Lorentz

Lorentz was impressed by the success of Fresnel’s beautiful theory of the propagation of light in moving transparent substances[36] He:

• designed his equations so as to accord with that theory
• showed that this might be done by:
  • drawing a distinction between matter and aether
  • assuming that a moving ponderable body cannot communicate its motion to the aether which surrounds it, or even to the aether which is entangled in its own particles

In this way, part of the aether can be in motion relative to any other part.

Such an aether is simply space endowed with certain dynamical properties.

The general plan of Lorentz’ investigation was to reduce all the complicated cases of electromagnetic action to one simple and fundamental case, in which the field contains only free aether with solitary electrons dispersed in it.

The theory which he adopted in this fundamental case was a combination of Clausius’ theory of electricity with Maxwell’s theory of the aether.

Suppose that e(x, y, z) and e′(x′, y′, z′) are two electrons. In the theory of Clausius,[37] the kinetic potential of their mutual action is

so when any number of electrons are present, the part of the kinetic potential which concerns any one of them—say, e—may be written

where a and φ denote potential functions, defined by the

ρ denoting the volume-density of electric charge, and v its velocity, and the integration being taken over all space.

We shall now reject Clausius’ assumption that electrons act instantaneously at a distance, and replace it by the assumption that they act on each other only through the mediation of an aether which fills all space, and satisfies Maxwell’s equations, This modification may be effected in Clausius’ theory without difficulty; for, as we have seen,[38] if the state of Maxwell’s aether at any point is defined by the electric vector d and magnetic vector h,[39] these vectors may be expressed in terms of potentials a and φ by the equations

and the functions a and φ may in turn be expressed in terms of the electric charges by the equations

where the bars indicate that the values of (ρvx)′ and (ρ)′ refer to the instant (t - r/c). Comparing these formulae with those given above for Clausius’ potentials, we see that the only change which it is necessary to make in Clausius’ theory is that of retarding the potentials in the way indicated by L. Lorenz.[40] The electric and magnetic forces, thus defined in terms of the position and motion of the charges, satisfy the Maxwellian equations

The theory of Lorentz is based on these 4 aethereal equations of Maxwell, together with the equation which determines the ponderomotive force on a charged particle. This, which we shall now derive, is the contribution furnished by Clausius’ theory.

The Lagrangian equations of motion of the electron e are

and two similar equations, where L denotes the total kinetic potential due to all causes, electric and mechanical. The ponderomotive force exerted on the electron by the electromagnetic field has for its x-component

which, since

reduces to

so that the force in question is

This was Lorentz’ expression for the ponderomotive force on an electrified corpuscle of charge e moving with velocity v in a field defined by the electric force d and magnetic force h.

In Lorentz’ fundamental case, which has thus been examined, account has been taken only of the ultimate constituents of which the universe is supposed to be composed, namely, corpuscles and the aether. We must now see how to build up from these the more complex systems which are directly presented to our experience.

The electromagnetic field in ponderable bodies, which to our senses appears in general to vary continuously, would present a different aspect if we were able to discern molecular structure; we should then perceive the individual electrons by which the field is produced, and the rapid fluctuations of electric and magnetie* force between them. As it is, the values furnished by our instruments represent averages taken over volumes which, though they appear small to us, are large compared with molecular dimensions.[41]

We shall denote an average value of this kind by a bar placed over the corresponding symbol.

Lorentz supposed that the phenomena of electrostatic charge and of conduction-currents are due to the presence or motion of simple electrons such as have been considered above. The part of

arising from these is the measurable density of electrostatic charge; this we shall denote by ρ1. If w denote the velocity of the ponderable matter, and if the velocity v of the electrons be written w + u, then the quantity

, so far as it arises from electrons of this type, may be written

The former of these terms represents the convection-current, and the latter the conduction-current.

Consider next the phenomena of dielectrics. Following Faraday, Thomson, and Mossotti,[42] Lorentz supposed that each dielectric molecule contains corpuscles charged vitreously and also corpuscles charged resinously. These in the absence of an external field are so arranged as to neutralize each other’s electric fields outside the molecule.

For simplicity, we may suppose that in each molecule only one corpuscle, of charge e, is capable of being displaced from its position; it follows from what has been assumed that the other corpuscles in the molecule exert the same electrostatic action as a charge – e situated at the original position of this corpuscle.

Thus if e is displaced to an adjacent position, the entire molecule becomes equivalent to an electric doublet, whose moment is measured by the product of e and the displacement of e. The molecules in unit volume, taken together, will in this way give rise to a (vector) electric moment per unit volume, P, which may be compared to the (vector) intensity of magnetization in Poisson’s theory of magnetism.[43] As in that theory, we may replace the doublet-distribution P of the scalar quantity ρ by a volume-distribution of ρ, determined by the equation[44

This represents the part of

due to the dielectric molecules.

Moreover, the scalar quantity ρwx, has also a doublet-distribution, to which the same theorem may be applied; the average value of the part of ρwx, due to dielectric molecules, is therefore determined by the equation

We have now to find that part of which is due to dielectric molecules. For a single doublet of moment p we have, by differentiation,

where the integration is taken throughout the molecule; so that

where the integration is taken throughout a volume V, which ​encloses a large number of molecules, but which is small compared with measurable quantities; and this equation may be written

Now, if Ṗ refers to differentiation at a fixed point of space (as opposed to a differentiation which accompanies the moving body), we have

so that

and therefore

This equation determines the part of

The general equations of the aether thus become, when the averaging process is performed,

convection-current + conduction-current

In order to assimilate those to the ordinary electromagnetic equations, we must evidently write

, the electric force;

, the electric induction;

, the electric vector.

The equations then become (writing ρ for ρ1, as there is no longer any need to use the subscript),

where

S = conduction-current + convection-current + Ḋ + curl [P.w].

The term Ḋ in S evidently represents the displacementcurrent of Maxwell; and the term curl [P.w] will be recognized as a modified form of the term curl [D.w], which was first introduced into the equations by Hertz.[45]

Hertz supposed this term to represent the generation of a magnetic force within a dielectric which is in motion in an electric field, and that Heaviside[46] by adducing considerations relative to the energy, showed that the term ought to be regarded as part of the total current, and inferred from its existence that a dielectric which moves in an electric field is the seat of an electric current, which produces a magnetic field in the surrounding space.