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Q. Assuming the sun to be a spherical body of radius $R \, $ at a temperature of $T \, K$ , evaluate the total radiant power, incident on earth, at a distance $r$ from the sun. ( $r_{0}$ is the radius of the earth and $\sigma $ is Stefan's constant)

NTA AbhyasNTA Abhyas 2020Thermal Properties of Matter

Solution:

From Stefan's law, the rate at which energy is radiated by sun at its surface is $P=\sigma \times 4\pi R^{2}T^{4}$
Solution
[Sun is a perfect black body as it emits radiations of all wavelengths and so for it $e$ =1]
The intensity of this power at earth's surface is
$I = \frac{P}{4 \pi r^{2}} = \frac{\sigma \times 4 \pi R^{2} T^{4}}{4 \pi r^{2}} = \frac{\sigma R^{2} T^{4}}{r^{2}}$
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
$= \, \pi r_{0}^{2}\times I$
= $\frac{\pi r_{0}^{2} \times \sigma R^{2} T^{4}}{r^{2}}=\frac{\pi r_{0}^{2} R^{2} \sigma T^{4}}{r^{2}}$