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# If $T$ is surface temperature of sun, $R$ is the radius of sun, $r$ is radius of earth’s orbit and $S$ is solar constant, then total radiant energy of sun per unit time from the sphere of radius $r,$ then(A) $\pi {{r}^{2}}S$ ( B) $4\pi {{r}^{2}}S$ (C) $\sigma \dfrac{4}{3}\pi {{R}^{3}}{{T}^{4}}$ (D) $\sigma 4\pi {{r}^{2}}{{T}^{4}}$

Last updated date: 10th Aug 2024
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Hint to calculate here, the total radiant energy of sun per unit time(from sphere of radius $r$ ) ,we can use here the Stefan’s-Boltzmann law.
It states that total energy radiated per unit surface area of the black body across all wavelengths per unit time, is directly proportional to the fourth power of the blackbody’s thermodynamic temperature $T$.
Formula used
$\text{Energy radiated per unit surface area per unit time}=\sigma {{T}^{4}}\times 4\pi {{r}^{2}}$

Complete step by step solution : Using Stefan’s Boltzmann law.
$\text{Energy radiated per unit surface area per unit time}=\sigma {{T}^{4}}$
Where $\sigma$ is Stefan’s Boltzmann constant
Here, surface area of earth$=4\pi {{r}^{2}}$
Thus, total energy radiated per unit time,
$L=\sigma {{T}^{4}}\times 4\pi {{r}^{2}}$

Thus, option (d) is correct.

\begin{align} & \sigma =\dfrac{2{{\pi }^{5}}{{k}^{4}}}{15{{c}^{2}}{{h}^{3}}}=5\cdot 670373\times {{10}^{-8}}\text{ }W{{m}^{-2}}{{k}^{-4}} \\ & k:\text{Boltzmann constant} \\ & h:\text{Plank }\!\!'\!\!\text{ s constant} \\ & c:\text{speed of light in a vacuum} \\ \end{align}