Question

A sphere $P$(emissivity $=1$) of radius $2R$ and and another sphere $Q$(emissivity $=1/2$) of radius $R$ are placed in vacuum at some distance. There are no other objects. The temperature of the sphere $Q$ is maintained at $200K$ by the means of a heater. A fraction $1/32$ of the power emitted by the sphere $Q$ falls on the sphere $P$. If the equilibrium temperature of the sphere $P$ is $10T$ (in $K$), find the value of $T$ :

Answer 1

In equilibrium power released by sphere-1 is equal to the power absorbed by sphere-2.
$P_1={\sigma }_1{ϵ}_1{A}_1{T_1}^4$
$P_2=\frac{1}{32}{\sigma }_2{ϵ}_2{A}_2{T_2}^4$
$P_1=P_2$
$\therefore \sigma \times \left(1\right)4\pi \times (2R{)}^2{T}^4=\frac{1}{32}\times \sigma \times (\frac{1}{2})\times 4\pi \times (R{)}^2(200{)}^4$
$\therefore 4T^4=\frac{1}{64}(200{)}^4$
$\Rightarrow (4T{)}^4=(200{)}^4$
$\Rightarrow 4T=200$
$\Rightarrow T=50K$
Given equilibrium temp is 10T, $T$ will be 5.