A black sphere has radius R whose rate of radiation is E at temperature T. If radius is made R/2 and temperature 3T, the rate of radiation will be

Step-by-step solution

Stefan-Boltzmann's Law gives the radiated power as P = A T^4 , where is the Stefan-Boltzmann constant, A is surface area, and T is absolute temperature. For a sphere with radius R , the area is A = 4 R^2 , yielding P = (4 R^2) T^4 . Given initial radiation rate E = (4 R^2) T^4 . With radius halved to R/2 and temperature tripled to 3T , the new radiation becomes: E' = [4 (R/2)^2 ] (3T)^4 = (4 R^2/4) (81T^4) = 81 R^2 T^4 . From the initial condition, R^2 T^4 = E/4 , so E' = 81(E/4) = 81E 4 . The new radiation rate is