2. A small source of homogeneous light is reflected in three mirrors, in such a manner that the images are equally bright and form an equilateral triangle abc, whose centre of gravity is o. A, B, C, O are the projections of a, b, c, o upon a screen which is parallel to the plane a, b, c. Show that the intensity at any point P in the line OA, is to that at O, in the ratio 1-sin2 3πpd/2λh: 1, where h is the distance of the screen, and PO= p, oa = d. CHAPTER III. COLOURS OF THIN AND THICK PLATES. 25. WHEN light is incident upon a thin film of a transparent substance, such as a soap-bubble, brilliant colours are observed. The explanation of this phenomenon is, that the light upon incidence upon the outer surface of the film, is separated into two portions, the first of which is reflected by the outer surface, whilst the second portion is refracted. The refracted portion is reflected from the second surface of the film, and afterwards refracted by the outer surface; and since the thickness of the film is very small, the difference of the paths of the two portions is comparable with the wave-length, and the two streams are therefore in a condition to interfere. Accordingly if sunlight is employed, a series of brilliantly coloured bands is observed. In order to obtain a mathematical theory of these bands, we shall suppose that two plates of glass cut from the same piece, are placed parallel to one another with a thin stratum of air between them; and we shall investigate the intensity of the reflected light. 26. It will be proved in a future Chapter, that when light is reflected or refracted at the surface of a transparent medium, the intensities of the reflected and refracted light are altered in a manner, which depends upon the angle of incidence and the index of refraction. The mathematical formulæ, which determine the intensities in these two cases, depend partly upon the particular dynamical theory which we adopt, and partly upon the state of polarization of the light. If however the angle of incidence of the light, which is refracted from the plate into the stratum of air, is less than the critical angle, we can achieve our object PRINCIPLE OF REVERSION. 29 without the assistance of any dynamical theories, by the aid of a principle due to Stokes', called the Principle of Reversion. Let S be the surface of separation of two uncrystallized media; let A be the point of incidence of a ray travelling along IA in the first medium, and let AR, AF be the reflected and refracted rays; also let AR' be the direction of the reflected ray for a ray incident along FA in the second medium. Then the principle asserts, that if the two rays AR, AF be reversed, so that RA and FA are reflected and refracted at A, they will give rise to the incident ray AI. Let A be the amplitude of the incident light; and let Ab, Ac be the amplitudes of the reflected and refracted light, when the first medium is glass and the second is air; also let Ae, Aƒ be the amplitudes of the reflected and refracted light, when light of amplitude A is refracted from air into glass. Then if AR be reversed, it will give rise to Ab reflected along AI, Abc refracted along AR'. Similarly if AF be reversed, it will give rise to Ace reflected along AR', Acf refracted along AI. Since the two rays superposed along AR' must destroy one another, whilst the two rays superposed along AI must be equivalent to the incident ray, we obtain b+e=0, b2 + cf = 1....... 27. We are now in a position to calculate the intensity. ..(1). be the incident vibration at A; let i be the angle of incidence, r that of refraction, D the thickness of the stratum of air. 1 Camb. and Dublin Math. Journ. vol. iv. p. 1; and Math, and Phys. Papers, vol. II. p. 89. The light which is incident at A, is reflected and refracted, and a portion of the latter is reflected at B1, and the reflected portion is again reflected and refracted at A, and so on ad infinitum. It therefore follows, that the light refracted at A which is due to light incident at A1, is represented by Also if x be the wave-length in glass, the vibration at A due to the light which is reflected at A, is where is the retardation of the light which was refracted at A. Taking account of (1), and also of the infinite series of reflections and refractions at A, A, ...... etc., we obtain for the resulting vibration at A y = Ab [sin & − (1 − b3) {sin ($ + d) + b2 sin (p + 28) + b1 sin (p + 38) Summing this series we obtain y = 2Ab (1 + b2) sin2 18 sin & – Ab (1 − b2) sin 8 cos & 1-262 cos 8 + b 1 whence the intensity is equal to +......}]. Since AB, D sec r, AM = 2D tan r sin i, X'/λ = sin i/sin r, we obtain 8=4π-D cos r. ·(5). Although we have supposed at the commencement, that a thin film of air is contained between two plates of glass very near one another, it is evident that the preceding investigation will apply equally well to a soap-bubble. If D= 0, then 80 and I=0; accordingly when a soap-bubble becomes exceedingly thin just before bursting, it appears to be black. The intensity will also vanish when 2D cos r = nλ, where n is an integer; but since this condition depends upon λ, it follows that when sunlight is employed, the intensity will never become absolutely zero, but the film will be coloured. In order to obtain the intensity of the transmitted light, it can be shown in a precisely similar manner, that the vibration which emerges at B, is represented by y= Acf (sin + e2 sin (+8) + e sin (p + 28) + ..............}. Summing this series, and taking account of (1), we shall find that the intensity is A2 (1-62)2 .(6). .(7) or the sum of the intensities of the reflected and transmitted lights is equal to that of the incident light. This result is sometimes expressed by saying, that the reflected and transmitted lights are complementary to one another. It must however be borne in mind, that (7) is not strictly accurate for ordinary transparent media, inasmuch as a portion of the light is always absorbed in transmission through the plate; it only becomes true in the limit for perfectly transparent substances. Newton's Rings. 28. The coloured rings produced by thin plates were first investigated experimentally by Newton, who produced them by pressing a convex lens down upon a flat piece of glass; the experiment may also be performed by pressing a prism upon the face of a convex lens. Since the curvature of the lens is exceedingly small in comparison with the wave-length of light, the two surfaces |