Question

A highly conducting solid sphere of radius $R$ , density $\rho$ and specific heat $s$ is kept in an evacuated chamber. A parallel beam of thermal radiation of intensity $I$ is incident on its surface.
Consider the sphere to be a perfectly black body and its temperature at certain instant $t=0$ is considered as $T_0$ [Take Stefan's constant as $\sigma$]
The maximum attainable temperature of the sphere is

• A. ${\left(\frac{I}{4\sigma }\right)}^{\frac{1}{4}}$
• B. ${\left(\frac{4I}{\sigma }\right)}^{\frac{1}{4}}$
• C. ${\left(\frac{I}{2\sigma }\right)}^{\frac{1}{4}}$
• D. ${\left(\frac{I}{3\sigma }\right)}^{\frac{1}{4}}$

Answer 1

The correct option is A ${\left(\frac{I}{4\sigma }\right)}^{\frac{1}{4}}$


$T_{\text{max}}=?$
Power incident $=I\pi R^2$
Power emitted $=e\sigma A{T}^4$
Given, $e=1,A=4\pi R^2$
$\frac{dQ}{dt}=\pi R^2[I-4\sigma T^4]$
$ms\frac{dT}{dt}=\pi R^2[I-4\sigma T^4]$

For maximum temperature $\frac{dT}{dt}=0$

$\frac{dT}{dt}=\frac{\pi R^2}{ms}(I-4\sigma T^4)=0$
$I=4\sigma T^4$
$T_{\text{max}}=T={\left(\frac{I}{4\sigma }\right)}^{\frac{1}{4}}$