with temperature. In this case the solution of the equation reduces to the form
| θ＝x(𝑙−x)C2R0/2lqk. | (9) |
By a property of the parabola, the mean temperature is 23rds of the maximum temperature, we have therefore
| (R−R0)/𝑎R0＝𝑙C2R0/12qk, | (10) |
which gives the conductivity directly in terms of the quantities actually observed. If the dimensions of the bar are suitably chosen, the distribution of temperature is always very nearly parabolic, so that it is not necessary to determine the value of the critical current C2=hpl/𝑎R0 very accurately, as the correction for external loss is a small percentage in any case. The chief difficulty is that of measuring the small change of resistance accurately, and of avoiding errors from accidental thermo-electric effects. In addition to the simple measurements of the conductivity (M‘Gill College, 1895–1896), some very elaborate experiments were made by King (Proc. Amer. Acad., June 1898) on the temperature distribution in the case of long bars with a view to measuring the Thomson effect. Duncan (M‘Gill College Reports, 1899), using the simple method under King’s supervision, found the conductivity of very pure copper to be 1·007 for a temperature of 33° C.
(b) The method of Kohlrausch, as carried out by Jaeger and Dieselhorst (Berlin Acad., July 1899), consists in observing the difference of temperature between the centre and the ends of the bar by means of insulated thermo-couples. Neglecting the external heat-loss, and the variation of the thermal and electric conductivities 𝑘 and 𝑘′, we obtain, as before, for the difference of temperature between the centre and ends, the equation
| θmax−θ0＝C2R𝑙/8qk＝EC𝑙/8qk＝E2𝑘′/8𝑘, | (11) |
where E is the difference of electric potential between the ends. Lorenz, assuming that the ratio 𝑘/𝑘′ =𝑎θ, had previously given
| θ2max−θ02＝E2/4𝑎, | (12) |
which is practically identical with the preceding for small differences of temperature. The last expression in terms of 𝑘/𝑘′ is very simple, but the first is more useful in practice, as the quantities actually measured are E, C, 𝑙, q, and the difference of temperature. The current C was measured in the usual way by the difference of potential on a standard resistance. The external heat-loss was estimated by varying the temperature of the jacket surrounding the bar, and applying a suitable correction to the observed difference of temperature. But the method (𝑎) previously described appears to be preferable in this respect, since it is better to keep the jacket at the same temperature as the end-blocks. Moreover, the variation of thermal conductivity with temperature is small and uncertain, whereas the variation of electrical conductivity is large and can be accurately determined, and may therefore be legitimately utilized for eliminating the external heat-loss.
From a comparison of this work with that of Lorenz, it is evident that the values of the conductivity vary widely with the purity of the material, and cannot be safely applied to other specimens than those for which they were found.
19. Conduction in Gases and Liquids.—The theory of conduction of heat by diffusion in gases has a particular interest, since it is possible to predict the value on certain assumptions, if the viscosity is known. On the kinetic theory the molecules of a gas are relatively far apart and there is nothing analogous to friction between two adjacent layers A and B moving with different velocities. There is, however, a continual interchange of molecules between A and B, which produces the same effect as viscosity in a liquid. Faster-moving particles diffusing from A to B carry their momentum with them, and tend to accelerate B; an equal number of slower particles diffusing from B to A act as a drag on A. This action and reaction between layers in relative motion is equivalent to a frictional stress tending to equalize the velocities of adjacent layers. The magnitude of the stress per unit area parallel to the direction of flow is evidently proportional to the velocity gradient, or the rate of change of velocity per cm. in passing from one layer to the next. It must also depend on the rate of interchange of molecules, that is to say, (1) on the number passing through each square centimetre per second in either direction, (2) on the average distance to which each can travel before collision (i.e. on the “mean free path”), and (3) on the average velocity of translation of the molecules, which varies as the square root of the temperature. Similarly if A is hotter than B, or if there is a gradient of temperature between adjacent layers, the diffusion of molecules from A to B tends to equalize the temperatures, or to conduct heat through the gas at a rate proportional to the temperature gradient, and depending also on the rate of interchange of molecules in the same way as the viscosity effect. Conductivity and viscosity in a gas should vary in a similar manner since each depends on diffusion in a similar way. The mechanism is the same, but in one case we have diffusion of momentum, in the other case diffusion of heat. Viscosity in a gas was first studied theoretically from this point of view by J. Clerk Maxwell, who predicted that the effect should be independent of the density within wide limits. This, at first sight, paradoxical result is explained by the fact that the mean free path of each molecule increases in the same proportion as the density is diminished, so that as the number of molecules crossing each square centimetre decreases, the distance to which each carries its momentum increases, and the total transfer of momentum is unaffected by variation of density. Maxwell himself verified this prediction experimentally for viscosity over a wide range of pressure. By similar reasoning the thermal conductivity of a gas should be independent of the density. This was verified by A. Kundt and E. Warburg (Jour. Phys. v. 118), who found that the rate of cooling of a thermometer in air between 150 mm. and 1 mm. pressure remained constant as the pressure was varied. At higher pressures the effect of conduction was masked by convection currents. The question of the variation of conductivity with temperature is more difficult. If the effects depended merely on the velocity of translation of the molecules, both conductivity and viscosity should increase directly as the square root of the absolute temperature; but the mean free path also varies in a manner which cannot be predicted by theory and which appears to be different for different gases (Rayleigh, Proc. R.S., January 1896). Experiments by the capillary tube method have shown that the viscosity varies more nearly as θ34, but indicate that the rate of increase diminishes at high temperatures. The conductivity probably changes with temperature in the same way, being proportional to the product of the viscosity and the specific heat; but the experimental investigation presents difficulties on account of the necessity of eliminating the effects of radiation and convection, and the results of different observers often differ considerably from theory and from each other. The values found for the conductivity of air at 0° C. range from ·000048 to ·000057, and the temperature coefficient from ·0015 to ·0028. The results are consistent with theory within the limits of experimental error, but the experimental methods certainly appear to admit of improvement.
The conductivity of liquids has been investigated by similar methods, generally variations of the thin plate or guard-ring method. A critical account of the subject is contained in a paper by C. Chree (Phil. Mag., July 1887). Many of the experiments were made by comparative methods, taking a standard liquid such as water for reference. A determination of the conductivity of water by S. R. Milner and A. P. Chattock, employing an electrical method, deserves mention on account of the careful elimination of various errors (Phil. Mag., July 1899). Their final result was 𝑘＝·001433 at 20° C., which may be compared with the results of other observers, G. Lundquist (1869), ·00155 at 40° C.; A. Winkelmann (1874), ·001104 at 15° C.; H. F. Weber (corrected by H. Lorberg), ·00138 at 4° C., and ·00152 at 23·6° C.; C. H. Lees (Phil. Trans., 1898), ·00136 at 25° C., and ·00120 at 47° C.; C. Chree, ·00124 at 18° C., and ·00136 at 19·5° C. The variations of these results illustrate the experimental difficulties. It appears probable that the conductivity of a liquid increases considerably with rise of temperature, although the contrary would appear from the work of Lees. A large mass of material has been collected, but the relations are obscured by experimental errors.
See also Fourier, Theory of Heat; T. Preston, Theory of Heat, cap. vii.; Kelvin, Collected Papers; O. E. Meyer, Die kinetische Theorie der Gase; A. Winkelmann, Handbuch der Physik. (H. L. C.)
CONE (Gr. κῶνος), in geometry, a surface generated by a line (the generator) which always passes through a fixed point (the vertex) and through the circumference of a fixed curve (the directrix). The two sheets of the surface, on opposite sides of the vertex, are called the “nappés” of the cone. The solid formed between the vertex and a plane cutting the surface is also called a “cone”; this is contained by a conical surface and the plane of section. Euclid defines a “right cone” as the
