Two spherical stars, A and B emit black body radiation. The radius of star A is 400 times that of star B and star A emits power 10^4 times emitted from star B. _A and _B are the wavelengths at which the peak occurs in their respective radiation curves such that _A = x , _B . The value of x is

Step-by-step solution

Using the Stefan-Boltzmann law for the power emitted by a black body: P = A T^4 = (4 R^2) T^4 The ratio of power emitted by star A and star B is: P_A P_B = ( R_A R_B )^2 ( T_A T_B )^4 Substituting the given values P_A = 10^4 P_B and R_A = 400 R_B : 10^4 = (400)^2 ( T_A T_B )^4 10^4 = 16 10^4 ( T_A T_B )^4 ( T_A T_B )^4 = 1 16 T_A T_B = 1 2 According to Wien's displacement law, T = b (constant), which implies 1 T . Therefore, the ratio of peak wavelengths is: _A _B = T_B T_A = 2 Given _A = x _B , we get x = 2 . Answ