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.

Solution & Explanation

1. Calculate the Total Radiant Power Emitted by the Sun

Assuming the Sun is a perfect spherical blackbody of radius $$R$$ at an absolute temperature of $$T \,\, \text{K}$$, its surface area ($$A_{\text{sun}}$$) is given by:

$$A_{\text{sun}} = 4\pi R^2$$

According to the Stefan-Boltzmann Law, the total radiant energy emitted by a blackbody per unit time (total power, $$P_0$$) is proportional to the fourth power of its absolute temperature:

$$P_0 = \sigma \cdot A_{\text{sun}} \cdot T^4 = 4\pi R^2 \sigma T^4$$

Where $$\sigma$$ is the Stefan-Boltzmann constant.

2. Determine the Radiant Energy Flux at Earth's Distance ($$r$$)

As this total power radiates uniformly outward into space, it spreads over an imaginary concentric sphere of radius $$r$$ (the distance from the Sun to the Earth).

The intensity ($$I$$) or radiant power per unit area reaching Earth's orbital distance is:

$$I = \frac{P_0}{4\pi r^2} = \frac{4\pi R^2 \sigma T^4}{4\pi r^2} = \frac{R^2 \sigma T^4}{r^2}$$

3. Calculate the Total Radiant Power Incident on Earth

Let $$r_0$$ be the radius of the Earth. From the perspective of the incoming parallel solar radiation, the Earth intercepts this energy flux not across its entire spherical surface area, but across its effective circular cross-sectional target area ($$A_{\text{intercept}} = \pi r_0^2$$).

The total radiant power ($$P_{\text{earth}}$$) incident on the Earth is the product of the solar intensity at that distance and this cross-sectional intercept area:

$$P_{\text{earth}} = I \cdot A_{\text{intercept}}$$

$$P_{\text{earth}} = \left( \frac{R^2 \sigma T^4}{r^2} \right) \cdot (\pi r_0^2)$$

$$P_{\text{earth}} = \frac{\pi r_0^2 R^2 \sigma T^4}{r^2}$$

Concept Check: Even though the Earth is a sphere, it presents a flat, circular shadow area of $$\pi r_0^2$$ to the oncoming wavefronts. The power collected scales directly with the size of this shadow disc and drops off inversely with the square of the distance ($$r^2$$) from the Sun due to geometric spreading.

Correct Option Key: Option C ($$\frac{\pi r_0^2 R^2 \sigma T^4}{r^2}$$)