Explanation

(b) From Stefan’s law, the rate at which energy is radiated by sun at its surface is$P = \sigma \times 4\pi r^{2}T^{4}$
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[Sun is a perfectly black body as it emits radiations of all wavelengths and so for it $e$=1.]
The intensity of this power at earth’s surface(under the assumption $r$>>$r_{0})$ is
$I = \frac{P}{4\pi R^{2}} = \frac{\sigma \times 4\pi r^{2}T^{4}}{4\pi R^{2}} = \frac{\sigma R^{2}\sigma T^{4}}{r^{2}}$
The area of earth which receives this energy is only one-half of total surface area of earth, whose projection would be $\pi r_{0}^{2}$.
$\therefore$ Total radiant power as received by earth
$= \ \pi r_{0}^{2} \times I$
= $\frac{\pi r_{0}^{2} \times \sigma R^{2}T^{4}}{r^{2}} = \frac{\pi r_{0}^{2}R^{2}\sigma T^{4}}{r^{2}}$