What are we trying to do?

In this chapter we are going to discuss the problem of obtaining the best description of our data in terms of some theory, which involves parameters whose values are initially unknown. Thus we could have data on the number of road accidents per year over the last decade; or we could have measured the length of a piece of metal at different temperatures. In either of these cases, we may be interested to see (i) whether the data lie on a straight line, and if so (ii) what are its gradient and intercept (see Fig. 2.1).

These two questions correspond to the statistics subjects known as Hypothesis Testing and Parameter Fitting. Logically, hypothesis testing precedes parameter fitting, since if our hypothesis is incorrect, then there is no point in determining the values of the free parameters (i.e. the gradient and intercept) contained within the hypothesis. In fact, we will deal with parameter fitting first, since it is easier to understand. In practice, one often does parameter fitting first anyway; it may be impossible to perform a sensible test of the hypothesis before its free parameters have been set at their optimum values.

Various methods exist for parameter determination. The one we discuss here is known as least squares. In order to fix our ideas, we shall assume that we have been presented with data of the form shown in Fig. 2.1, and that it corresponds to some measurements of the length of our bar yiobs at various known temperatures xi.

In Section 1.5 we discussed the Gaussian distribution. In this and the next appendix, we describe the binomial and Poisson distributions.

Let us imagine that we throw an unbiassed die 12 times. Since the probability of obtaining a 6 on a single throw is 1/6, we would expect on average to end up with two 6's. However, we would not be surprised if in fact we obtained 6 once or three times (or even not at all, or four times). In general, we could calculate how likely we are to end up with any number of 6's, from none to the very improbable 12.

These possibilities are given by the binomial distribution. It applies to any situation where we have a fixed number N of independent trials, in each of which there are only two possible outcomes, success which occurs with probability p, or failure for which the probability is 1 — p. Thus, in the example of the previous paragraph, the independent trials were the separate throws of the die of which there were N = 12, success consisted of throwing a 6 for which the probability p =1/6, while failure was obtaining any other number with probability 5/6.

The requirement that the trials are independent means that the outcome of any given trial is independent of the outcome of any of the others. This is true for a die because what happens on the next throw is completely unrelated to what came up on any previous one.

In the chapters which follow, we assume that you are already familiar with the basic mathematics of scalar and vector fields in three dimensions, the properties of the ∇ operator, the integral theorems which hold for these fields, and so forth. In this prologue, we remind you of some basic definitions, and outline (without proof) those mathematical theorems of which we shall make extensive use. We also establish our notation and sign conventions.

We envisage space filled with electromagnetic fields, and at any instant we describe these fields mathematically using functions which may be scalar functions of position (like the potential Φ(r)) or vector functions of position (like the electric field E(r)). We shall assume that the functions which appear in the theory are continuous, and have derivatives existing as required, except perhaps at special points or on special surfaces. Singularities in the mathematics will usually correspond to singularities in the physics. For example, the electrostatic potential of a point charge Q at the origin is Q/4πε0r, and this function satisfies our conditions except at r = 0, which is the position of the point charge.

We sometimes focus on these fields in limited regions of space, say inside a volume V enclosed by a surface S, or over a surface S(Γ) bounded by a curve Γ.

Volume integrals

Volume integrals will often arise naturally in the theory, for example when we calculate the total charge or total energy in some volume V of space.

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.