Assuming the sun to be a spherical body of radius  $R$  at a temperature of  ${}^{\text{'}}T^{\text{'}}K$, evaluate the total radiant power, incident on earth, at a distance $\left(r\right)$ from the sun (take radius of earth as $r_0$   )

From Stefan’s law, the rate at which energy is radiated by sun at its surface is  $P=\sigma \left(4\pi R^2\right)T^4$   [sun is a perfectly black body so, $e=1$  ]
The intensity of this power at earth’s surface (under the assumption $r>>r_0$ ) is
 

$I=\frac{P}{4\pi R^2}=\frac{\sigma \left(4\pi R^2\right)T^4}{4\pi r^2}=\frac{\sigma R^2{T}^4}{r^2}$ 

The area of earth which receives this energy is only one half of total surface area of earth, whose projection would be  $\left(\pi r_0^2\right)$
The total radiant power as received by earth $=\pi r_0^2\times I$ 

$=\pi r_0^2\left[\frac{\sigma R^2{T}^4}{r^2}\right] =\frac{\pi {{\displaystyle r}}_0^2{R}^2\sigma T^4}{r^2}$