Question

Two spherical stars A and B emit blackbody radiation. The radius of A is 400 times that of B and A emits $10^4$ times the power emitted from B. The ratio $\left(\frac{{\lambda }_A}{{\lambda }_B}\right)$ of their wavelength ${\lambda }_A$ and ${\lambda }_B$ at which the peaks occur in their respective radiation curves is

Answer 1

Answer: 2

Emissive power
$P=\sigma A{T}^4$
$\frac{P_1}{P_2}=P=\frac{\sigma A_1{T}_1^4}{\sigma A_2{T}_2^4}\left(A=4\pi R^2\right)$
$\frac{10^4}{1}=400\times 400\times {\left(\frac{{\lambda }_B}{{\lambda }_A}\right)}^4$ [using Weins displacement law ${\lambda }_{max}\propto \frac{1}{T}$]
$\frac{10^4}{{\left(20\right)}^4}={\left(\frac{{\lambda }_B}{{\lambda }_A}\right)}^4\Rightarrow \frac{{\lambda }_B}{{\lambda }_A}=\frac{10}{20}$
$\Rightarrow \frac{{\lambda }_B}{{\lambda }_A}=\frac{1}{2}$
$\Rightarrow \frac{{\lambda }_A}{{\lambda }_B}=\frac{2}{1}$