Q.Compare the rates of emission of heat by a blackbody maintained at 727 ºC and at 227 ºC, if the blackbodies are surrounded by an enclosure (black) at 27 ºC. What would be the ratio of their rates of loss of heat?

Stefan's law gives the total radiant energy from a black body as $\sigma A T^4$ per second; multiply by time for the total energy.

By Stefan's law, the power radiated by a perfectly black body of surface area $A$ at absolute temperature $T$ is

$P = \sigma A T^{4}$

Given $A = 200\text{ cm}^2 = 200\times10^{-4}\text{ m}^2 = 0.02\text{ m}^2$, $T = 127^\circ\text{C} = 400\text{ K}$, and $\sigma = 5.67\times10^{-8}\text{ W m}^{-2}\text{K}^{-4}$:

$T^4 = (400)^4 = 2.56\times10^{10}\text{ K}^4$

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Rate of loss of heat by radiation follows Stefan's law with a surrounding-temperature correction.

By Stefan's law (with surroundings at temperature $T_0$), the net rate of loss of heat is $R \propto (T^4-T_0^4)$.

Convert to kelvin: $T_1=827+273=1100$ K, $T_2=427+273=700$ K, $T_0=27+273=300$ K.

$\frac{R_1}{R_2} = \frac{T_1^4-T_0^4}{T_2^4-T_0^4} = \frac{(1100)^4-(300)^4}{(700)^4-(300)^4}$

Computing: $1100^4 = 1.4641\times10^{12}$, $700^4=2.401\times10^{11}$, $300^4=8.1\times10^{9}$. …