Q. Two spherical black bodies of radii $r_{1}$ and $r_{2}$ and with surface temperature $T_{1}$ and $T_{2}$ respectively radiate the same power. Then the ratio of $r_{1}$ and $r_{2}$ will be
A) $\left( \frac{T_{2}}{T_{1}} \right)^{2}$
B) $\left( \frac{T_{2}}{T_{1}} \right)^{4}$
C) $\left( \frac{T_{1}}{T_{2}} \right)^{2}$
D) $\left( \frac{T_{1}}{T_{2}} \right)^{4}$
Explanation
(a) For black body, $P = A\varepsilon\sigma T^{4}$. For same power $A \propto \frac{1}{T^{4}}$
$\Rightarrow \left( \frac{r_{1}}{r_{2}} \right)^{2} = \left( \frac{T_{2}}{T_{1}} \right)^{4} \Rightarrow \frac{r_{1}}{r_{2}} = \left( \frac{T_{2}}{T_{1}} \right)^{4}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.