Question

Assuming the sun to be a spherical body of radius $R$ at a temperature $T\left(\text{K}\right)$, Evaluate the total radiant power incident on the Earth. ($r$ is the distance between the sun and the earth, $R_0$ is the radius of Earth and $\sigma$ is Stefan’s Constant)

Answer 1

Step 1: Given parameters

$\text{Spherical radius}=R$

$\text{Temperature}=T$

$\text{Distance between the Sun and the Earth}=r$

$\text{Radius of}\mathrm{Earth}=R_0$

Step 2: Calculate the total Radiant Power incident on the Earth

Assuming the sun as a perfect black body, energy radiated per sec by the sun using Stefan's law is given by,
$P=\sigma A{T}^4$

Where, $A$is the area of the sun, $P$ is energy radiated per sec, $\sigma$ is Stefan’s Constant, and,$T$ is the temperature in kelvin.

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

The intensity of this power at the Earth's surface is
$$\begin{gathered} I=\frac{P}{4\pi r^2} \\ =\frac{\sigma \times 4\pi R^2{T}^4}{4\pi r^2} \\ =\frac{\sigma R^2{T}^4}{r^2} \end{gathered}$$​,
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:

$$\begin{gathered} P_E=\pi {r_0}^{2}I \\ =\frac{\pi {r_0}^2\sigma R^2{T}^4}{r^2} \end{gathered}$$

Hence, the total radiant power as received by the Earth is $P_E=\frac{\pi {r_0}^2\sigma R^2{T}^4}{r^2}$.