Question

Assuming the sun to be a spherical body of radius $R$ at a temperature $T$ K, evaluate the total radiant power incident on earth. ($r$ is the distance between the sun and the earth, $r_0$ is the radius of earth and $\sigma$ is stefans constant) :

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

Answer 1

The correct option is B $\frac{\pi r_0^2{R}^2\sigma T^4}{r^2}$

Assuming sun as a perfect blackbody, energy radiated per sec by sun using Stefan's law is:

$P=\sigma A{T}^4$ (Where A is the area of the sun, P is energy radiated per second)

$\Rightarrow P=\sigma \times 4\pi R^2{T}^4$.................(1)

The intensity of this power at earth's surface is (assuming $r>>r_o$)

$I=\frac{P}{4\pi r^2}$

$\Rightarrow I=\frac{\sigma \times 4\pi R^2{T}^4}{4\pi r^2}$, (Putting the value from the equation (1))

$\Rightarrow I=\frac{\sigma R^2{T}^4}{r^2}$

Since the earth is very far from the sun, out of the total energy radiated, a small fraction of it is received by the earth. Earth can be considered as a small disc whose radius is the radius of the earth.

The surface area of the disc is $\pi {r_0}^2$, hence total radiant power as received by the earth is:

$P_E=\pi r_0^2\times I$

$P_E=\frac{\pi r_o^2\times \sigma R^2{T}^4}{r^2}$