A solid at temperature T₁ is kept in an evacuated chamber at temperature T₂ > T₁. The rate of increase of temperature of the body is proportional to

(a) T₂ − T₁
(b) $T_{2}^{2} - T_{1}^{2}$
(c) $T_{2}^{3} - T_{1}^{3}$
(d) $T_{2}^{4} - T_{1}^{4}$

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Solution

(d) $T_{2}^{4} - T_{1}^{4}$

From Stefan-Boltzmann law, the energy of thermal radiation emitted per unit time by a blackbody of surface area A is given by
$u = σ A T^{4}$
Here, $σ$ is Stefan-Boltzmann constant.
Since the temperature of the solid is less than the surroundings, the temperature of the solid will increase with time until it reaches equilibrium with the surroundings. The rate of emission from the solid will be proportional to ${T_{1}}^{4}$ and rate of emission from the surroundings will be proportional to ${T_{2}}^{4}$ .
So, the net rate of increase in temperature will be proportional to $T_{2}^{4} - T_{1}^{4}$ .

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