Page images
PDF
EPUB

the concentrated rays of the electric lamp. Such rays falling on a thin metallic disk, delicately suspended in a vacuum, might perhaps produce an observable mechanical effect. When a disturbance of any kind consists of terms involving sines or cosines of angles which vary with the time, the maximum energy is double of the mean energy. Hence, if P is the maximum electromotive force, and the maximum magnetic force which are called into play during the propagation of light,

[blocks in formation]

With Pouillet's data for the energy of sunlight, as quoted by Thomson, Trans. R. S. E., 1854, this gives in electromagnetic mea

sure

P60000000, or about 600 Daniell's cells per mètre;

[ocr errors]

=

0.193, or rather more than a tenth of the horizontal magnetic force in Britain.

Propagation of a Plane Wave in a Crystallized Medium.

794.] In calculating, from data furnished by ordinary electromagnetic experiments, the electrical phenomena which would result from periodic disturbances, millions of millions of which occur in a second, we have already put our theory to a very severe test, even when the medium is supposed to be air or vacuum. But if we attempt to extend our theory to the case of dense media, we become involved not only in all the ordinary difficulties of molecular theories, but in the deeper mystery of the relation of the molecules to the electromagnetic medium.

To evade these difficulties, we shall assume that in certain media the specific capacity for electrostatic induction is different in different directions, or in other words, the electric displacement, instead of being in the same direction as the electromotive force, and proportional to it, is related to it by a system of linear equations similar to those given in Art. 297. It may be shewn, as in Art. 436, that the system of coefficients must be symmetrical, so that, by a proper choice of axes, the equations become

[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

where K1, K2, and K, are the principal inductive capacities of the medium. The equations of propagation of disturbances are therefore

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

dt2 dz dt)

795.] If l, m, n are the direction-cosines of the normal to the wave-front, and V the velocity of the wave, and if

lx+my+nz - Vt = w,

(3)

and if we write F", G", H", 4" for the second differential coefficients of F, G, II, respectively with respect to w, and put

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

where a, b, c are the three principal velocities of propagation, the

[merged small][ocr errors][ocr errors][merged small][merged small][merged small][subsumed][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

Hence, either = 0, in which case the wave is not propagated at all; or, U= 0, which leads to the equation for V given by Fresnel; or the quantities within brackets vanish, in which case the vector whose components are F", G", H" is normal to the wave-front and proportional to the electric volume-density. Since the medium is a non-conductor, the electric density at any given point is constant, and therefore the disturbance indicated by these equations is not periodic, and cannot constitute a wave. We may therefore consider "0 in the investigation of the wave.

797.] The velocity of the propagation of the wave is therefore completely determined from the equation U = 0, or

[ocr errors][subsumed][subsumed][merged small][merged small]

There are therefore two, and only two, values of V2 corresponding to a given direction of wave-front.

If λ, μ, v are the direction-cosines of the electric current whose components are u, v, w,

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][ocr errors][merged small][merged small]

or the current is in the plane of the wave-front, and its direction in the wave-front is determined by the equation

[blocks in formation]

These equations are identical with those given by Fresnel if we define the plane of polarization as a plane through the ray perpendicular to the plane of the electric disturbance.

According to this electromagnetic theory of double refraction the wave of normal disturbance, which constitutes one of the chief difficulties of the ordinary theory, does not exist, and no new assumption is required in order to account for the fact that a ray polarized in a principal plane of the crystal is refracted in the ordinary manner *.

Relation between Electric Conductivity and Opacity.

798.] If the medium, instead of being a perfect insulator, is a conductor whose conductivity per unit of volume is C, the disturbance will consist not only of electric displacements but of currents of conduction, in which electric energy is transformed into heat, so that the undulation is absorbed by the medium.

If the disturbance is expressed by a circular function, we may F = e-P2 cos (nt — qz),

write

for this will satisfy the equation

(1)

dz2

d2 F
- μκ

[blocks in formation]

=

[blocks in formation]

(2)

dt

[blocks in formation]

* See Stokes' Report on Double Refraction'; Brit. Assoc. Reports, 1862, p. 255.

(3)

(4)

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]

Let R be the resistance, in electromagnetic measure, of a plate whose length is 7, breadth 6, and thickness z,

[ocr errors][merged small][merged small][merged small]

The proportion of the incident light which will be transmitted by this plate will be

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]

799.] Most transparent solid bodies are good insulators, and all good conductors are very opaque. There are, however, many exceptions to the law that the opacity of a body is the greater, the greater its conductivity.

Electrolytes allow an electric current to pass, and yet many of them are transparent. We may suppose, however, that in the case of the rapidly alternating forces which come into play during the propagation of light, the electromotive force acts for so short a time in one direction that it is unable to effect a complete separation between the combined molecules. When, during the other half of the vibration, the electromotive force acts in the opposite direction it simply reverses what it did during the first half. There is thus no true conduction through the electrolyte, no loss of electric energy, and consequently no absorption of light.

800.] Gold, silver, and platinum are good conductors, and yet, when formed into very thin plates, they allow light to pass through them. From experiments which I have made on a piece of gold leaf, the resistance of which was determined by Mr. Hockin, it appears that its transparency is very much greater than is consistent with our theory, unless we suppose that there is less loss of energy when the electromotive forces are reversed for every semivibration of light than when they act for sensible times, as in our ordinary experiments.

801.] Let us next consider the case of a medium in which the conductivity is large in proportion to the inductive capacity.

In this case we may leave out the term involving K in the equations of Art. 783, and they then become

[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

Each of these equations is of the same form as the equation of the diffusion of heat given in Fourier's Traité de Chaleur.

802.] Taking the first as an example, the component F of the vector-potential will vary according to time and position in the same way as the temperature of a homogeneous solid varies according to time and position, the initial and the surface-conditions being made to correspond in the two cases, and the quantity 4μ C being numerically equal to the reciprocal of the thermometric conductivity of the substance, that is to say, the number of units of volume of the substance which would be heated one degree by the heat which passes through a unit cube of the substance, two opposite faces of which differ by one degree of temperature, while the other faces are impermeable to heat *.

The different problems in thermal conduction, of which Fourier has given the solution, may be transformed into problems in the diffusion of electromagnetic quantities, remembering that F, G, H are the components of a vector, whereas the temperature, in Fourier's problem, is a scalar quantity.

Let us take one of the cases of which Fourier has given a complete solution †, that of an infinite medium, the initial state of which is given.

The state of any point of the medium at the time t is found by taking the average of the state of every part of the medium, the weight assigned to each part in taking the average being

[ocr errors]
[blocks in formation]

where is the distance of that part from the point considered. This average, in the case of vector-quantities, is most conveniently taken by considering each component of the vector separately.

* See Maxwell's Theory of Heat, p. 235.

+ Traité de la Chaleur, Art. 384. The equation which determines the temperature, v, at a point (x, y, z) after a time t, in terms of ƒ (a, B, y), the initial temperature at the point (a, B, y), is

[blocks in formation]
« PreviousContinue »