The Followers Of Maxwell

The most notable imperfection in the electromagnetic theory of light, as presented in Maxwell’s original memoirs, was the absence of any explanation of reflection and refraction.

Before the publication of Maxwell’s Treatise, however, a method of supplying the omission was indicated by Helmholtz.[1]

The principles on which the explanation depends are that tho normal component of the electric displacement D, the tangential components of the electric force E, and the magnetic vector B or H, are to be continuous across the interface at which the reflexion takes place;

The optical difference between the contiguous bodies being represented by a difference in their dielectric constants, and the electric vector being assumed to be at right angles to the plane of polarization.[2]

The analysis required is a mere transcription of MacCullagh’s theory of reflexion,[3] if the derivate of MacCullagh’s displacement e with respect to the time be interpreted as the magnetic force; μ curl e as the electric force, and curl e as the electric displacement.

The mathematical details of the solution were not given by Helmholtz himself, but were supplied a few years later in the inaugural dissertation of H. A. Lorentz.[4]

In the years immediately following the publication of Maxwell’s Treatise, a certain amount of evidence in favour of his theory was furnished by experiment.

That an electric field is closely concerned with the propagation of light was demonstrated in 1875, when John Kerr[5] showed that dielectrics subjected to powerful electrostatic force acquire the property of double refraction, their optical behaviour being similar to that of uniaxal crystals whose axes are directed along the lines of force.

Other researches undertaken at this time had a more direct bearing on the questions at issue between the hypothesis of Maxwell and the older potential theories. In 1875-6 Helmholtz[6] and his pupil Schiller[7] attempted to discriminate between the various doctrines and formulae relative to unclosed circuits by performing a crucial experiment.

It was agreed in all theories that a ring-shaped magnet, which returns into itself so as to have no poles, can exert no ponderomotive force on other magnets or on closed electric currents.

Helmholtz[8] had, however, shown in 1873 that according to the potential-theories such a magnet would exert a ponderomotive force on an unclosed current.

The matter was tested by suspending a magnetized steel ring by a long fibre in a closed metallic case, near which was placed a terminal of a Holtz machine.

No ponderomotive force could be observed when the machine was put in action so as to produce a brush discharge from the terminal: from which it was inferred that the potential-theories do not correctly represent the phenomena, at least when displacement-currents and convection-currents (such as that of the electricity carried by the electrically repelled air from the terminal) are not taken into account.

The researches of Helmholtz and Schiller brought into prominence the question as to the effects produced by the translatory motion of electric charges.

The convection of electricity is equivalent to a current had been suggested long before by Faraday.[9] “If,” he wrote in 1838, “a ball be electrified positively in the middle of a room and be then moved in any direction, effects will be produced as if a current in the same direction had existed.”

To decide the matter a new experiment inspired by Helmholtz was performed by H. A. Rowland[10] in 1876.

The electrified body in Rowland’s disposition was a disk of ebonite, coated with gold leaf and capable of turning rapidly round a vertical axis between two fixed plates of glass, each gilt on one side.

The gilt faces of the plates could be earthed, while the ebonite disk received electricity from a point placed near its edge; each coating of the disk thus formed a condenser with the plate nearest to it.

An astatic needle was placed above the upper condenser-plate, nearly over the edge of the disk; and when the disk was rotated a magnetic field was found to be produced.

This experiment, which has since been repeated under improved conditions by Rowland and Hutchinson,[11] H. Penders[12], and Eichenwald,[13] shows that the “convection-current” produced by the rotation of a charged disk, when the other ends of the lines of force are on an earthed stationary plate parallel to it, produces the same magnetic field as an ordinary conduction-current flowing in a circuit which coincides with the path of the convection-current.

When two disks forming a condenser are rotated together, the magnetic action is the sum of the magnetic actions of cach of the disks separately.

It appears, therefore, that electric charges cling to the matter of a conductor and move with it, so far as Rowland’s phenomenon is concerned.

The first examination of the matter from the point of view of Maxwell’s theory was undertaken by J. J. Thomson,[14] in 1881, If an electrostatically charged body is in motion, the change in the location of the charge must produce a continuous alteration of the electric field at any point in the surrounding medium; or, in the language of Maxwell’s theory, there must be displacement-currents in the medium.

It was to these displacement-currents that Thomson, in his original investigation, attributed the magnetic effects of moving charges. The particular system which he considered was that formed by a charged spherical conductor, moving uniformly in a straight line.

It was assumed that the distribution of electricity remains uniform over the surface during the motion, and that the electric field in any position of the sphere is the same as if the sphere were at rest; these assumptions are true so long as quantities of order (v/c)2 are neglected, where v denotes the velocity of the sphere and c the velocity of light.

Thomson’s method was to determine the displacement-currents in the space outside the sphere from the known values of the electric field, and then to calculate the vectorpotential due to these displacement-currents by means of the formula

where S′ denotes the displacement-current at (x′y′z′). The magnetic field was then determined by the equation

A defect in this investigation was pointed out by FitzGerald, who, in a short but most valuable note,[15] published a few months afterwards, observed that the displacement-currents of Thomson do not satisfy the circuital condition.

This is most simply seen by considering the case in which the system consists of two parallel plates forming a condenser; if one of the plates is fixed, and the other plate is moved towards it, the electric field is annihilated in the space over which the moving plate travels: this destruction of electric displacement constitutes a displacement-current, which, considered alone, is evidently not a closed current.

The detect, as FitzGerald showed, may be immediately removed by assuming that a moving charge itself is to be counted as a current-element: the total current, thus composed of the displacement-currents and the convection-current, is circuital. Making this correction, FitzGerald found that the magnetic force due to a sphere of charge e moving with velocity v along the axis of z is curl(0, 0, ev/r)—a formula which shows that the displacement-currents have no resultant magnetic effect, since the term ev/r would be obtained from the convection-current alone.

The expressions obtained by Thomson and FitzGerald were correct only to the first order of the small quantity v/c. The effect of including terms of higher order was considered in 1889 by Oliver Heaviside,[16] whose solution may be derived in the following manner:—

Suppose that a charged system is in motion with uniform velocity v parallel to the axis of z; the total current consists of the displacement-current Ė/4πc2 where E denotes the electric force, and the convection-current ρv where ρ denotes the volume-density of electricity. So the equation which connects magnetic force with electric current may be written

Eliminating E between this and the equation

and remembering that H is here circuital, we have

If, therefore, a vector-potential a be defined by the equation

the magnetic force will be the curl of a; and from the equation for a it is evident that the components ax and ay are zero, and that az, is to be determined from the equation

Now, let (x, y, ζ) denote coordinates relative to axes which are parallel to the axes (x, y, z), and which move with the charged bodies; then (az, is a function of (x, y, ζ) only; so we have

and the preceding equation is readily seen to be equivalent to

where ζ1 denotes (1 - v2/c2)-

ζ. But this is simply Poisson’s equation, with ζ1 substituted for z; so the solution may be transcribed from the known solution of Poisson’s equation: it is

the integrations being taken over all the space in which there are moving charges; or

If the moving system consists of a single charge e at the point ζ = 0, this gives

where sin2 θ = (x2 + y2)/r2.

It is readily seen that the lines of magnetic force due to the moving point-charge are circles whose centres are on the line of motion, the magnitude of the magnetic force being

The electric force is radial, its magnitude being

The fact that the electric vector due to a moving point-charge is everywhere radial led Heaviside to conclude that the same solution is applicable when the charge is distributed over a perfectly conducting sphere whose centre is at the point, the only change being that E and H would now vanish inside the sphere.

This inference was subsequently found[17] to be incorrect: a distribution of electric charge on a moving sphere could in fact not be in equilibrium if the electric force were radial, since there would then be nothing to balance the mechanical force exerted ou the moving charge (which is equivalent to a current) by the magnetic field. The moving system which gives rise to the same field as a moving point-charge is not a sphere, but an oblate spheroid whose polar axis (which is in the direction of motion) bears to its equatorial axis the ratio {{Wikimath|(1 - v2/c2) :1.[18]

The energy of the field surrounding a charged sphere is greater when the sphere is in motion than when it is at rest. To determine the additional energy quantitatively (retaining only the lowest significant powers of v/c), we have only to integrate, throughout the space outside the sphere, the expression H2/8π, which represents the electrokinetic energy per unit volume: the result is e2v2/3a, where e denotes the charge, v the velocity, and a the radius of the sphere.

It is evident from this result that the work required to be done in order to communicate a given velocity to the sphere is greater when the sphere is charged than when it is uncharged; that is to say, the virtual mass of the sphere is increased by an amount 2e2/3a, owing to the presence of the charge. This may be regarded as arising from the self-induction of the convection-current which is formed when the charge is set in motion.

J. Larmor[19] and W. Wien[20] suggested that the inertia of ordinary ponderable matter may ultimately prove to be of this nature, the atoms being constituted of systems of electrons.[21]

It may, however, be remarked that this view of the origin of mass is not altogether consistent with the principle that the electron is an indivisible entity.

For the so-called self-induction of the spherical electron is really the mutual induction of the convection-currents produced by the elements of electric charge which are distributed over its surface.

The calculation of this quantity presupposes the divisibility of the total charge into elements capable of acting severally in all respects as ordinary electric charges; a property which appears scarcely consistent with the supposed fundamental nature of the electron.

After the first attempt of J. J. Thomson to determine the field produced by a moving electrified sphere, the mathematical development of Maxwell’s theory proceeded rapidly. The problems which admit of solution in terms of known functions are naturally those in which the conducting surfaces involved have simple geometrical forms—planes, spheres, and cylinders.[22]

A result which was obtained by Horace Lamb,[23] when investigating electrical motions in a spherical conductor, led to interesting consequences. Lamb found that if a spherical conductor is placed in a rapidly alternating held, the induced currents are almost entirely confined to a superficial layer; and his result was shortly afterwards generalized by Oliver Heaviside,[24] who showed that whatever be the form of a conductor rapidly alternating currents do not penetrate far into its substance.[25]

The reason for this may be readily understood: it is virtually an application of the principle[26] that a perfect conductor is impenetrable to magnetic lines of force. No perfect conductor is known to exist; but[27] if the alternations of magnetic force to which a good conductor such as copper is exposed are very rapid, the conductor bas not time (so to speak) to display the imperfection of its conductivity, and the magnetic field is therefore unable to extend far below the surface.

The same conclusion may be reached by different reasoning.[28]

When the alternations of the current are very rapid, the ohmic resistance ceases to play a dominant part, and the ordinary equations connecting electromotive force, induction, and current are equivalent to the conditions that the currents shall be so distributed as to make the electrokinetic or magnetic energy a minimum. Consider now the case of a single straight wire of circular cross-section.

The magnetic energy in the space outside the wire is the same whatever be the distribution of current in the cross-section (so long as it is symmetrical about the centre), since it is the same as if the current were flowing along the central axis, so the condition is that the magnetic energy in the wire shall be a minimum; and this is obviously satisfied when the current is concentrated in the superficial layer, since then the magnetic force is zero in the substance of the wire.