# University of New Zealand. — Mathematical Physics. — Paper e (1). Heat

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## Papere (1). Heat.

 1 Equal masses of ice and of ice-cold water are enclosed together within a vessel impervious to heat, the whole being at 0°. State and discuss quantitatively what will happen (a) when the pressure is indefinitely diminished, (b) when the pressure is indefinitely increased. 2 Define an isothermal line and an isothermal surface. Show that if a gas changes its volume without undergoing any change of temperature, the quantities of heat emitted are in arithmetical progression, whilst the volumes are in a geometrical progression. 3 Deduce the law of Royle from the kinetic theory of gases. 4 Give an outline of the chief points in Fourier's theory of the conduction of heat. 5 A hollow spherical shell has inner and outer radii r and r' respectively, the inner and outer surfaces being maintained page 2 at absolute temperatures T and T' respectively. Suppose the specific conductivity of the material of the shell to be inversely proportional to its absolute temperature. Find the variations of temperature (in the steady distribution) along any radius between the outer and the inner surface. Does it make any difference in the internal distribution whether the inner or the outer surface be the hotter? 6 A hollow wire is heated by having an electric current passed through it, the strength of the current being caused to vary in a simple periodic manner, such as is represented by the equation where I is the maximum strength of current, and T the period of the variations. Assuming that Newton's law of cooling is true, that the value for the coefficient of surface emission of heat is constant, and that the electric resistance of the wire is directly proportional to the absolute temperature, obtain an expression for the fluctuations in the temperature of the wire. 7 How do you explain the existence of a difference between the specific heat of the air, as measured when heated without being allowed to expand and when heated while being allowed to expand without mechanical constraint other than that of the external atmospheric pressure? Is any such difference observed in the case of water, or of ice? 8 How do you account for the following facts? A large glass globe full of air is connected to a manometer, and also is provided with a large stop-cock opening into the air. Some air having been pumped out of the globe the manometer gauge shows a certain reduction of internal pressure. The stop-cock is then opened for an instant, and then closed again. The pressure during this operation returns to that of the external atmosphere, but a few seconds after closing the stop-cock a reduction of internal pressure is again observed, though of a less amount than had existed before opening the cock. 9 Why do a pound of ice-cold water and a pound of boiling water when mixed together not give 50° C as the resulting temperature?