The total radiant energy per unit area, normal to the direction of incidence, received at a distance `R` from the centre of a star of radius `r` whose outer surface radiates as a black body at a temperature `T K` is given by
(where `sigma` is Stefan's constant)

Options

A. `rho r^(2) T^(4)//R^(2)`
B. `rho R^(2)T^(4)//4pi R^(2)`
C. `rho r^(2)T^(4)//4pi r^(2)`
D. `4pi rho r^(2)T&^(4)//R^(2)`

Explanation

To solve the problem of finding the total radiant energy per unit area received at a distance \( R \) from the center of a star, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Black Body Radiation**:
A star behaves like a black body radiator. According to Stefan-Boltzmann law, the total energy radiated per unit area of a black body is given by:
\[ E = \sigma T^4 \]
where \( \sigma \) is Stefan's constant and \( T \) is the temperature of the black body.

2. **Calculate the Total Energy Radiated by the Star**:
The total energy radiated by the star can be calculated by multiplying the energy per unit area by the surface area of the star. The surface area \( A \) of a sphere (the star) is given by:
\[ A = 4\pi r^2 \]
Thus, the total energy \( E_{\text{total}} \) radiated by the star is:
\[ E_{\text{total}} = \sigma T^4 \cdot 4\pi r^2 \]

3. **Determine the Intensity at Distance \( R \)**:
The intensity \( I \) (which is energy per unit area) at a distance \( R \) from the star is given by the total energy radiated divided by the area over which this energy is spread. The area of a sphere with radius \( R \) is:
\[ A_R = 4\pi R^2 \]
Therefore, the intensity \( I \) can be expressed as:
\[ I = \frac{E_{\text{total}}}{A_R} = \frac{\sigma T^4 \cdot 4\pi r^2}{4\pi R^2} \]

4. **Simplify the Expression**:
The \( 4\pi \) terms cancel out:
\[ I = \frac{\sigma T^4 r^2}{R^2} \]

5. **Final Result**:
The total radiant energy per unit area received at a distance \( R \) from the star is:
\[ I = \frac{\sigma r^2 T^4}{R^2} \]