Question

Assuming the sun to be a spherical body of radius $R$ at a temperature of $TK$ , 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)

• 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 $P=\sigma \times 4\pi R^2{T}^4$

[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
$=\pi r_0^2\times I$
= $\frac{\pi r_0^2\times \sigma R^2{T}^4}{r^2}=\frac{\pi r_0^2{R}^2\sigma T^4}{r^2}$