THE GENESIS OF AN ELECTRO-MAGNETIC

FIELD.

N the development of a branch of mathematical physics the first stage often consists of a study of the permanent states of a system, for instance the states of equilibrium, states of steady motion, and states of periodic motion; sometimes the development does not proceed much further than this, but frequently the oscillations about the permanent states are considered in full detail. In later stages of development efforts are made to elucidate the way in which the permanent states are attained, to find the conditions that they should be attained, and so forth. In chemical dynamics and in the theory of the conduction of heat a state of equilibrium is generally approached gradually in one direction without over-reaching the mark, while in ordinary dynamics and in the theory of electricity a permanent state is generally attained as a result of a series of damped oscillations.

In nearly every case in which the approach to a permanent state has been discussed, the system under consideration is supposed to be started with an initial motion, and indeed this seems to be necessary, for instance the transition from one permanent state to another could not otherwise be made, while it frequently happens that a given initial motion would have arisen in the natural (or continuous) order of events from motion of a violent character.

In the theory of electromagnetism the discontinuities

which render possible the transition from one permanent state to another are propagated as waves of discontinuity. The theory of such waves has been developed by many writers, and Prof. A. E. H. Love has worked out the details in the case of the transition from the electrostatic field of an electric doublet to the periodic electromagnetic field of a vibrating electric doublet, considering also the case in which the vibrations are damped. In this and in many other investigations in electromagnetic theory an electrostatic field is regarded as the simplest initial field, and this is generally imagined to fill the whole of space and to have existed for ever. A field which fills the whole of space seems, however, to require the existence of an infinite ether or medium to support it, and as the idea of an infinite medium is repugnant to some minds, it may be worth while to consider the question whether an electrostatic field, which does not fill the whole of space, but is bounded by a moving surface of discontinuity, can arise from a state of affairs in which there is initially no electromagnetic field at all.

An answer to this question may be derived from a careful study of the different solutions of Maxwell's equations for the propagation of electric waves. These equations when written in the symmetrical form adopted by Hertz and Heaviside are as follows:

c curl E- -SH/St, div H = 0,

where E and H are the electric and magnetic intensities and c the velocity of light.

In a type of solution which we shall regard as fundamental the complex vector H+ iE is of the form M = mf (a, B) where the vector m depends on both position and time, while f is an arbitrary scalar function of two quantities a and ẞ which are functions of both position and time. Quantities a and ẞ cannot both be real; they have constant values for certain points, which move along straight lines

with the velocity of light, and which may be conveniently called "light-particle." The fact that the vector mf (a, ẞ) provides us with a solution of the equations, whatever the arbitrary function f may be, suggests that the elements of disturbance associated with the different light-particles can be regarded as independent of one another and this is just what a further study of the above solution indicates.' It appears in fact that the collection of light-particles which lie at any instant within a small volume carry with them a certain amount of energy which remains unaltered during their motion.

The electric and magnetic intensities E and H are, moreover, at right angles to the direction of motion of the light-particles at a point and so the flow of energy, as indicated by Poynting's vector, is in the direction of motion of the light-particle. It should be mentioned that E and H are also perpendicular to one another and equal in magnitude so that the field is a "self-conjugate" or simple radiant field in which there is a simple propagation of energy but no accumulation or expenditure of energy at any ordinary point of space. In such a radiant field the energy may, perhaps, be regarded as energy of motion and as analogous to kinetic energy, although the view is unorthodox.

Now mathematicians have thought for a long time that all energy is really kinetic. The idea that potential energy can be regarded as kinetic energy of concealed cyclic motion was put forward in 1888 by Sir J. J. Thomson in his remarkable book The Applications of Dynamics to Physics and Chemistry and was adopted independently by Hertz in his work on the principles of mechanics.

Sir Joseph Thomson says: "This view which regards all potential energy as really kinetic has the advantage of keeping before us the idea that it is one of the objects of

1 It is indicated to some extent by the general theory of the characteristics and bi-characteristics of linear partial differential equations. Cf. Hadamard, Propagation des Ondes.

Physical Science to explain natural phenomena by means of the properties of matter in motion. When we have done this we have got a complete physical explanation of any phenomenon and any further explanation must be rather metaphysical than physical. It is not so, however, when we explain the phenomenon as due to changes in the potential energy of the system; for potential energy cannot be said, in the strict sense of the term, to explain anything. It does little more than embody the results of experiments in a form suitable for mathematical investigations.”

Since the energy in an electrostatic field is generally regarded as potential energy, it is clear that an electrostatic field ought not to be regarded as fundamental in electromagnetic theory, and it is now necessary for us to see if the type of field we have chosen as fundamental fulfils the requirements which Thomson considers as characteristic of a type of motion which can be regarded as fundamental in an attempt to eliminate the idea of potential energy.

In a second passage Thomson says: "As all the energy is kinetic its magnitude remains constant by the principle of the Conservation of Energy, and so the principle of Least Action takes the very simple form that with a given quantity of energy any material system will by its unguided motion go along the path which will take it from one configuration to another in the least possible time." The requirements are evidently fulfilled, and so the next step is to choose a and ẞ so that the radiant field under consideration is of a simple character.

Let S be a point which moves in an arbitrary manner with a velocity less than the velocity of light and let the light-particles start from the different positions of S. If P be an arbitrary point in space there is just one position of S, viz., So from which a light-particle can start so as to reach Pat time t. The time at which this particle must leave

So is a suitable value of a for the position P at time t and it is clear that a is constant for all the space-time points covered by the light-particle. To obtain a suitable value of ẞ we must find a complex quantity which will specify the direction of motion of the light-particle. Let a sphere of unit radius be described with So as center and let us imagine a steady irrotational two-dimensional motion of an incompressible fluid on the surface or on a portion of the surface, then the complex variable q + i whose real and imaginary parts are constant along the equipotentials and stream-lines respectively is a suitable value of ẞ. The imaginary fluid motion may, of course, vary in an arbitrary manner as S。 varies; in other words, ẞ depends on both the time of creation and the direction of motion of the lightparticle which arrives at P at time t.

Let us now consider the simple case in which the streamlines are cut out by planes through two points A and A. on the sphere. These lines may be regarded either as lines of electric or magnetic force. In the former case the corresponding radiant field possesses the following characteristics:

The field is produced by the creation at the moving point S of pairs of oppositely electrified light-particles and the rectilinear motion of these charged particles in different directions with the velocity of light. A pair of oppositely electrified light-particles may perhaps be supposed to have been derived from a neutral particle traveling initially with the velocity of light.

This creation of electricity may take place continuously for any length of time, and the rate at which electricity of one sign is produced may either vary in an arbitrary manner or may remain constant, while the directions of rectilinear motion of the charged light-particles may also

2 In the latter case the electromagnetic field may be limited to a certain portion of space.