`R_(s)` `d` and `R_(e)` are the radius of sun , distance between sun and earth and radius of earth respectively. If temperature of sun is T. Then amount of radiation incident on earth is –

Options

A. `((R_(s))/(d))^(2) sigmaT^(4)`
B. `((R_(s))/(d))^(2) sigmaT^(4) (2 pi R_(e)^(2))`
C. `((R_(s))/(d))^(2) sigma T^(4) (pi R_(e)^(2))`
D. `((R_(s))/(R_(e))) sigma T^(4) (pi d^(2))`

Answer

To find the amount of radiation incident on Earth from the Sun, we can use Stefan's Law and the concept of radiative power.

### Step-by-Step Solution:

1. **Understanding the Problem:**
- We have the radius of the Sun \( R_s \), the distance from the Sun to the Earth \( d \), the radius of the Earth \( R_e \), and the temperature of the Sun \( T \).
- We need to calculate the amount of radiation incident on the Earth.

2. **Using Stefan's Law:**
- According to Stefan's Law, the power radiated by a black body is given by:
\[ P = \sigma A T^4 \]
- Here, \( \sigma \) is the Stefan-Boltzmann constant, \( A \) is the surface area, and \( T \) is the temperature.

3. **Calculating the Power Radiated by the Sun:**
- The Sun can be approximated as a sphere, so its surface area \( A_s \) is:
\[ A_s = 4 \pi R_s^2 \]
- Thus, the total power \( P_r \) radiated by the Sun is:
\[ P_r = \sigma (4 \pi R_s^2) T^4 \]

4. **Radiation Spreading Out:**
- The power emitted by the Sun spreads out in all directions. At a distance \( d \) from the Sun, this power is distributed over the surface area of a sphere with radius \( d \):
\[ A_{sphere} = 4 \pi d^2 \]

5. **Calculating the Intensity (Power per Unit Area):**
- The intensity \( S \) (power per unit area) at distance \( d \) from the Sun is given by:
\[ S = \frac{P_r}{A_{sphere}} = \frac{\sigma (4 \pi R_s^2) T^4}{4 \pi d^2} \]
- Simplifying this gives:
\[ S = \frac{\sigma R_s^2 T^4}{d^2} \]

6. **Calculating the Power Incident on Earth:**
- The Earth can be treated as a disk when considering the incident radiation. The area \( A_e \) of the Earth (considered as a disk) is:
\[ A_e = \pi R_e^2 \]
- The total power \( P_{incident} \) incident on the Earth is:
\[ P_{incident} = S \cdot A_e = \left(\frac{\sigma R_s^2 T^4}{d^2}\right) \cdot (\pi R_e^2) \]
- Therefore, the final expression for the power incident on Earth is:
\[ P_{incident} = \frac{\sigma R_s^2 T^4 \pi R_e^2}{d^2} \]

### Final Answer:
The amount of radiation incident on Earth is:
\[ P_{incident} = \frac{\sigma R_s^2 T^4 \pi R_e^2}{d^2} \]