787.]

VELOCITY OF LIGHT.

387

zero except within a certain space S. Then their values at O at the time t will be zero, unless the spherical surface described about O as centre with radius Vt lies in whole or in part within the space S. If O is outside the space & there will be no disturbance at O until Vt becomes equal to the shortest distance from 0 to the space S. The disturbance at O will then begin, and will go on till Vt is equal to the greatest distance from 0 to any part of S. The disturbance at O will then cease for ever.

786.] The quantity V, in Art. 793, which expresses the velocity of propagation of electromagnetic disturbances in a non-conducting medium is, by equation (9), equal to

1

Κμ

1

μπ
v2

so that V, or the velocity

If the medium is air, and if we adopt the electrostatic system of measurement, K = 1 and of propagation is numerically equal to the number of electrostatic units of electricity in one electromagnetic unit. If we adopt the electromagnetic system, K = and μ 1, so that the equation V v is still true.

=

1

v2

On the theory that light is an electromagnetic disturbance, propagated in the same medium through which other electromagnetic actions are transmitted, V must be the velocity of light, a quantity the value of which has been estimated by several methods. On the other hand, v is the number of electrostatic units of electricity in one electromagnetic unit, and the methods of determining this quantity have been described in the last chapter. They are quite independent of the methods of finding the velocity of light. Hence the agreement or disagreement of the values of V and of v furnishes a test of the electromagnetic theory of light.

787.] In the following table, the principal results of direct observation of the velocity of light, either through the air or through the planetary spaces, are compared with the principal results of the comparison of the electric units :

It is manifest that the velocity of light and the ratio of the units are quantities of the same order of magnitude. Neither of them

can be said to be determined as yet with such a degree of accuracy as to enable us to assert that the one is greater or less than the other. It is to be hoped that, by further experiment, the relation between the magnitudes of the two quantities may be more accurately determined.

In the meantime our theory, which asserts that these two quantities are equal, and assigns a physical reason for this equality, is certainly not contradicted by the comparison of these results such as they are.

788.] In other media than air, the velocity V is inversely proportional to the square root of the product of the dielectric and the magnetic inductive capacities. According to the undulatory theory, the velocity of light in different media is inversely proportional to their indices of refraction.

There are no transparent media for which the magnetic capacity differs from that of air more than by a very small fraction. Hence the principal part of the difference between these media must depend on their dielectric capacity. According to our theory, therefore, the dielectric capacity of a transparent medium should be equal to the square of its index of refraction.

But the value of the index of refraction is different for light of different kinds, being greater for light of more rapid vibrations. We must therefore select the index of refraction which corresponds to waves of the longest periods, because these are the only waves whose motion can be compared with the slow processes by which we determine the capacity of the dielectric.

789.] The only dielectric of which the capacity has been hitherto determined with sufficient accuracy is paraffin, for which in the solid form M.M. Gibson and Barclay found *

Dr. Gladstone has found the following values of the index of refraction of melted paraffin, sp. g. 0.779, for the lines A, D and H:-

from which I find that the index of refraction for waves of infinite

length would be about

The

square root of K is

1.422.

1.405.

The difference between these numbers is greater than can be ac

* Phil. Trans. 1871, p. 573.

790.]

PLANE WAVES.

389

counted for by errors of observation, and shews that our theories of the structure of bodies must be much improved before we can deduce their optical from their electrical properties. At the same time, I think that the agreement of the numbers is such that if no greater discrepancy were found between the numbers derived from the optical and the electrical properties of a considerable number of substances, we should be warranted in concluding that the square root of K, though it may not be the complete expression for the index of refraction, is at least the most important term in it.

Plane Waves.

790.] Let us now confine our attention to plane waves, the front of which we shall suppose normal to the axis of z. All the quantities, the variation of which constitutes such waves, are functions of z and t only, and are independent of x and y. Hence the equations of magnetic induction, (A), Art. 591, are reduced to

or the magnetic disturbance is in the plane of the wave.

This

agrees with what we know of that disturbance which constitutes light.

μβ

Putting μa, μẞ and μy for a, b and c respectively, the equations of electric currents, Art. 607, become

Hence the electric disturbance is also in the plane of the wave, and if the magnetic disturbance is confined to one direction, say that of x, the electric disturbance is confined to the perpendicular direction, or that of

y.

But we may calculate the electric disturbance in another way, for if f, g, h are the components of electric displacement in a nonconducting medium

If P, Q, R are the components of the electromotive force

(15)

(16)

and since there is no motion of the medium, equations (B), Art. 598,

Comparing these values with those given in equation (14), we find

The first and second of these equations are the equations of propagation of a plane wave, and their solution is of the well-known form

F = ƒ1 (≈ − Vt) +ƒ2 (z+Vt),

where A and B are functions of z. His therefore either constant or varies directly with the time. In neither case can it take part in the propagation of waves.

electric displacement ✦

791.] It appears from this that the directions, both of the magnetic and the electric disturbances, lie in the plane of the wave. The mathematical form of the disturbance therefore, agrees with that of the disturbance which constitutes light, in being transverse to the direction of propagation.

magnetic force +

=

If we suppose G = 0, the disturbance will correspond to a plane-polarized ray of light.

The magnetic force is in this case paral

Fig. 66.

netic force is therefore in a plane perpen

dicular to that which contains the electric force.

The values of the magnetic force and of the electromotive force at a given instant at different points of the ray are represented in Fig.66,

793.]

ENERGY AND STRESS OF RADIATION.

391

for the case of a simple harmonic disturbance in one plane. This corresponds to a ray of plane-polarized light, but whether the plane of polarization corresponds to the plane of the magnetic disturbance, or to the plane of the electric disturbance, remains to be seen. See Art. 797.

Energy and Stress of Radiation.

792.] The electrostatic energy per unit of volume at any point of the wave in a non-conducting medium is

In virtue of equation (8) these two expressions are equal, so that at every point of the wave the intrinsic energy of the medium is half electrostatic and half electrokinetic.

Let p be the value of either of these quantities, that is, either the electrostatic or the electrokinetic energy per unit of volume, then, in virtue of the electrostatic state of the medium, there is a tension. whose magnitude is p, in a direction parallel to x, combined with a pressure, also equal to p, parallel to y and z. See Art. 107.

In virtue of the electrokinetic state of the medium there is a tension equal to p in a direction parallel to y, combined with a pressure equal to p in directions parallel to a and z. x See Art. 643.

Hence the combined effect of the electrostatic and the electrokinetic stresses is a pressure equal to 2p in the direction of the propagation of the wave. Now 2p also expresses the whole energy in unit of volume.

Hence in a medium in which waves are propagated there is a pressure in the direction normal to the waves, and numerically equal to the energy in unit of volume.

793.] Thus, if in strong sunlight the energy of the light which falls on one square foot is 83.4 foot pounds per second, the mean energy in one cubic foot of sunlight is about 0.0000000882 of a foot pound, and the mean pressure on a square foot is 0.0000000882 of a pound weight. A flat body exposed to sunlight would experience this pressure on its illuminated side only, and would therefore be repelled from the side on which the light falls. It is probable that a much greater energy of radiation might be obtained by means of