series including a large number of different wave-lengths. It may be perfectly correct, that as regards the mechanism causing the limits of interference the above explanation is true, and we should in that case be justified in saying that the regularity of the motion is interfered with by molecular impacts, but this is the same thing as saying that the homogeneity of the light sent out is destroyed by these impacts. Without forming any hypothesis as to the mechanical cause which produces the irregularity, we simply state the facts if we ascribe the limit of the power of interference to the want of homogeneity of the source. The reason for the absence of homogeneity may then be left over for an independent discussion. Experimental investigation of the retardation at which interference is still possible coincides therefore with the investigation as to the homogeneity of the light. Its importance begins when we come to study the details of the structure of the radiations of luminous gases which in ordinary language are described as being homogeneous. The explanation which depends on the assumption of sudden changes of phase by molecular impacts, is also generally used to account for the absence of interference of two different sources, and here again the mechanical explanation may be perfectly correct. But it is important to realize that two independent sources sending out homogeneous light of exactly the same wave-length are capable of interfering with each other in exactly the same way as if the light were derived from the same source. If two waves, spreading out from e.g. two mercury lamps, which give out nearly homogeneous radiations are found not to interfere with each other, it only means that the homogeneity is not sufficient. When an impulsive motion of definite type is analysed by Fourier's series, it is found that there is a certain definite relation of phase between the waves of different periods. Some explanation may be necessary to bring this into agreement with our previous view of white light, according to which there could be no relation of phase between any two wave-lengths however near each other. It is obvious at once that no phase relation can exist when the light is such that its average intensity is constant. In fact, such a relation is inconceivable. As it exists for a single impulse, it must be destroyed by the succession of impulses which converts the instantaneous source of light into one of constant average intensity. 186. Talbot's Bands. If the spectrum formed by a prism or grating is observed, half the pupil of the eye being covered with a thin plate of mica or glass, the spectrum is seen to be traversed by dark bands, provided the plate be inserted on that side on which the blue of the spectrum appears. These bands were first observed by Fox Talbot. Instead of viewing the spectrum directly we may use a telescope, the plate being inserted on the side of the thin edge of the prism forming the spectrum, so as to cover a portion of the aperture of the object glass. Similar bands have been observed by Powell, who used a hollow glass prism with its refracting edge pointing downwards and filled with some highly refractive liquid, into which he inserted a plate of glass with its lower edge parallel to the edge of the prism and so that its plane approximately bisected the angle of the prism. The plate was only partially inserted, so as to leave the lower portion clear. The bands only appeared when the refractive index of the liquid was greater than that of the glass, but Stokes showed that when the refractive index of the glass was the greater of the two the band could still be observed, only in this case it was necessary to place the plate in the thinner part of the prism, leaving the thicker portion clear. A simple explanation of these bands is sometimes based on the consideration that the two portions of the light, which, in the absence of the interposed plate, would reach the retina in the same phase, are retarded relatively to each other by the plate, so that interference may take place. This explanation is obviously incomplete, for it leaves out of account the essential fact that the effects are only observed when the plate is inserted on one side and not on the other. A more complete explanation taking account of this want of symmetry has been given by Airy and Stokes, and involves an elaborate mathematical process. A very simple treatment may be given if, instead of basing the calculation on Fourier's analysis, we consider the source of the light to be due to a succession of impulsive velocities. In Fig. 76 (Art. 59) we have a wave-front consisting of a simple impulse which reaches the grating so that the points A1, A2, A3, etc. are simultaneously disturbed. At the plane HK, the disturbance will reach the points C1, C2, C3, in succession, and if a lens be placed with its axis at right angles to HK, the disturbance will pass the focus of the lens at regular intervals of time, as already explained in Art. 183. The question now is: How can the impulses which succeed each other at the focus of the lens, be made to interfere with each other? Clearly only by retarding those which reach the focus first or by accelerating those which reach it last. A plate of appropriate thickness introduced on the left-hand side of the figure as it is drawn could be made to answer the purpose. If, on the contrary, the same plate be introduced on the right-hand side, it would only retard those impulses which already arrive late, and therefore no interference could take place. This is all that need be said in explanation of the bands, but a more detailed consideration leads to a simple expression for the calculation of the thickness of the plate which shows the bands most distinctly. It is easily seen that the best thickness is secured when the whole series of impulses is divided into two equal portions, the impulses arriving in pairs simultaneously at the focus. The retardation must therefore be such that the retarded impulse coming from the first line of the grating, and the unretarded impulse coming from the central line, arrive together. This means that the retardation is Nλ, if N is the total number of lines in the grating, and the plate should be pushed sufficiently far into the beam to affect half its width. The wave-length A here means the wave-length of that homogeneous train of waves which has its first principal maximum at the focus of the lens, so that the retardation of each impulse compared with the next is λ. If the retardation is either greater or smaller, some of the impulses arrive too soon or too late to overlap others, and the bands are less clear. If the retardation has more than twice its most effective value, the series of impulses from the first half of the grating pass through the focus later than those coming from the second half, and hence there cannot be any interference. If at a certain point of the spectrum corresponding to a wavelength λ there is a maximum of light, the relative retardation of the two interfering impulses must be equal to mλ, m being an integer; the next adjoining band towards the violet will appear at a wave-length X' such that mλ= (m + 1) X'. Hence for the distance between the bands with the best thickness of interposed plate, m=1N, and hence (A-X')'=2/N where in the denominator may with sufficient If X" be that wave-length nearest to λ accuracy be replaced by λ. at which there is a minimum of light, it follows that If a linear homogeneous source of light of wave-length A be examined by means of a grating, the central image extends to a wavelength A, such that where N, as before, is the total number of lines on the grating. Hence the following proposition:-If, in observing Talbot's bands, the best thickness of retarding plate be chosen, the distance between each maximum and the nearest minimum is equal to the distance between the central maximum and the first minimum of the diffractive image of homogeneous light, observed in the same region of the spectrum with the same optical arrangement. This proposition holds for all orders of spectra; but the appropriate thickness of the retarding plate increases in the same proportion as the order. If we use prisms instead of a grating, the number of lines N must be replaced by the quantity which corresponds to it as regards resolving power, viz., tdμ/dλ where t is the aggregate effective thickness of the prisms. It follows that the retardation which gives the best interference bands with prisms, is Atdu/dλ. The above explanation was given, with further details, in a recent communication to the Philosophical Magazine*. 187. Roentgen radiation. The radiations which are produced by the impact of kathode rays on solid bodies, possess, as shown by their discoverer, some properties which at first seem to distinguish them from the transverse waves of light. Roentgen could find no trace of refraction or polarization, and he tentatively suggested that the rays he had discovered were due to longitudinal vibrations. The absence of diffraction and interference was also felt to be a difficulty in ascribing the observed phenomena to a wave-motion. It was however almost immediately pointed out that according to Sellmeyer's equation waves of very short length would not suffer any refraction, the velocity of propagation through all media being the same, and that the want of homogeneity would be quite sufficient to account for the apparent absence of interference. The absence of polarization by reflexion or refraction follows from the equality of the velocities of propagation. Some years later Professor J. J. Thomson calculated the electromagnetic disturbance sent out by the impulsive stoppage of an electrified particle moving with great velocity. Such an electromagnetic wave possesses all the properties we have ascribed to white light, and would only be distinguishable from it by the smallness of the linear quantity involved. Thus in the hypothetical law of radiation expressed by (8), the disturbance is the more sudden the greater the value of c, and the linear quantity, V/c, is a measure of the distance in space over which the disturbance is appreciable. If this view as to the nature of the Roentgen rays is right, it would still be correct to say that these rays are due to very short waves, the spectrum being continuous and extending over a considerable range of wave-lengths. If there is any distinction between the statements that Roentgen rays consist of short *Phil. Mag. Vol. vii. p. 1 (1904). + Nature, Vol. LIII. p. 268 (1896). waves and that they are due to irregular impulses, it can only lie in the fact that the first statement excludes long waves and the second does not. The matter is introduced here because the mechanical stoppage of a moving electrified particle may serve as a good example of the kind of wave we imagine white light to consist of. 188. The radiation of a black body. All bodies of sufficient thickness send out radiations which are independent of the nature of the body, and therefore identical with those sent out by a perfectly black surface. The experimental investigation of the law of complete radiation of a black body presents considerable difficulties which have only recently been partially overcome. According to Stefan's law the intensity of the total radiation is proportional to the fourth power of the absolute temperature. This law, for which there is some theoretical justification, has been found to be correct so far as our observations are able to go. We cannot enter here into a complete account of the subject, but must refer to the papers of W. Wien*, and others, whose work has been summarized in an important paper by H. Rubens and F. Kurlbaum†. The distribution of energy in the spectrum seems to be best represented by a formula first given by M. Planck. If C is a constant, and the energy lying between the wave-lengths A and A+ dλ be Edλ, If this formula is examined it will be found that for a given value of the absolute temperature T the value E is a maximum, and that this maximum satisfies the condition Amax. T= const. If the quantity E is calculated for that value of λ at which it is a maximum it is found to be proportional to the fifth power of the absolute temperature. These two important relations were derived by W. Wien from theoretical considerations. The manner in which the complete radiation of a body is independent of its nature, and yet vibrations of individual molecules as observed by spectroscopic analysis are entirely characteristic of the nature of the body, forms a great difficulty, in the satisfactory treatment of the subject of radiation. Attempts have been made to show that the white light of the complete radiation is an entirely distinct phenomenon from the homogeneous vibration which we observe in the spectra of gases. But the theory of exchanges shows that there must be some connexion Wied. Ann. Vol. LVIII. p. 62 (1896). |