A solid aluminium sphere and a solid copper sphere, the latter having twice the radius, are heated to identical temperatures and subsequently allowed to cool in identical surrounding temperatures. Assuming that the emissivity of both spheres is equal, determine the ratio of the rate of fall of temperature of the aluminium sphere to the rate of fall of temperature of the copper sphere. The specific heat capacity of al

Step-by-step solution

The rate of fall of temperature of a body is given by the Stefan-Boltzmann law: dT dt = e A (T^4 - T_s^4) mc For a solid sphere, the mass is m = V = ( 4 3 R^3 ) and the surface area is A = 4 R^2 . Substituting these into the equation gives: dT dt = e (4 R^2) (T^4 - T_s^4) ( 4 3 R^3 ) c = 3 e (T^4 - T_s^4) c R Since the temperatures, surrounding temperatures, and emissivities are identical for both spheres, the rate of fall of temperature r is inversely proportional to the product of density, specific heat capacity,