We have so far in this book regarded the sources ρ, J of the electromagnetic field as given. However, the charges and currents in material media are themselves driven by the fields, so that we need to describe the electrical and magnetic responses of materials to an electromagnetic field. At the atomic level, the Coulomb forces between electrons and atomic nuclei are responsible for their binding into atoms and molecules, and the large scale structure of materials. A description at this level involves the complicated quantum mechanics of the constituent particles of the materials, and is the province of condensed matter physics and material science. For the most part, we shall rather be concerned with the macroscopic electrical and magnetic properties of materials, which can often be described phenomenologically by a small number of parameters, such as the electrical conductivity. These parameters are usually obtained by direct experiment on the material concerned.

We begin in this chapter with a description of conductors in electrostatic equilibrium.

Electrostatic equilibrium

A conductor is a material in which there are electrons, or ions, free to migrate and transport charge in response to an electric field. In a metal or semiconductor the charge carriers are electrons. The principal property of any homogeneous conductor in electrostatic equilibrium is that the electric field E(r) = 0 at all interior points r. If E(r) were not zero, the mobile charge carriers would move in response to the mean Coulomb force, until a charge distribution was established for which the condition held.

Electric charge and conservation of charge

The basic constituents of matter, electrons and atomic nuclei, are all endowed with electric charge. It is through the electromagnetic fields generated by these charges that electrons and nuclei interact to form atoms and molecules and, hence, all materials. An electron carries a negative charge – e, and an atomic nucleus a positive charge Ze, where Z is an integer ranging from Z = 1 for hydrogen to Z = 92 for uranium (and higher for some unstable nuclei). The SI unit of charge is the coulomb (C) and e ≈ 1.602 × 10-19 C.

The assignation of negative and positive sign is no more than a convention, which was set in the eighteenth century by the American physicist and statesman Benjamin Franklin. It is, however, a profound law of nature that, in an isolated system, the net total charge will never change: charge is conserved. In much of physics and chemistry neither an electron nor an atomic nucleus is ever created or destroyed, and charge conservation follows from this. More generally, the processes of nuclear physics do create and annihilate electrons, and transmute nuclei, but no known physical process can change the net total charge of an isolated system.

Charge density

At the present limits of experimental resolution, electrons seem to be ‘point particles’, in the sense that no intrinsic size or structure has yet been discerned for them.

Oersted's discovery of the magnetic effect of currents not only stimulated renewed interest in electricity and magnetism, but also led to the development of sensitive instruments which used the deflection of magnets to measure currents; a simple galvanometer was devised by Schweigger in 1820, and the more sensitive astatic galvanometer by Nobili in 1825. Previously, electric currents could only be detected by the observation of sparks, or by their chemical effects in electrolysis. The latter method was quantitative, but not well suited to the detection of small currents. The use of the galvanometer was important in the experimental work of Faraday at the Royal Institution in London, where in 1831–2 he carried out a now famous series of experiments on the induction of electric currents by magnetic fields.

Faraday found that a current was induced to flow round a closed conducting circuit when a nearby magnet was moved, or the current in a nearby circuit was changed, or when the circuit was moved in a fixed magnetic field. In all these cases he established that the induced current was proportional to the rate of change of magnetic flux through the circuit.

Faraday's law of induction

The current flow that Faraday observed when a coil of wire was moving in a static magnetic field B(r) can be understood in terms of effects we have already discussed.