Assuming the sun to have a spherical outer surface of radius `r` radiating like a black body at temperature `t^(@)C`. The power received by a unit surface (normal to the incident rays) at a distance `R` from the centre of the sun is
where `sigma` is the Stefan's constant.

Options

A. `r(2) sigma (t+273)^(4)//R^(2)`
B. `4 pi r^(2) sigmat^(4)//R^(2)`
C. `r^(2) sigma (t+273)^(4)//4piR^(2)`
D. `16pi^(2)r^(2) sigma t^(4)//R^(2)`

Explanation

To solve the problem, we need to find the power received by a unit surface (normal to the incident rays) at a distance \( R \) from the center of the Sun, given that the Sun behaves like a black body radiating at a temperature \( t \) (in degrees Celsius).

### Step-by-Step Solution:

1. **Convert Temperature to Kelvin**:
The temperature \( t \) in degrees Celsius needs to be converted to Kelvin. The conversion formula is:
\[ T = t + 273 \]

2. **Calculate the Total Power Radiated by the Sun**:
The total power \( P \) radiated by the Sun can be calculated using the Stefan-Boltzmann law:
\[ P = \sigma A T^4 \]
where \( A \) is the surface area of the Sun. The surface area of a sphere is given by:
\[ A = 4\pi r^2 \]
Thus, substituting for \( A \):
\[ P = \sigma (4\pi r^2) (T^4) = 4\pi r^2 \sigma (t + 273)^4 \]

3. **Calculate the Power per Unit Area at Distance \( R \)**:
The power received per unit area at a distance \( R \) from the center of the Sun can be found by dividing the total power by the surface area of a sphere of radius \( R \):
\[ \text{Power per unit area} = \frac{P}{4\pi R^2} \]
Substituting the expression for \( P \):
\[ \text{Power per unit area} = \frac{4\pi r^2 \sigma (t + 273)^4}{4\pi R^2} \]

4. **Simplify the Expression**:
The \( 4\pi \) terms cancel out:
\[ \text{Power per unit area} = \frac{r^2 \sigma (t + 273)^4}{R^2} \]

5. **Final Expression**:
Thus, the power received by a unit surface at a distance \( R \) from the center of the Sun is:
\[ \text{Power per unit area} = \sigma \frac{r^2}{R^2} (t + 273)^4 \]

### Final Answer:
The power received by a unit surface at a distance \( R \) from the center of the Sun is:
\[ \text{Power per unit area} = \sigma \frac{r^2}{R^2} (t + 273)^4 \]