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Assuming the sun to be a spherical body of radius $R$ at a temperature of $TK$ , evaluate the total radiant power, incident on the earth, at a distance $r$ from the sun, is (Where $r_0$ is the radius of the earth and $\sigma $ is Stefan s constant.)
$\frac{{{R^2}\sigma {T^4}}}{{{r^2}}}$
$\frac{{4\pi r_0^2{R^2}\sigma {T^4}}}{{{r^2}}}$
$\frac{{\pi r_0^2{R^2}\sigma {T^4}}}{{{r^2}}}$
$\frac{{r_0^2{R^2}\sigma {T^4}}}{{4\pi {r^2}}}$
Solution
Energy radiated per second by the sun,
$\mathrm{E}=\sigma \mathrm{T}^{4} 4 \pi \mathrm{R}^{2}$
This energy falls uniformly on the inner surface of spheres centred around the sun. If $r$ is the distance of the earth from the sun, then energy falling per second on unit area of the sphere of radius ris, $\frac{2 \pi R^{2} \sigma T^{4}}{2 \pi r^{2}}=\frac{\sigma R^{2} T^{4}}{r^{2}}$
It is only from the front hemisphere of the sun that the energy is falling on the earth and it is only on the front half of the earth.
Therefore, $\frac{4 \pi R^{2}}{2}$ and $\frac{4 \pi r^{2}}{2}$ have to be used.
The radiant power incident on the earth is given by $:$
$\mathrm{Q}=\pi r_{0}^{2} \times \frac{\sigma \mathrm{R}^{2} \mathrm{T}^{4}}{\mathrm{r}^{2}}=\frac{\pi r_{0}^{2} \mathrm{R}^{2} \sigma \mathrm{T}^{4}}{\mathrm{r}^{2}}$
