Common Light. 240. The distinguishing feature of common light is, that it exhibits no trace of polarization; and the theory of sources of light given in § 232 furnishes an explanation of the reason why it is, that the light emitted from an incandescent substance is unpolarized. The molecules of an incandescent body are in a violent state of vibration; each molecule may therefore be regarded as a centre of disturbance, which produces ethereal waves. The most general form of the waves produced by any molecule is given by (45), but for simplicity, we shall confine our attention to the first term of this series for which n = 0. It therefore follows from (46), that at a distance from the molecule, which is large compared with the wave-length of light, the displacements would be represented by the equation 1 u = [ + + 2 = {Ar2 = 3 (da + By + Cz) a}] cos 2 (btr) - (Ax λ with symmetrical expressions for v and w, where A, B, C are proportional to the direction cosines of the direction of vibration of the molecule. This expression represents a spherical wave of light, whose direction of vibration lies in the plane passing through the line of vibration of the molecule, and the line joining the latter with the eye of the observer. But owing to a variety of causes, amongst which may be mentioned collisions, which are continually taking place between the molecules, the line of vibration of any particular molecule is perpetually changing, so that the angular motion of this line is most irregular. These changes take place in all probability with a rapidity, which is comparable with the period of waves of light, so that it is impossible for the eye to take cognizance of any particular direction. Moreover the light which is received from an incandescent body, is due to the superposition of the waves. produced by an enormous number of vibrating molecules, the lines of vibration of each of which are different, and are continually changing. Hence the actual path which any particle of ether describes during a complete period is an irregular curve, whose form changes many million times in a second. We thus see why it is that common light is unpolarized. 241. We can now understand why interference fringes cannot be produced by means of light coming from two different sources. For the production of these fringes requires, that there should be a fixed relation between the phases of the two streams; but inasmuch as the two streams are affected by two distinct sets of irregularities, no fixed phase relation between them is possible. If however the two streams come from the same source, the irregularities by which the two streams are affected are identical, and consequently a fixed phase relation will exist between them. EXAMPLES. 1. A luminous point is surrounded by an atmosphere containing a number of small equal particles of dust, the density of whose distribution varies inversely as the nth power of the distance from the point, and which scatter the light incident upon them. Show that except in the immediate vicinity of the luminous point, the (n + 1)/(n + 3)th part of the whole light scattered by the dust will be polarized. 2. Establish the truth of Stokes' expression for the effect of an element of an infinite plane wave at a point Q, by integration over the whole wave-front. If the wave be finite, and all points of its boundary be at the same distance a from Q, prove that the displacement at Q will be where x is the distance of Q from the wave at the plane of resolution. 3. In a biaxal crystal the ratios of the axes of the ellipsoid of elasticity are slightly different for different colours, so that the angles between the optic axes for yellow and violet are a, a + p. The normal to a wave-front of white light in such a crystal makes angles 01, 0, with the mean optic axis, and the planes through the normal and the optic axes make an angle w with one another. Show that the directions of polarization lie within a small angle CHAPTER XIV. GREEN'S THEORY OF DOUBLE REFRACTION. 242. THE theory of double refraction proposed by Green', is the theory of the propagation of waves in an aolotropic elastic medium. We have stated in Chapter XI., that the potential energy of such a medium is a homogeneous quadratic function of the six components of strain; and we shall now proceed to examine this statement. Let O be any point of the medium, and let OA, OB, OC be the sides of an elementary parallelopiped of the medium when unstrained. Then any strain which acts upon the medium, will produce the following effects upon the element. (i) Every point of the element will experience a bodily displacement. (ii) The three sides OA, OB, OC will be elongated or contracted. (iii) The element will be distorted into an oblique parallelopiped. Let u, v, w be the component displacements at 0; e, f, g the extensions of OA, OB, OC; a, b, c the angles which the faces OCA, OAB, OBC make with their original positions. Since a bodily displacement of the medium as a whole, cannot produce any strain, it follows that the potential energy due to strain cannot be a function of u, v, w; but since any displacement, which 1 Trans. Camb. Phil. Soc. 1839; Math. Papers, p. 291. POTENTIAL ENERGY. 253 produces an alteration of the forms (ii) or (iii) must necessarily endow the medium with potential energy, it follows that the potential energy due to strain, must be a function of the six strains e, f, g, a, b, c. 243. The most general form of the potential energy W, is given by the equation where W is a homogeneous n-tic function of the strains. It is evident that W cannot contain a constant term of the form Wo for when the medium is unstrained, the potential energy is zero. The most general expression for W1 is W1 = Ee+Ff Gg + Aa + Bb + Cc, 1 where E, F... are constants. Now Green supposed, that if the medium were subjected to external pressure, the first three terms of W, might come in; but it appears to me that this hypothesis is untenable. For if P be the stress of type e, then accordingly if W contained a term W1, stresses would exist, when the medium is free from strain. If the medium were absolutely incompressible, the stresses might undoubtedly contain terms independent of the strains. For if a portion of such a medium were enclosed in a rectangular box, and stresses E, F, G, A, B, C were applied to the sides of the box, of such magnitude as to preserve its rectangular form, no displacement, and consequently no strain would be produced, on account of the incompressibility of the medium; but the internal stresses would contain terms. depending on the values of the surface stresses. These surface stresses could not however give rise to any terms in the potential energy, inasmuch as they do no work. If on the other hand, the medium were compressible, the effect of the surface stresses would be to produce displacements, and consequently strains depending upon them, in the interior of the medium; hence the internal stresses P, Q,... could not contain any terms independent of the strains, and the term W, could not exist. We have already pointed out, that in order to get rid of the pressural or dilatational wave, it is unnecessary to make the extravagant assumption, that the medium is incompressible; all that it is necessary to ass is, that the constants upon which compressibility depends, 1 1 large in comparison with those upon which distortion depends. Under these circumstances, we conclude that W, is zero, and that the internal stresses do not contain any terms independent of the strains. Also since the terms W3, W... would introduce quadratic and cubic terms into the equations of motion, they will be neglected. 244. The potential energy is therefore a homogeneous quadratic function of the six strains, and accordingly contains twentyone terms. Biaxal crystals, however, have three rectangular planes of symmetry; and as Green's object was to construct a theory which would explain double refraction, he assumed that the medium possessed this property. Whence the expression for W reduces to the following nine terms, and may be written. 2 W = Ee2 + Ff2 + Gg2 + 2E'fg +2F'ge + 2G'ef + Aa2 + Bb2 + Cc2........ where e, f, g, a, b, c are the six strains. .(1), The coefficients in the expression for W are all constants, depending on the physical properties of the medium. The first three, E, F, G are called by Rankine' coefficients of longitudinal elasticity; the second three, E, F, G' are called coefficients of lateral elasticity; whilst the last three, A, B, C are the three principal rigidities. 245. The waves which are capable of being propagated in an isotropic medium, have already been shown to consist of two distinct types, which are propagated with different velocities; viz. longitudinal waves, which involve dilatation unaccompanied by distortion; and transversal waves, which involve distortion unaccompanied by dilatation. Waves of the first type depend upon the dilatation 8, and do not involve rotation; hence the rotations En, are zero, and the displacements are the differential coefficients of a single function 4. Waves of the second type depend upon the rotations §, n, C, and do not involve dilatation; hence 8 is zero, and the displacements must therefore satisfy the equation du dv dw which is the condition, that the displacement should be perpendicular to the direction of propagation. 1 Miscellaneous Scientific Papers, p. 107. |