Maxwell Versus Thomson

It was inevitable that a theory so novel and so capacious as that of Maxwell should involve conceptions which his contemporaries understood with difficulty and accepted with reluctance.

Of these the most difficult and unacceptable was the principle that the total current is always a circuital vector; or, as it is generally expressed, that “all currents are closed.”

According to the older electricians, a current which is employed in charging a condenser is not closed, but terminates at the coatings of the condenser, where charges are accumulating.

Maxwell, on the other hand, taught that the dielectric between the coatings is the seat of a process—the displacement-current—which is proportional to the rate of increase of the electric force in the dielectric; and that this process produces the same magnetic effects as a true current, and forms, so to speak, a continuation, through the dielectric, of the charging current, so that the latter may be regarded as flowing in a closed circuit.

Another characteristic feature of Maxwell’s theory is the conception—for which, as we have seen, lie was largely indebted to Faraday and Thomson—that magnetic energy is the kinetic energy of a medium occupying the whole of space, and that electric energy is the energy of strain of the same medium. By this conception electromagnetic theory was brought into such close parallelism with the elastic-solid theories of the aether, that it was bound to issue in an electromagnetic theory of light.

Maxwell’s views were presented in a more developed form in a memoir entitled “A Dynamical Theory of the Electromagnetic Field,” which was read to the Royal Society in 1864;[31] in this the architecture of his system was displayed, stripped of the scaffolding by aid of which it had been first erected.

As the equations employed were for the most part the same as had been set forth in the previous investigation, they need only be briefly recapitulated. The magnetic induction μH, being a circuital vector, may be expressed in terms of a vector-potential A by the equation

μH = curl A.

The electric displacement D is connected with the volume-density ρ of free electric charge by tho electrostatic equation

div D = ρ.

The principle of conservation of electricity yields the equation

div ι = -∂ρ/∂t,

where ι denotes the conduction-current.

The law of induction of currents—namely, that the total electromotive force in any circuit is proportional to the rate of decrease of the number of lines of magnetic induction which pass through it—may be written

-curl E = μḢ;

from which it follows that the electric force E must be expressible in the form

E = - Å + grad,

where ψ denotes some scalar function. The quantities A and ψ which occur in this equation are not as yet completely determinate; for the equation by which A is defined in terms of the magnetic induction specifies only the circuital part of A; and as the irrotational part of A is thus indeterminate, it is evident that ψ also must be indeterminate, Maxwell decided the matter by assuming[32] A to be a circuital vector; thus

div A = 0,

and therefore

div E = -∇2ψ,

from which equation it is evident that represents the electrostatic potential.

The principle which is peculiar to Maxwell’s theory must now be introduced.

Currents of conduction are not the only kind of currents; even in the older theory of Faraday, Thomson, and Mossotti, it had been assumed that electric charges are set in motion in the particles of a dielectric when the dielectric is subjected to an electric field; and the predecessors of Maxwell would not have refused to admit that the motion of these charges is in some sense a current.

Suppose, then, that S denotes the total current which is capable of generating a magnetic field: since the integral of the magnetic force round any curve is proportional to the electric current which flows through the gap enclosed by the curve, we have in suitable units curl H = 4πS.

In order to determine S, we may consider the case of a condenser whose coatings are supplied with electricity by a conduction-current ι per unit-area of coating. If ± σ denote the surface-density of electric charge on the coatings, we have

i = ∂σ/∂t, and σ = D,

where D denotes the magnitude of the electric displacement D in the dielectric between the coatings; so ι = Ḋ. But since the total current is to be circuital, its value in the dielectric must be the same as the value ι which it has in the rest of the circuit; that is, the current in the dielectric has the value Ḋ. We shall assume that the current in dielectrics always has this value, so that in the general equations the total current must be understood to be ι + Ḋ.

The above equations, together with those which express the proportionality of E to D in insulators, and to ι in conductors, constituted Maxwell’s system for a field formed by isotropic bodies which are not in motion. When the magnetic field is due entirely to currents (including both conduction-currents and displacement-currents), so that there is no magnetization, we have

…

so that the vector-potential is connected with the total current by an equation of the same form as that which connects the scalar potential with the density of electric charge.

To these potentials Maxwell inclined to attribute a physical significance; he supposed ψ to be analogous to a pressure subsisting in the mass of particles in his model, and A to be the measure of the electrotonic state.

The two functions are, however, of merely analytical interest, and do not correspond to physical entities. For let two oppositely-charged conductors, placed close to each other, give rise to an electrostatic field throughout all space. In such a field the vector-potential A is everywhere zero, while the scalar potential ψ has a definite value at every point.

Now let these conductors discharge each other; the electrostatic force at any point of space remains unchanged until the point in question is reached by a wave of disturbance, which is propagated outwards from the conductors with the velocity of light, and which annihilates the field as it passes over it. But this order of events is not reflected in the behaviour of Maxwell’s functions ψ and A; for at the instant of discharge, ψ is everywhere annihilated, and A suddenly acquires a finite value throughout all space.

As the potentials do not possess any physical significance, it is desirable to remove them from the equations. This was afterwards done by Maxwell himself, who[33] in 1868. proposed to base the electromagnetic theory of light solely on the equations

…

together with the equations which define S in terms of E, and B in terms of H.