Question

Two spheres $P$ [$e=1$] of radius $2R$ and $Q$ [$e=\frac{1}{2}$] of radius $R$ are placed at some distance from each other in vacuum and are isolated from other objects. The temperature of the sphere $Q$ is maintained at $200\text{K}$ by means of a heater. It is observed that ${\frac{1}{32}}^{th}$ of the power emitted by the sphere $Q$ falls on the sphere $P$. If the equilibrium temperature of the sphere $P$ is $10x$ Kelvin, then the value of $x$ is

Answer 1

Let the temperature of sphere $P$ be $T$.


Given, ${\frac{1}{32}}^{th}$ of the power emitted by $Q$ falls on $P$.
By Stefan's law, we can say that
$(P_{\text{rad}}{)}_Q=eA\sigma T^4=\frac{1}{2}4\pi R^2\sigma (T_Q{)}^4$
$(P_{\text{abs}}{)}_P=eA\sigma (T{)}^4=\left[4\pi \right(2R{)}^2\left]\sigma \right(T{)}^4$
Given, $\frac{1}{32}{\left(P_{\text{rad}}\right)}_Q={\left(P_{\text{abs}}\right)}_P$
$\Rightarrow \frac{1}{32}\times \frac{1}{2}\times 4\pi R^2\sigma \times (200{)}^4=4\pi R^2\sigma \times 4\times T^4$
$\Rightarrow T^4=\frac{1}{32}\times \frac{1}{2}\times \frac{(200{)}^4}{4}$
$\Rightarrow T={\left(\frac{(200{)}^4}{4^4}\right)}^{\frac{1}{4}}$
$\Rightarrow T=\frac{200}{4}=50\text{K}$
Equilibrium temperature of sphere $P$ is $10x=50\text{K}$ $\Rightarrow x=5$