Lorentz Modification

Lorentz replaced D by P in the vector-product. This implied that the moving dielectric does not carry along the aethereal displacement, which is represented by the term E/4πc2 in D, but only carries along the charges which exist at opposite ends of the molecules of the ponderable dielectric, and which are represented by the term P. The part of the total current represented by the term curl [P.w] is generally called the current of dielectric convection.

That a magnetic field is produced when an uncharged dielectric is in motion at right angles to the lines of force of a constant electrostatic field had been shown experimentally in 1888 by Röntgen.[47]

His experiment consisted in rotating a dielectric disk between the plates of a condenser; a magnetic field was produced, equivalent to that which would be produced by the rotation of the “fictitious charges” on the two faces of the dielectric, i.e., charges which bear the same relation to the dielectric polarization that Poisson’s equivalent surfacedensity of magnetism[48] bears to magnetic polarization.

If U is the difference of potential between the opposite coatings of the condenser, and ε the specific inductive capacity of the dielectric, the surface-density of electric charge on the coatings ​is proportional to ±εU, and the fictitious charge on the surfaces of the dielectric is proportional to

If a plane condenser is charged to a given difference of potential, and is rotated in its own plane, the magnetic field produced is proportional to ε if (as in Rowland’s experiment[49]) the coatings are rotated while the dielectric remains at rest, but is in the opposite direction, and is proportional to (ε - 1) if (as in Röntgen’s experiment) the dielectric is rotated while the coatings remain at rest.

If the coatings and dielectric are rotated together, the magnetic action (being the sum of these) should be independent of ε—a conclusion which was verified later by Eichenwald.[50]

Hitherto we have taken no account of the possible magnetization of the ponderable body. This would modify the equations in the usual manner,[51] so that they finally take the form

where S denotes the total current formed of the displacementcurrent, the convection-current, the conduction-current, and the current of dielectric convection. Moreover, since

we have

which vanishes by virtue of the principle of conservation of electricity. Thus

or the total current is a circuital vector, Equations (I) to (V) are the fundamental equations of Lorentz’ theory of electrons.

We have now to consider the relation by which the polarization P of dielectrics is determined. If the dielectric is moving with velocity w, the ponderomotive force on unit electric charge moving with it is (as in all theories)[52]

In order to connect P with E′, it is necessary to consider the motion of the corpuscles. Let e denote tho charge and m the mass of a corpuscle, (ξ, η, ζ, ) its displacement from its position of equilibrium, k2(ξ, η, ζ, ) the restitutive force which retains it in the vicinity of this point; then the equations of motion of the corpuscle are

and similar equations in η and ζ. When the corpuscle is set in motion by light of frequency n passing through the medium, the displacements and forces will be periodic functions of nt—say,

Substituting these values in the equations of motion, we obtain

and therefore

Thus, if N denote the number of polarizable molecules per unit volume, the polarization is determined by the equation

In the particular case in which the dielectric is at rest, this equation gives But, as we have seen[53] D bears to E the ratio μ2/4πc2, where μ ​denotes the refractive index of the dielectric; and therefore the refractive index is determined in terms of the frequency by the equation

This formula is equivalent to that which Maxwell and Sellmeier[54] had derived from the elastic-solid theory. Though superficially different, the derivations are alike in their essential feature, which is the assumption that the molecules of the dielectric contain systems which possess free periods of vibration, and which respond to the oscillations of the incident light. The formula may be derived on electromagnetic principles without any explicit reference to electrons; all that is necessary is to assume that the dielectric polarization has a free period of vibration.[55]

When the luminous vibrations are very slow, so that n is small, μ2 reduces to the dielectric constant ε[56]; so that the theory of Lorentz leads to the expression

for the specific inductive capacity in terms of the number and circumstances of the electrons.[57] Returning now to the case in which the dielectric is supposed to be in motion, the equation for the polarization may be written

from this equation, Fresnel’s formula for the velocity of light in a moving dielectric may be deduced. For, let the axis of z be taken parallel to the direction of motion of the dielectric, which is supposed to be also the direction of propagation of the light; and, considering a plane-polarized wave, take the axis of x parallel to the electric vector, so that the magnetic vector must be parallel to the axis of y. Then equation (III) above becomes equation (IV) becomes (assuming B equal to II, as is always the case in optics), The equation which defines the electric induction gives

and equations (1) and (2) give

Eliminating D., Px, and Ily, we have or, neglecting w/cº,

Substituting

{\displaystyle E_{x}=e^{n(t-z/V){\sqrt {-1}}}}, so that V denotes the velocity of light in the moving dielectric with respect to the fixed aether we have

​or (neglecting w2/c2)

which is the formula of Fresnel.[59]

The hypothesis of Fresnel, that a ponderable body in motion carries with it the excess of aether which it contains as compared with space free from matter, is thus seen to be transformed in Lorentz’ theory into the supposition that the polarized molecules of the dielectric, like so many small condensers, increase the dielectric constant, and that it is (so to speak) this augmentation of the dielectric constant which travels with the moving matter.

One evident objection to Fresnel’s theory, namely, that it required the relative velocity of aether and matter to be different for light of different colours, is thus removed; for the theory of Lorentz only requires that the dielectric constant should have different values for light of different colours, and of this a satisfactory explanation is provided by the theory of dispersion.

The correctness of Lorentz’ hypothesis, as opposed to that of Hertz (in which the whole of the contained aether was supposed to be transported with the moving body), was afterwards confirmed by various experiments.

In 1901 R. Blondlot[60] drove a current of air through a magnetic field, at right angles to the lines of magnetic force.

The air-current was made to pass between the faces of a condenser, which were connected by a wire, so as to be at the same potential. An electromotive force E′ would be produced in the air by its motion in the magnetic field; and, according to the theory of Hertz, this should produce an electric induction D of amount (ε/4πc2)E′ (where ε denotes the specific inductive capacity of the air, which is practically unity); so that, according to Hertz, the faces of the condenser should become charged. According to Lorentz theory, on the other hand, the electric induction D is determined by the equation

where E denotes the electric force on a charge at rest, which is zero in the present case. Thus, according to Lorentz’ theory, the charges on the faces would have only (ε - 1)/ε of the values which they would have in Hertz’ theory; that is, they would be practically zero. The result of Blondlot’s experiment was in favour of the theory of Lorentz.

An experiment of a similar character was performed in 1905 by H. A. Wilson.[61] In this, the space between the inner and outer coatings of a cylindrical condenser was filled with the dielectric ebonite.

When the coatings of such a condenser are maintained at a definite difference of potential, charges are induced on thein; and if the condenser be rotated on its axis in a magnetic field whose lines of force are parallel to the axis, these charges will be altered, owing to the additional polarization which is produced in the dielectric molecules by their motion in the magnetic field.

As before, the value of the additional charge according to the theory of Lorentz is (ε - 1)/ε times its value as calculated by the theory of Hertz. The result of Wilson’s experiments was, like that of Blondlot’s, in favour of Lorentz.

The reconciliation of the electromagnetic theory with Fresnel’s law of the propagation of light in moving bodies was a distinct advance.