based on this hypothesis was brought forward by Lord Rayleigh*, but abandoned because observations made by Stokes, and afterwards by Glazebrook, decided in favour of Fresnel's surface. Instead of taking the inertia as variable, we may adopt the very plausible hypothesis that the rigidity is different in different directions. Thus different values of n in the first two equations (11) would give two plane waves propagated with different velocities, along the axis of z. A general theory cannot however be formed by a simple modification of the equations holding for isotropic media. According to Greent, there may be twenty-one different coefficients defining the properties of crystalline media, which shows the complication we might be led into if we wished to attack the problem in its most general form.

133. Equations of the Electromagnetic Field. The line integral of the magnetic force round a closed curve is numerically equal to the electric current through the curve multiplied by 4. It is shown in treatises on Electricity that the mathematical expression of this law is contained in the three equations:

where a, ẞ, y are the components of magnetic force, and u, v, w the components of current density. The factor 4 depends on the units chosen, which are those of the electromagnetic system.

Another proposition which embodies Faraday's laws of electromagnetic induction states that if a closed curve encloses lines of magnetic induction which vary in intensity, an electromotive force acts round the curve, and the line integral of the electric force round the closed curve is equal to the rate of diminution of the total magnetic induction through the circuit. This leads to the equations

where ua, uß, μy are the components of magnetic induction, μ being the permeability, and P, Q, R those of electric force.

*Collected Works, Vol. 1. p. 111.

+ Collected Works, p. 245.

The two sets of equations may be taken to represent experimental facts and to be quite independent of any theory, although equations (13) may be deduced from (12) with the help of the principle of the conservation of energy. Both sets of equations would be equally true if we considered electric and magnetic forces to be due to action at a distance.

There are some additional equations to be considered.

Differentiating equations (12) with respect to x, y, z respectively and adding, we find

Similarly we derive from (13), if μ be constant and a, ẞ, y periodic,

134. Maxwell's Theory. The fundamental principle of Maxwell lies in his conception of an electric current in dielectrics and the way in which this current is made to depend on electric force. His views are best explained by an analogy taken from the theory of stress and strain. A stress in an elastic solid produces certain displacements which are proportional to the stress. If the stresses increase, the displacements increase, and the change of displacements constitutes a transference of matter. This flow or current of matter is proportional to the rate of change of the elastic stress. Taking this as a guide we may imagine the medium to yield in some unknown manner to the application of electric force, and if so, the rate of change of that force will be proportional to a "flow" which according to Maxwell is identical in all its effects with an electric current.

dE

dt'

If the electric force is E, the electric current is proportional to and if the law that the total flow is the same across all crosssections of a circuit holds good for these so-called "displacement currents or "polarization currents," it can be shown that the current is equal to K 4, where K is the specific inductive capacity of

,,

dE
dt

the medium. In a conductor, the current would, according to Ohm's law, be CE, where C is the conductivity. If we imagine a medium to possess both specific inductive capacity and conductivity, we must introduce an expression which includes both cases and put the current equal to

Confining ourselves at present to non-conductors and resolving along the three coordinate axes, we have

These equations allow us to combine (12) and (13) so as to obtain two fresh sets containing respectively only the magnetic and the electric forces.

135. Differential equation for propagation of electric and magnetic disturbances in dielectric media.

Equations (12) with the help of (17) become

Differentiate each of the equations (13) with respect to the time, eliminate P, Q, R, by means of (18), and use (15), when the following sets of equations, involving only magnetic forces, will be obtained:

We may eliminate the magnetic forces in a similar manner and obtain

These equations show that the magnetic and electric forces are propagated with a velocity 1/√Kp. In the electromagnetic system of units, μ= 1 in vacuo, and differs very little from that value in any known dielectric. K the specific inductive capacity is, in vacuo, unity when the electrostatic system of units is employed, but in the electromagnetic system K is numerically equal to 1/v2, if v is equal to the number of electrostatic units of quantity which are contained in an electromagnetic unit. This number, which gives the velocity of propagation of electromagnetic waves in vacuo, may be determined by experiment, and is found, within the errors of experiment, to be equal to the velocity of light in vacuo. Both velocities differ from 3 × 1010 probably by not more than one part in a thousand.

Maxwell's theory, which is embodied in equations (19) or (20), leads therefore to the remarkable conclusion that an electromagnetic disturbance is propagated with a finite velocity which is equal to the velocity of light. This conclusion has been amply verified by the

celebrated experiments of Hertz. Kirchhoff* had already in 1857 pointed out that a longitudinal electric disturbance is propagated in a wire with a velocity equal to that of light, but it was left to Maxwell to discover the reason for this coincidence.

If both the disturbance of light and the electromagnetic wave are propagated through the same medium with the same velocity, the conclusion is irresistible that both phenomena are identical in character. This conclusion constitutes the so-called "Electromagnetic Theory of Light." The electromagnetic theory of light establishes for the propagation of a luminous disturbance, equations which in several instances, as will appear, fit the facts better than the older elastic solid theory, but it should not be forgotten that it furnishes no explanation of the nature of light. It only expresses one unknown quantity (light) in terms of other unknown quantities (magnetic and electric disturbances), but magnetic and electric stresses are capable of experimental investigation, while the elastic properties of the medium through which, according to the older theory, light was propagated, could only be surmised from the supposed analogy with the elastic properties of material media. Hence it is not surprising that the electromagnetic equations more correctly represent the actual phenomena. Whatever changes be introduced in future, in our ideas of the nature of light, the one great achievement of Maxwell, the proof of the identity of luminous and electromagnetic disturbances, will never be overthrown.

136. Refraction. We have so far only considered the propagation of waves in vacuo. According to equations (20), the squares of the velocities of propagation in two media having identical magnetic permeabilities, ought to be inversely as their specific inductive capacities. If therefore K, be the inductive capacity of the vacuum, K, that of any dielectric, the “refractive index" ought to be equal to √K1/K。. This relation is approximately verified in the case of a few gases, as shown in the following table, which contains the square roots of specific inductive capacities (D) as measured by Klemencict, and the refractive indices (n) of the same gases for Sodium light, as measured by G. W. Walker. Both constants are reduced to a temperature of 0° C., and a pressure of 760 mm.

The discrepancy for sulphur dioxide is already well marked.

For solids and liquids the relation altogether fails. Thus water has a specific inductive capacity which is 80 times greater than that of air, and its refractive index should therefore be equal to 9, or six times larger than its actual value. But these discrepancies are not surprising, for we have left a factor out of consideration, which to a great extent dominates the phenomenon of refraction, and that is absorption. The theoretical relationship really applies only to waves of infinite length, but in most cases we know nothing of the refractive index for very long waves. The subject will be further discussed in the next Chapter.

137. Direction of Electric and Magnetic Forces at right angles to each other. If we confine ourselves for the sake of simplicity to waves, parallel to the plane of xy, we must take in equations (13) and (17) all quantities to be independent of x and y: these equations then become

It follows that there is no component of either the electric or the magnetic force normal to the plane of the wave, and that therefore the whole of the disturbance is in that plane. If the electric disturbance is in one direction only, so that e.g. Q=0, it follows that a = 0, or that the magnetic disturbance is also rectilinear, and at right angles to the electric disturbance. We have therefore for the simplest case of a plane wave, two vectors representing the electric and magnetic forces respectively, and these vectors are at right angles to each other and to the direction of propagation.

More generally let the components P and Q of a plane wave-front parallel to ay be

This shows that also in this more general case the electric and magnetic forces are at right angles to each other.