Two spherical stars, A and B emit black body radiation. The radius...

Using the Stefan-Boltzmann law for the power emitted by a black body:

$P = \sigma A T^4 = \sigma (4\pi R^2) T^4$

The ratio of power emitted by star A and star B is:

$\dfrac{P_A}{P_B} = \left(\dfrac{R_A}{R_B}\right)^2 \left(\dfrac{T_A}{T_B}\right)^4$

Substituting the given values $P_A = 10^4 P_B$ and $R_A = 400 R_B$:

$10^4 = (400)^2 \left(\dfrac{T_A}{T_B}\right)^4$

$10^4 = 16 \times 10^4 \left(\dfrac{T_A}{T_B}\right)^4$

\left(\dfrac{T_A}{T_B}\right)^4 = \dfrac{1}{16}$<br><br>$\dfrac{T_A}{T_B} = \dfrac{1}{2}$<br><br>According to Wien's displacement law,$\lambda T = b$(constant), which implies$\lambda \propto \dfrac{1}{T}$.<br><br>Therefore, the ratio of peak wavelengths is:<br><br>$\dfrac{\lambda_A}{\lambda_B} = \dfrac{T_B}{T_A} = 2$<br><br>Given$\lambda_A = x \lambda_B$, we get$x = 2$.<br><br>Answer:$2