A spherical blackbody of radius 10 m is maintained at a temperature T , radiating power P with a peak emission wavelength 5000 Å . If the body is compressed uniformly such that its radius becomes R/2 , while its total radiated power is simultaneously increased to 16P by heating, what is the new wavelength _m' at which the peak emission occurs?

Step-by-step solution

According to the Stefan-Boltzmann law, the radiated power is P = A T^4 = (4 R^2) T^4 . For the new state, P' = 16P and R' = R 2 . 16P = (4 ( R 2 )^2 ) (T')^4 16P = 1 4 (4 R^2) (T')^4 16P = 1 4 P ( T' T )^4 ( T' T )^4 = 64 T' = (64)^ 1/4 T = 2 2 T According to Wien's displacement law, _m T = constant . _m' T' = _m T _m' = _m ( T T' ) = 5000 1 2 2 = 2500 2 Answer: 2500 2 Å