$\text{T}_2^4-\text{T}_1^4$
Explanation:
From stefan-Boltzmann law, the energy of thermal radiation per unit time by a blackbody of surface are A is given by,
$\text{u}-\sigma\text{AT}^4$
Here, $\sigma$ is stefan-Boltzmann constant.
Since the temperature of the solid is less than the surrouncings, the temperature of the solid will increase with time until it reaches equilibrium with the surroundings. The rate of emission from the soild will be proportional to $\text{T}_1^4$ and rate of emission from the surroundings will be proportional to $\text{T}_2^4.$
So, the net rate of increase in temperature will be proportional to $\text{T}_2^4-\text{T}^4_1.$
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