Question

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 :
where $r_0$ is the radius of the earth and $\sigma$ is Stefan’s constant.

• A. $4\pi r_0^2{R}^2\sigma T^4/r^2$
• B. $\pi r_0^2{R}^2\sigma T^4/r^2$
• C. $r_0^2{R}^2\sigma T^4/4\pi r^2$
• D. $R^2\sigma T^4/r^2$

Answer 1

Answer: (B)

From Stefan’s law, the rate at which energy is radiated by sun at its surface is
$P=\sigma \times 4\pi R^2\times T^4$
[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
$I=\frac{p}{4\pi r^2}=\frac{\sigma \times 4\pi R^2{T}^4}{4\pi r^2}=\frac{\sigma R^2{T}^4}{r^2}$
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
$=\pi r_0^2\times I$
$=\frac{\pi r_0^2\times \sigma R^2{T}^4}{r^2}$
$=\frac{\pi r_0^2{R}^2\sigma T^4}{r^2}$