James Clerk Maxwell

Since the time of Descartes, natural philosophers have speculated how electric and magnetic influences are transmitted through space.

In the mid-19th century, speculation assumed a definite form, issued in a rational theory.

Karl Friedrich Gauss (1777-1855) wrote a letter[1] to Weber of date March 19, 1845, remarking that he had long ago proposed to himself to supplement the known forces which act between electric charges by other forces, such as would cause electric actions to be propagated between the charges with a finite velocity.

But he expressed himself as determined not to publish his researches until he should have devised a mechanism by which the transmission could be conceived to be effected; and this he had not succeeded in doing.

More than one attempt to realize Gauss’s aspiration was made by his pupil Riemann. In a fragmentary note,[2] t which appears to have been written in 1853, but which was not published until after his death, Riemann proposed an aether whose elements should be endowed with the power of resisting compression, and also (like the elements of MacCullagh’s aether) of resisting changes of orientation.

The former property he conceived to be the cause of gravitational and electrostatic effects, and the latter to be the cause of optical and magnetic phenomena. The theory thus outlined was apparently not developed further by its author; but in a short investigation[3] which was published posthumously in 1867,[4] he returned to the question of the process by which electric action is propagated through space. In this memoir he proposed to replace Poisson’s equation for the electrostatic potential, namely,

by the equation

according to which the changes of potential due to changing electrification would be propagated outwards from the charges with a velocity c. This, so far as it goes, is in agreement with the view which is now accepted as correct; but Riemann’s hypothesis was too slight to serve as the basis of a complete theory. Success came only when the properties of the intervening medium were taken into account.

In that power to which Gauss attached so much importance, of devising dynamical models and analogies for obscure physical phenomena, perhaps no one has ever excelled W. Thomson[5]; and to him, jointly with Faraday, is due the credit of having initiated the theory of the electric medium. In one of his earliest papers, written at the age of seventeen,[6] Thomson compared the distribution of electrostatic force, in a region containing electrified conductors, with the distribution of the flow of heat in an infinite solid: the equipotential surfaces in the one case correspond to the isothermal surfaces in the other, and an electric charge corresponds to a source of heat.[7]

It may, perhaps, seem as if the value of such an analogy as this consisted merely in the prospect which it offered of comparing, and thereby extending, the mathematical theories of heat and electricity. But to the physicist its chief interest lay rather in the idea that formulae which relate to the electric field, and which had been deduced from laws of action at a distance, were shown to be identical with formulae relating to the theory of heat, which had been deduced from hypotheses of action between contiguous particles.

In 1846, Thomson had taken his degree as second wrangler at Cambridge. He investigated[8] the analogies of electric phenomena with those of elasticity by examining the equations of equilibrium of an incompressible elastic solid which is in a state of strain.

He showed that the distribution of the vector which represents the elastic displacement might be assimilated to the distribution of the electric force in an electrostatic system. He showed that this is not the only analogy which may be perceived with the equations of elasticity; for the elastic displacement may equally well be identified with a vector a, defined in terms of the magnetic induction B by the relation

The vector a is equivalent to the vector-potential which had been used in the memoirs of Neumann, Weber, and Kirchhoff, on the induction of currents. But Thomson arrived at it independently by a different process, without being aware of the identification.

The results of Thomson’s memoir suggested a picture of the propagation of electric or magnetic force. Might it not take place in somewhat the same way as changes in the elastic displacement are transmitted through an elastic solid?

He did not puruse these suggestions further. But they helped to inspire James Clerk Maxwell, another young Cambridge man, to take up the matter a few years later.

Maxwell eventually solved the problem.

He was born in 1831, the son of a landed proprietor in Dumfriesshire.

He was educated at Edinburgh, and at Trinity College, Cambridge, of which society he became in 1855 a Fellow; and not long after his election to Fellowship, he communicated to the Cambridge Philosophical Society the first of his endeavours[9] to form a mechanical conception of the electro-magnetic field.

Maxwell read Faraday’s Experimental Researches and was gifted with a physical imagination akin to Faraday’s.

He had been profonudly impressed by the theory of lines of force.

At the same time, he was a trained mathematician. The distinguishing feature of almost all his researches was the union of the imaginative and the analytical faculties to produce results partaking of both natures.

This first memoir may be regarded as an attempt to connect the ideas of Faraday with the mathematical analogies which had been devised by Thomson.

Maxwell considered first the illustration of Faraday’s lines of force which is afforded by the lines of flow of a liquid.

The lines of force represent the direction of a vector; and the magnitude of this vector is everywhere inversely proportional to the cross-section of a narrow tube formed by such lines.

This relation between magnitude and direction is possessed by any circuital vector; and in particular by the vector which represents the velocity at any point in a fluid, if the fluid be incompressible.

It is therefore possible to represent the magnetic induction B, which is the vector represented by Faraday’s lines of magnetic force, as the velocity of an incompressible fluid.

Such an analogy had been indicated some years previously by Faraday himself,[10] who had suggested that along the lines of magnetic force there may be a “dynamic condition,” analogous to that of the electric current, and that, in fact, “the physical lines of magnetic force are currents.”

The comparison with the lines of flow of a liquid is applicable to electric as well as to magnetic lines of force. In this case the vector which corresponds to the velocity of the fluid is, in free aether, the electric force E. But when different dielectrics are present in the field, the electric force is not a circuital vector, and therefore cannot be represented by lines of force; in fact, the equation

is now replaced by the equation

where ε denotes the specific inductive capacity or dielectric constant at the place (x, y, z).

This equation shows that the vector εE is circuital.

This vector, which will be denoted by D, bears to E a relation similar to that which the magnetic induction B bears to the magnetic force H. It is the vector D which is represented by Faraday’s lines of electric force, and which in the hydrodynamical analogy corresponds to the velocity of the incompressible fluid.

In comparing fluid motion with electric fields it is necessary to introduce sources and sinks into the fluid to correspond to the electric charges; for D is not circuital at places where there. is free charge. The magnetic analogy is therefore somewhat the simpler.

In the latter half of his memoir Maxwell discussed how Faraday’s “electrotonic state” might be represented in mathematical symbols. This problem be solved by borrowing from Thomson’s investigation of 1847 the vector a, which is defined in terms of the magnetic induction by the equation

if, with Maxwell, we call a the electrotonic intensity, the. equation is equivalent to the statement that “the entire electrotonic intensity round the boundary of any surface measures the number of lines of magnetic force which pass through that surface.”

The electromotive force of induction at the place (x, y, z) is -∂a/∂t: as Maxwell said, “the electromotive force on any element of a conductor is measured by the instantaneous rate of change of the electrotonic intensity on that element.”

From this it is evident that a is no other than the vector-potential which had been employed by Neumann, Weber, and Kirchhoff, in the calculation of induced currents; and we may take[11] for the electrotonic intensity due to a current i′ flowing in a circuit s′ the value which results from Neumann’s theory, namely,

It may, however, be remarked that the equation

taken alone, is insufficient to determine a uniquely; for we can choose a so as to satisfy this, and also to satisfy the equation

where ψ denotes any arbitrary scalar. There are, therefore, an infinite number of possible functions a. With the particular value of a which has been adopted, we have

so the vector-potential a which we have chosen is circuital.

In this memoir the physical importance of the operators curl and div first became evident[12]; for, in addition to those applications which have been mentioned, Maxwell showed that he connexion between the strength ι of a current and the magnetic field H, to which it gives rise, may be represented by the equation

this equation is equivalent to the statement that “the entire magnetic intensity round the boundary of any surface measures the quantity of electric current which passes through that surface.”