Q.A small blackened solid copper sphere of radius 2.5 cm is placed in an evacuated chamber. The temperature of the chamber is maintained at 100 ºC. At what rate must energy be supplied to the copper sphere to maintain its temperature at 110 ºC? (Take Stefan's constant $\sigma = 5.670\times10^{-8}$ J s⁻¹ m⁻² K⁻⁴, $\pi = 3.1416$, and treat the sphere as a blackbody.)

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