A solid at temperature T_1 is placed in an evacuated chamber that...

Since the chamber is evacuated, heat transfer between the solid and the chamber occurs exclusively through radiation.

According to the Stefan-Boltzmann law, the rate at which the solid absorbs heat from the chamber is $P_{abs} = \sigma e A T_2^4$, and the rate at which it emits heat is $P_{emit} = \sigma e A T_1^4$.

The net rate of heat gained by the solid is given by:

$\dfrac{dQ}{dt} = P_{abs} - P_{emit} = \sigma e A (T_2^4 - T_1^4)$

The rate of increase of the solid's temperature is related to the net heat gained by:

$\dfrac{dT_1}{dt} = \dfrac{1}{ms} \dfrac{dQ}{dt} = \dfrac{\sigma e A}{ms} (T_2^4 - T_1^4)$

Thus, the rate of increase of the solid's temperature is proportional to $T_2^4 - T_1^4$.