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

Emissive power
$P = σAT^{4}$
$\frac{P_{1}}{P_{2}} = P = \frac{σA_{1}T_{1}^{4}}{σA_{2}T_{2}^{4}}\quad\quad\left(A = 4\pi R^{2}\right)$
$\frac{10^{4}}{1} = 400 × 400 × \left(\frac{\lambda_{B}}{\lambda_{A}}\right)^{4}\quad$ [using Weins displacement law $λ_{max} \,∝ \frac{1}{T}$]
$\frac{10^{4}}{\left(20\right)^{4}} = \left(\frac{\lambda_{B}}{\lambda_{A}}\right)^{4}\,\,\Rightarrow \frac{\lambda_{B}}{\lambda_{A}} = \frac{10}{20}$
$\Rightarrow \frac{\lambda_{B}}{\lambda_{A}} = \frac{1}{2}$
$\Rightarrow \frac{\lambda_{A}}{\lambda_{B}} = \frac{2}{1}$