectangular current loop in a uniform field B. (Since any current loop could be built up from infinitesimal rectangles, with all the “internal” sides canceling, as indicated in Fig. 6.1, there is no real loss of generality here; but if you prefer to start from scratch with an arbitrary shape, see Prob. 6.2.) Center the loop at the origin, and tilt it an angle θ from the z axis towards the y axis (Fig. 6.2). Let B point in the z direction.