Question

A spherical black body has a radius $R$ and steady surface temperature $T$, heat sources in it ensure the heat evolution at a constant rate and distributed uniformly over its volume.
If the radius is decreased by half, what would be the new temperature of the surface at steady state ?
Assume surrounding to be at absolute zero and heat evolution rate through unit volume remains the same.

• A. $\sqrt[4]{42}T$
• B. $\sqrt[4]{4\frac{1}{2}}T$
• C. $\sqrt{2}T$
• D. $\sqrt{\frac{1}{2}}T$

Answer 1

The correct option is B $\sqrt[4]{4\frac{1}{2}}T$

Let $P$ be heat radiated per unit volume.
Net heat getting generated in complete volume of sphere = rate of heat radiated by its surface.
$P\left(\frac{4}{3}\pi r^3\right)=\sigma e\left(4\pi r^2\right)T^4$
$\Rightarrow T^4\propto r$
when radius is halved let surface temperature be $T^{\prime }$
$\frac{T^{\prime }}{T}={\left(\frac{r/2}{r}\right)}^{1/4}={\left(\frac{1}{2}\right)}^{1/4}$
$T^{\prime }=\sqrt[4]{4\frac{1}{2}}T$