The somewhat important question relating to the amount of light reflected at normal incidence has been investigated directly by E. Hagen and H. Rubens*. Some of their results are embodied in Table XII. A comparison with Drude's numbers shows generally a good agreement. The exceptions are Platinum and especially Copper. The alloy of Brandes and Schünemann is of practical importance owing to its permanence and resistance to deterioration when exposed to moist or impure air. Drude has also determined v, for red light and found that all metals except lead, gold, and copper, refract the red more than the yellow light. The coefficient of extinction was determined directly by Rathenau t. There is another difficulty which consists in the fact, already pointed out by Maxwell in his Treatise, that the simple theory according to which v。。 should be (neglecting B) equal to CVX gives too large values for the coefficient of extinction. The introduction of A and B does not entirely remove the difficulty, but as was first shown by Drude, a satisfactory explanation may be arrived at, if we adopt the theory of electrons throughout. The whole electric current according to this theory is carried by ions which possess effective inertia inasmuch as they possess energy proportional to the square of the velocity, the energy being that of the magnetic field they create. In the ordinary theory of conductors, this energy is only partially taken account of, the average magnetic field established by the moving electron being used in the calculations. If instead of an evenly distributed electric fluid, we imagine electricity to be concentrated within the electron, the magnetic field in the immediate neighbourhood of that electron will be very much larger than the average energy in each element of volume. Hence the inertia, or the coefficient of self-induction, by whichever name the factor in question may be called, is underrated in the ordinary treatment. The error committed depends on the nearness of the moving electrons, and on their linear dimensions, but if their distance apart is great compared with their diameter, the additional energy per unit volume is where i is the current density and σ stands for p/Net, N being the number of moving electrons per unit volume, and ρ the apparent mass. If each molecule supplies one electron, which carries the conduction current, σ is of the order of magnitude 5 × 10-11 and has the dimensions of a surface. * Drude's Annalen, vol. 1. p. 352. + Quoted in Winkelmann, Encyclopædia der Wissenschaften, vol. II. p. 838. A. Schuster, Phil. Mag. vol. 1. p. 227 (1901). The effect of the inertia is the same as that of an electric force o di dt opposing the current. If C be the conductivity this electric force prodi dt duces a current density Co current. where only relates to the conduction The equation of electric current (33) now becomes, if we denote by w' the component of the conduction current, or with the help of (34), omitting the last term on the right-hand side of that equation, Net) - oC V2R. 4π Differentiating with respect to the time and applying (34) again, we find D {4πCR + (1 + σCD) (K dR + 4π Nel If the motion is periodic and contains e-iwt as factor, we may substitute –¿w for D, and for ¿ we may use its equivalent (37) in terms of R. The equation then becomes ...(75), 2vxo = V2 (NBe+2) where in the last term, is written for 2π/w. T The effect of or is therefore to diminish the product vok, and hence to diminish that part of the absorption which depends on conductivity. Equations (75) are, allowing for a change of notation, identical with those obtained by Drude. If we disregard A and B, we obtain If the numerical values of C and w are introduced and the quantity σ is estimated, it is found that oCo is large, and equations (75) become with sufficient accuracy 4π V2 + NAV2e, σω 2V0o= V2 (22+NBe). As B is always positive, it follows that This relation allows us to calculate the number of electrons which take part in conduction currents, and it is found that this number in the different metals is of the same order of magnitude as the number of molecules*. 157. Reflecting powers of metals for waves of low frequency. Maxwell's theory has received an important confirmation in the work recently published by Hagen and Rubenst, on the relation between the optical and electrical qualities of metals. These investigations relate to waves of low frequency. Neglecting σ, we may write equations (76) v2 - K2 = 1, Vo Ko = CVX. As is supposed to be large, both v and κ must be large and nearly equal. The second equation gives, neglecting the difference between the two quantities, To test this formula for long waves, Hagen and Rubens measure the reflecting powers at normal incidence. For the intensity of the reflected light, we have obtained the expression (64), which by substitution becomes (k-1)/(k+1) for large values of Ko Writing R for the reflecting powers of a metal in per cent., 100-R' gives the intensity of the light which enters the metal, the * Schuster, Phil. Mag., Vol. VII. p. 151 (1904). Ann. d. Physik, Vol. XI. p. 873 (1903) and Phil. Mag. Vol. VII. p. 157 (1904). intensity of incident light being 100, and the formula to be verified becomes In Table XIII., columns 2, 3, the observed and computed values of 100 - R' are tabulated for λ = 12μ (μ = 101 cms.). For larger values of λ, R' approaches 100% asymptotically with increasing wave-lengths, and the difficulty of experimentally determining 100 – R' increases accordingly. Hagen and Rubens therefore measured the emissive power instead of the reflecting power of the metals. From a comparison of the radiation sent out by a metal with that sent out by a black body, the reflecting power may be directly deduced. In Table XIII., column 4 gives the conductivity at 170° C. calculated from the known conductivities at 18° and the temperature coefficient. Columns 5 and 6 give the observed and computed emissive powers at 170°. It will be noticed that the agreement is excellent in all cases except Aluminium which shows a considerable deviation, and Bismuth which forms a complete exception to the law. Professors Hagen and Rubens have also directly verified the fact that the quantity 100 R' indicates with increasing temperature a change corresponding to the change of electrical resistance, and they further point out the remarkable fact that it would be possible to undertake absolute determinations of electrical resistance solely by the aid of measurements on radiation. The agreement of Maxwell's theory in its simple original form with the result of the experiments just described, proves that for wave-lengths as great as 12, the free periods of vibration of the molecules do not affect the optical constants of metals. 158. Connexion between refractive index and density. The investigations of Sellmeyer and those based on similar principles all lead to the conclusion that for any one kind of molecule, μ2-1 is proportional to the density. We possess a good many experimental investigations on the changes observed when the density (D) is altered by pressure or by temperature, and these investigations have not been favourable to the constancy of (μ2 – 1)/D. The failure of the formula in the case of a change from the liquid to the gaseous state is not perhaps surprising because the molecule in the liquid state may be expected to be more complex and behave optically very differently from the molecule of the same substance in the state of a gas. Even changes of temperature and pressure may to some extent affect the absorbing power, and consequently the velocity of propagation of light-waves, so that our theory cannot pretend completely to include such changes. But though the failure of the theory is not surprising it must be pointed out that other formulae give better results. The simpler relationship which asserts the constancy of (μ-1)/D discovered empirically by Gladstone and Dale, has often been successfully applied, and two authors of similar name, H. A. Lorentz of Leyden*, and L. Lorenz† of Kopenhagen, have almost simultaneously published investigations leading to the result that (μ2 − 1)/(μ2 + 2)D is constant. The latter formula is capable of predicting with fair accuracy the refractive index of a gas, that of the liquid being known. Lord Rayleigh's investigations on the effect of spherical obstacles * Wied. Ann. Ix. p. 641. (1880.) |