A solid spherical body of radius R , density , and specific heat capacity s is initially at temperature T₀ . It is placed in a vacuum chamber at absolute zero and cools by emitting thermal radiation as a perfect blackbody (Stefan-Boltzmann constant ). If the rate of fall of temperature depends on its instantaneous temperature T as T^x , and the time t taken for its temperature to drop from T₀ to T₀ 2 is proportional

Step-by-step solution

The rate of heat loss from the spherical body by thermal radiation is given by the Stefan-Boltzmann law: dQ dt = A T^4 where A = 4 R^2 is the surface area of the sphere. The heat loss is also related to the rate of fall of temperature by: dQ dt = -m s dT dt where m = 4 3 R^3 is the mass of the sphere. Equating the two expressions: - ( 4 3 R^3 ) s dT dt = (4 R^2) T^4 - dT dt = 3 s R T^4 The rate of fall of temperature is proportional to T^4 , which gives x = 4 . To find the time taken for the temperature to drop fro