PHYSICAL PRINCIPLES.

It has been seen that, on moving a magnetic pole about in the presence of electric currents, there is a certain amount of work done on the pole by the forces of the field. If the conservation of energy is to be true of a field of this kind, the work done on the magnetic pole must be represented by the disappearance of an equal amount of energy in some other part of the field. If all the currents in the field remain steady, there is only one store of energy from which this amount of work can be drawn, namely the energy of the batteries which maintain the currents, so that these batteries must, during the motion of the magnetic poles, give up more than sufficient energy to maintain the currents, the excess amount of energy representing work performed on the poles. Or again, if the batteries supply energy at a uniform rate, part of this energy must be used in performing work on the moving poles, so that the currents maintained in the circuits will be less than they would be if the moving poles were at rest.

Let us suppose that we have an imaginary arrangement by which additional electromotive forces can be inserted into, or removed from, each circuit as required, and let us suppose that this arrangement is manipulated so as to keep each current constant.

[TO THE FIRST EDITION]

There is a certain well-defined range in Electromagnetic Theory, which every student of physics may be expected to have covered, with more or less of thoroughness, before proceeding to the study of special branches of developments of the subject. The present book is intended to give the mathematical theory of this range of electromagnetism, together with the mathematical analysis required in its treatment.

The range is very approximately that of Maxwell's original Treatise, but the present book is in many respects more elementary than that of Maxwell. Maxwell's Treatise was written for the fully-equipped mathematician: the present book is written more especially for the student, and for the physicist of limited mathematical attainments.

The questions of mathematical analysis which are treated in the text have been inserted in the places where they are first needed for the development of the physical theory, in the belief that, in many cases, the mathematical and physical theories illuminate one another by being studied simultaneously. For example, brief sketches of the theories of spherical, zonal and ellipsoidal harmonics are given in the chapter on Special Problems in Electrostatics, interwoven with the study of harmonic potentials and electrical applications: Stokes' Theorem is similarly given in connection with the magnetic vectorpotential, and so on. One result of this arrangement is to destroy, at least in appearance, the balance of the amounts of space allotted to the different parts of the subject.

PHYSICAL PRINCIPLES.

If two conductors charged with electricity to different potentials are connected by a conducting wire, we know that a flow of electricity will take place along the wire. This flow will tend to equalise the potentials of the two conductors, and when these potentials become equal the flow of electricity will cease. If we had some means by which the charges on the conductors could be replenished as quickly as they were carried away by conduction through the wire, then the current would never cease. The conductors would remain permanently at different potentials, and there would be a steady flow of electricity from one to the other. Means are known by which two conductors can be kept permanently at different potentials, so that a steady flow of electricity takes place through any conductor or conductors joining them. We accordingly have to discuss the mathematical theory of such currents of electricity.

We shall begin by the consideration of the flow of electricity in linear conductors, by a linear conductor being meant one which has a definite cross-section at every point. The commonest instance of a linear conductor is a wire.

Definition. The strength of a current at any point in a wire or other linear conductor, is measured by the number of units of electricity which flow across any cross-section of the conductor per unit time.

GENERAL THEORY OF DYNAMICAL SYSTEMS.

We have so far developed the theory of electromagnetism by starting from a number of simple data which are furnished or confirmed by experiment, and examining the mathematical and physical consequences which can be deduced from these data.

There are always two directions in which it is possible for a theoretical science to proceed. It is possible to start from the simple experimental data and from these to deduce the theory of more complex phenomena. And it may also be possible to start from the experimental data and to analyse these into something still more simple and fundamental. We may, in fact, either advance from simple phenomena to complex, or we may pass backwards from simple phenomena to phenomena which are still simpler, in the sense of being more fundamental.

As an example of a theoretical science of which the development is almost entirely of the second kind may be mentioned the Dynamical Theory of Gases. The theory starts with certain simple experimental data, such as the existence of pressure in a gas, and the relation of this pressure to the temperature and density of a gas. And the theory is developed by shewing that these phenomena may be regarded as consequences of still more fundamental phenomena, namely the motion of the molecules of the gas.

THE THREE DIVISIONS OF ELECTROMAGNETISM

1. The fact that a piece of amber, on being rubbed, attracted to itself other small bodies, was known to the Greeks, the discovery of this fact being attributed to Thales of Miletus (640–548 B.C.). A second fact, namely, that a certain mineral ore (lodestone) possessed the property of attracting iron, is mentioned by Lucretius. These two facts have formed the basis from which the modern science of Electromagnetism has grown. It has been found that the two phenomena are not isolated, but are insignificant units in a vast and intricate series of phenomena. To study, and as far as possible interpret, these phenomena is the province of Electromagnetism. And the mathematical development of the subject must aim at bringing as large a number of the phenomena as possible within the power of exact mathematical treatment.

2. The first great branch of the science of Electromagnetism is known as Electrostatics. The second branch is commonly spoken of as Magnetism, but is more accurately described as Magnetostatics. We may say that Electrostatics has been developed from the single property of amber already mentioned, and that Magnetostatics has been developed from the single property of the lodestone. These two branches of Electromagnetism deal solely with states of rest, not with motion or changes of state, and are therefore concerned only with phenomena which can be described as statical. The developments of the two statical branches of Electromagnetism, namely Electrostatics and Magnetostatics, are entirely independent of one another.

MOTION THROUGH THE ETHER.

The Michelson-Morley Experiment.

When we have spoken of a system at rest we have so far meant, for all practical purposes, a system at rest in our laboratories. But if we have been right in conjecturing that all electromagnetic phenomena have their seat in the ether, then a system at rest would most naturally be taken to mean a system at rest in the ether. We have so far made no clear distinction between the conceptions of rest in the ether and rest relative to the walls of a laboratory.

The view was at one time held that a moving body drags the ether along with it. If this were a true view the distinction just referred to would not arise; a body at rest relative to the walls of a laboratory would also be at rest in the ether. But in time it was found that this was not a true view; it could not be reconciled simultaneously with results of laboratory experiments such as Fizeau's water-tube experiment (cf. § 687 below), and with the astronomical theory of the aberration of light (cf. § 689 below). Finally it became established that the ether, if one existed at all, could not share in the motion of moving bodies ; it must be stagnant, and moving bodies must simply move through it without setting up mass-motions in it.

CONCEPTIONS USED IN THE SURVEY OF A FIELD OF FORCE

The Intensity at a point.

The space in the neighbourhood of charges of electricity, considered with reference to the electric phenomena occurring in this space, is spoken of as the electric field.

A new charge of electricity, placed at any point 0 in an electric field, will experience attractions or repulsions from all the charges in the field. The introduction of a new charge will in general disturb the arrangement of the charges on all the conductors in the field by a process of induction. If, however, the new charge is supposed to be infinitesimal, the effects of induction will be negligible, so that the forces acting on the new charge may be supposed to arise from the charges of the original field.

Let us suppose that we introduce an infinitesimal charge ∈ on an infinitely small conductor. Any charge e1, in the field at a distance r1 from the point 0 will repel the charge with a force ∈e/r12. The charge ∈ will experience a similar repulsion from every charge in the field, so that each repulsion will be proportional to ∈.

The resultant of these forces, obtained by the usual rules for the composition of forces, will be a force proportional to ∈–say a force R∈ in some direction OP. We define the electric intensity at 0 to be a force of which the magnitude is R, and the direction is OP.