Charles A Coulson
Edinburgh and London
The author of a book on electricity must decide from the first whether his approach is mainly mathematical or physical; efforts at a satisfactory synthesis of the two have not hitherto proved very successful. The present book is intended to outline from the very beginning a consistent mathematical account of the phenomena of electricity and magnetism. In many respects the field covered is similar to that of Maxwell's Classical Treatise on Electricity and Magnetism. The present book differs from his in being much shorter, in assuming a working knowledge of vector notation, and in making use where necessary of the atomic viewpoint of modern physics. The introduction of an atomic viewpoint is made possible by the fact that most students are now familiar with the main outline of atomic theory: it is also desirable since a habit of thinking in atomic terms and relating atomic behaviour to macroscopic phenomena should be encouraged as soon as possible.
The insistence that electrical phenomena have their origin in atomic properties leads to one important change from the usual: the theory of magnetism is developed entirely independently of the notion of magnetic poles. Since the basis of all magnetism is some form of atomic current, it is wiser to develop the theory of magnetism from the known laws of interaction of currents; there is no need to introduce the postulate of magnetic poles, except for pictorial purposes. Indeed, since an isolated magnetic pole does not exist, the consistency of the argument is improved if no reference is made to any hypothetical law of force between such entities.
On grounds of space it has been necessary to omit certain items: in particular there are very few details given of the experiments which are needed to test the theory at important points; nor is there any account of applied electricity, such as dynamos, motors, influence-machines, post-office boxes, etc. These may all be found in any standard physical text-book. The theory of the earth's magnetism has been only lightly touched upon, and electrolysis has been completely omitted, for this, like the theory of electrons, belongs more properly to the field of quantum theory, and is outside the range of this book. But with these omissions, an attempt has been made to build a self-consistent mathematical theory, introducing as few postulates as possible, and capable of explaining all the more familiar phenomena of electrostatics, magnetism and electrodynamics.
A word is necessary in the matter of units. These nearly always cause the student a lot of trouble, and experience has shown that the use of practical units from start to finish does not make the fundamental ideas any clearer. For these reasons the discussion of units and dimensions has been deferred to a final chapter, and very little direct reference to them, except in the broad distinction between the electromagnetic and electrostatic systems, is made in the main body of the text.
No-one really understands any branch of mathematics until he has worked a good many examples in it. Accordingly there are examples, many of them embodying important results, at the end of each chapter. An average student should be able to solve at least half of these, though in some of the others he would require help.
It is a pleasure to acknowledge the guidance that I have received from Professors E T Copson and G S Rushbrooke, who have enabled me to remove some errors and have pointed out several obscurities: to them, and to my wife who has helped me with the preparation of the manuscript, I offer my grateful thanks.
Yet this book would be incomplete without a reference to my former teacher, Mr E Cunningham, of St John's College, Cambridge, who first showed me how beautifully vector methods fitted the subject of electricity. Much of what is best in this account must be attributed, directly or indirectly, to his influence.
KING'S COLLEGE, London
§ 1. Electrostatics
THE fact that a piece of amber, when rubbed, will attract small particles of matter, was known 2500 years ago by Thales of Miletus (640-548 B.C.). From this simple experimental fact has developed the whole science of electrostatics, that is, the properties of electricity at rest. Indeed, the very word electricity is derived from the Greek word for amber. During the years since Thales, and especially in the last 150 years, more experimental knowledge has been accumulated. This knowledge has seldom been obtained in the most systematic order; so our best policy in this book will not be to report the various experiments stage by stage as they were first made, but rather to start with a general survey of our present knowledge. In later chapters we shall from time to time briefly refer to the crucial experiments needed to establish or confirm each particular point of the theory.
We know now that electricity consists of two kinds - positive and negative charges. Like charges repel each other, but opposite charges attract. If it were not for this latter fact our material universe would not hold together at all. The smallest negative charge which it is possible to obtain is that of the ordinary electron, discovered and measured for the first time by J J Thomson in 1897. The smallest positive charge has the same numerical value, and is found on the proton; the same charge exists on the positive electron, a much lighter particle than the proton, but this particle is not stable and we shall not need to discuss it. All charges are integral multiples of these fundamental units; but each unit is so very small that in any common electrical measurement the discreteness of electric charge will not affect us, and we may suppose that a given charge may be allowed any arbitrary numerical value. The smallness of the electronic unit in relation to ordinary measurements may be shown by the fact that in a 60-watt lamp at 200 volts approximately 2 × 1018 electronic units of charge flow along the filament per second. The masses of the electron and proton are extremely small, that of the electron being about 9 × 10-28 grams: the proton is about 1837 times as heavy. Neither the electron nor the proton is a strict mathematical point. One of the most important of the unsolved problems of electricity is the precise nature and size of these elementary particles; it is usual to assign a radius to them of the order of 10-13 cms. Since we cannot measure any distances as small as this, it will be quite in order for us to regard our charges as points, and we shall therefore refer, when necessary, to point charges.
Matter, as we understand it, consists of atoms: that is, of positive and negative charges associated together in small structures of about 10-8 cms. diameter. The positive charge rests on the heavy part, or nucleus, of each atom, and the negative charge is in the form of electrons that move round the nucleus in much the same way as planets round the sun. The precise manner in which this motion takes place need not concern us, and belongs to atomic theory. All that we need to know is that their motion can be somewhat changed, so that the atom becomes distorted, if the right kind of external force is applied. Normal matter consists of neutral atoms, i.e. atoms in which there is no excess of positive or negative charge. In 1 c.c. of an ordinary solid there are about 1023 such atoms. (In diamond, for example, the number is 4 × 1023.) But some of the atoms may have too few, or too many, electrons to balance the positive nucleus, and then we have net positive or net negative charges present. A study of the forces that these exert on each other, or on any neutral matter that may be present, forms the content of Chapters II-IV; this may be said to represent the science of electrostatics proper.
But this brings us to an important distinction. In the theory of electricity, in contrast with atomic physics, we are not primarily concerned with the forces exerted by one atom, or one electron, on another atom, since the forces and distances involved are far too small for us to measure individually in the laboratory. It is rather with the bulk effect, in which a large number of electrons and atoms are involved, that we shall be concerned. Thus if the smallest mass we can conveniently measure is taken to be 1/10 milligram, this would represent no less than 1023 electrons, or between 1018 and 1020 atoms, depending on the substance we are using. The distinction that we are making is between the macroscopic and macroscopic points of view. In the microscopic point of view we deal with individual atoms and electrons, which are the field of atomic physics and quantum theory. In the macroscopic point of view we average these forces over the large number of atoms in a tiny measurable volume; for that purpose it is often quite fair to neglect the peculiar individual atomic and inter-atomic effects revealed in a microscopic survey. For example, just as we referred earlier to point. charges, thereby neglecting the internal structure of an electron or proton, so also for many purposes we shall neglect the atomicity, or "graininess," of matter itself.
The question will now be asked: if we are not to take into account this detailed structure of matter, but are to use averages in which we have effectively smoothed it out to become homogeneous and continuous, what is the advantage of introducing the microscopic, or atomic, point of view at all? The answer is twofold. First, the microscopic viewpoint throws light on the fundamental physical processes; this enables us to view our subject as one whole and means that we shall not have to introduce from time to time apparently unrelated physical assumptions, for we shall see how our macroscopic equations arise quite naturally from simple microscopic properties of the atom and the electron. This is particularly important (see § 3) in discussing the relation between electric currents and magnetism. Secondly, the microscopic point of view prepares us for the more intimate detailed study of these atomic processes which is necessary if we are to understand fully the nature of our physical world.
§ 2. Electric currents
The study of electrostatics, or charges at rest, leads naturally to a study of electric currents, or charges in motion. The current may be caused by movement either of the positive or negative charges, or of both. Thus in a discharge tube positive ions (that is, atoms with an excess of positive charge) move in one direction, and electrons carrying a negative charge move in the opposite direction. It is possible by bending the beams magnetically to separate the two, and obtain a current composed solely of positive or negative charge. On the other hand, in metals, such as a copper wire, the charge is carried entirely by electrons. It makes no difference to our formulation of the laws of current flow, as we develop it in Chapter V, which type of carrier is bearing the charge, for in all cases the current is measured by the rate at which the charge flows, i.e. the net amount crossing unit area in unit time. The direction of the current is the direction in which the mean drift, or flow, of charge is taking place.
The distinction which we made in § 1 between microscopic and macroscopic measurement is important here. For on the microscopic point of view the charges are moving in all directions with all possible speeds; but on the macroscopic point of view, in which we consider merely the average motion of the charges within a very small volume, we determine a mean drift velocity, the magnitude and direction of which measure the electric current. The situation is rather like the flow of gas down a tube. According to the Kinetic Theory, the various particles of gas have all possible velocities in all directions; but the mean velocity lies along the direction of the tube, and for most purposes we may forget the random distribution of velocities and suppose that each particle of gas has, in fact, this mean "drift" velocity of flow down the tube.
For purposes of discussing the flow of current all substances may be placed in one of two categories-insulators and conductors. An insulator (e.g. amber, glass, shellac) is a substance in which it is practically impossible to cause any current to flow. The explanation is simple, for in these substances all the negative charges (or electrons) are firmly attached to corresponding positive charges. As we cannot easily separate them it follows that no net flow of charge can take place. A conductor, on the other hand, is a substance in which a certain number of electrons (or negative charges) are easily separated from their associated positive charges, and one or both can move under the influence of a force of the right kind. Thus in metals, such as copper or tungsten, there are a certain number of electrons, called metallic, or free, or conduction electrons, which are able to flow freely through the material and give rise to the current, while the positive charges remain fixed. But in electrolytes, such as the dilute sulphuric acid in an ordinary accumulator, each particle, or molecule, of electrolyte spontaneously separates into positively and negatively charged parts which can move independently of each other. In the case of sulphuric acid, protons move in one direction carrying a positive charge, and sulphate ions move in the opposite direction carrying a negative charge; the total current is the sum of these two separate currents.
§ 3. Magnetism
We have seen that a current may be measured by the quantity of charge flowing in unit time. This counting of charge, which is made with an electrometer, provides us with an electrostatic measure of current, and the result would naturally be expressed in electrostatic units, generally abbreviated to e.s.u.; we shall have more to say about these units later. But there is another entirely different way of measuring current; for when charges are flowing we discover a whole series of new phenomena, to which we give the name magnetism. In actual practice it is these magnetic, or more properly electromagnetic, effects that are most commonly used to measure currents, and in such cases our result will be expressed in electromagnetic units and written e.m.u. The same current may be measured in both units, and the relation between them, or, which is the same, the ratio of the corresponding units, is a matter of prime importance. We shall see in Chapter XIII how Maxwell was able to use its known value to show that light waves were essentially an electromagnetic phenomenon.
These magnetic effects, however, had been known in another connection for a very long time. Thus Lucretius mentions that certain mineral ores such as loadstone have the power of attracting small pieces of iron placed near them, and one of the earliest attempts at a perpetual motion machine makes use of this attraction. These forces were called magnetic forces, and the attracting materials were called permanent magnets. From this beginning there developed, quite independently of electrostatics, the science of magnetism, the study of which gained great importance when it was realised that the earth itself behaved like a large permanent magnet. It was the discovery of Oersted (1820) and Faraday (1831), that the same magnetic effects are produced by currents as by permanent magnets, which related the two hitherto distinct sciences. If electric currents are able to produce the same effects as permanent magnets it is natural to enquire whether the so-called permanent magnetism may not be due to currents of some form or other; and indeed it was not long before Ampère proposed the hypothesis that each elementary constituent of matter (as we should say, each atom) was really a minute electric current. This hypothesis is now accepted. If we go back to our earlier discussion of the atom we can soon see how this comes about, and what is the nature of Ampère's minute currents. We have seen that in an atom electrons move in orbits round the central nucleus; we have also seen that when charges move there is an electric current flowing. Combining these two facts it follows that in each atom there are indeed tiny electric currents. It may happen - and in fact very often does - that in each atom there are pairs of electrons moving in opposite directions, and in such a case the atomic currents cancel so that the atom is not a permanent magnet. It may equally well happen that although each atom is itself magnetic, these magnets are arranged in a given block of material in random directions; then again the substance is not a permanent magnet. But it may be that a majority of these tiny currents are oriented in the same direction, and then we do have a permanent magnet. In the case of substances which are not permanent magnets it is possible, by using the right kind of force, to alter the orbits of the electrons and in this way we may induce a temporary magnetism which vanishes when the disturbing force is removed. This is the phenomenon of induced magnetism. Thus the difference between substances which are or are not permanent magnets is not that they are made of essentially different material, but rather that with permanent magnets we have no way (or at any rate no simple way) of destroying the co-operative effect of the separate atoms, whereas with non-permanent magnets this co-operation is solely the result of forces exerted from outside, and automatically disappears when the force is removed.
(We have spoken of an electron in its orbit as being equivalent to a minute current. But there is another type of current, known to atomic physicists as spin, e.g. electron-spin or nuclear-spin, which we have not mentioned. It is good enough for our purposes to regard this as a flow of current inside a single electron or nucleus, giving rise to magnetic effects which may or may not average out to zero, similarly to the electron orbits previously described.)
Our study of the magnetic effects of currents and the properties of permanent magnets occupies Chapters VI-VIII. In olden days, following the historical sequence that we have already indicated, it was usual to develop the subject of magnetism quite separately from electrostatics, starting from the existence of permanent magnets. The two subjects were finally united through the phenomena of electromagnetism, rather in the way that the lintel of a door rests upon and links together the two door-posts. But we shall find it best to proceed from the electrostatic properties of currents to their electromagnetic properties, and this will then lead us naturally to our discussion of permanent magnetism.The great advantage of such a procedure is that every part of our scheme is immediately related to and follows from our knowledge of the structure of the atom, as we have indicated earlier in this section.
It turns out that there is a close parallelism between the laws of electrostatics and magnetism, so that the same type of mathematical analysis can be employed to solve problems in either field. We therefore break off, in Chapters IX and X, to discuss in detail a selected number of such problems, and to illustrate the technique required in their solution.
§ 4. Electrodynamics
We have so far been mainly concerned with steady currents. When currents do not remain constant we have to distinguish two cases, depending on whether the change is slow (known as quasi-steady currents) or rapid. In the first of these, associated particularly with Lenz and Faraday, we discover that if by any means we try to change the current flowing in any circuit then our apparatus behaves as if it were trying to prevent such change, i.e. it sets up new systems of currents whose effect, by themselves, is to counteract the original change. This is the phenomenon of electromagnetic induction which, in its developed form, underlies all the construction of dynamos, transformers and electric motors. Chapter Xl is devoted to a study of these quasi-steady
currents in general terms. Chapter XII outlines some of the extremely important applications of induction in the alternating current theory of electrical circuits. This is part of the whole field of wireless telegraphy.
We have by this time brought most of the common electric effects within the power of our equations, but not all. It was the peculiar genius of Maxwell that he recognised a previous omission. We have seen that under certain circumstances an atom may be deformed so that the positive and negative charges are slightly separated. While this separation is taking place there is a very small flow of charge, i.e. a small current. If the change is extremely rapid and the negative charge fluctuates from one side to the other of the positive charge then an oscillating current may be said to flow. This is one part of Maxwell's Displacement Current, and we shall see in Chapter XIII that although for quasi-steady or steady systems this new term is not effective, for sufficiently rapidly varying currents it often becomes dominant. Men it is included we are led to formulate' the general equations of the electromagnetic field, usually known as Maxwell's Equations. It is upon these equations that all later development rests. These equations show for the first time the possibility of electromagnetic waves, and by comparison with the velocity of light we infer that light itself is an electromagnetic phenomenon. A whole new field of experience, including such diverse phenomena as the laws of reflection and refraction and the energy radiated from aerials is brought within our scope. The equations themselves reveal a surprising symmetry with respect to electrostatic and magnetic effects. We have seen that electric charges in motion produce the same effect as magnets at rest; it now appears that magnets in motion produce the same result as charges at rest. The link between electrostatics and magnetism is not only that the mathematical technique is the same for both; rather this latter fact is itself the expression of an organic unity-they are both parts of the complete study of electrodynamics.
This is about as far as our macroscopic theories can profitably carry us. Further progress depends upon a more detailed study of microscopic, or atomic, properties. Such studies have been made, but they introduce us to two essentially new ideas - the quantum theory and the wave nature of matter. As such they are outside the scope of this book.