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

Explanation

To solve the problem of finding the power received by a unit surface area at a distance \( R \) from the center of the sun, we can follow these steps:

### Step 1: Understand the Concept of Intensity
Intensity (\( I \)) is defined as the power (\( P \)) received per unit area. According to Stefan-Boltzmann law, the total power radiated by a black body is given by:
\[ P = \sigma A T^4 \]
where \( \sigma \) is the Stefan-Boltzmann constant, \( A \) is the surface area of the black body, and \( T \) is the absolute temperature in Kelvin.

### Step 2: Calculate the Surface Area of the Sun
The surface area \( A \) of the sun, which we assume to be a sphere, is given by:
\[ A = 4 \pi r^2 \]
where \( r \) is the radius of the sun.

### Step 3: Convert the Temperature to Kelvin
The temperature given is \( t \) degrees Celsius. To convert this to Kelvin, we use the formula:
\[ T = t + 273 \]
For example, if \( t = 80 \) degrees Celsius, then:
\[ T = 80 + 273 = 353 \text{ K} \]

### Step 4: Calculate the Total Power Radiated by the Sun
Using the Stefan-Boltzmann law, the total power radiated by the sun can be expressed as:
\[ P = \sigma (4 \pi r^2) (T)^4 \]

### Step 5: Determine the Intensity at a Distance \( R \)
The intensity \( I \) at a distance \( R \) from the center of the sun is given by:
\[ I = \frac{P}{4 \pi R^2} \]
Substituting the expression for \( P \):
\[ I = \frac{\sigma (4 \pi r^2) (T)^4}{4 \pi R^2} \]

### Step 6: Simplify the Expression
The \( 4 \pi \) terms cancel out:
\[ I = \frac{\sigma r^2 T^4}{R^2} \]

### Final Expression
Thus, the power received by a unit surface area at a distance \( R \) from the center of the sun is:
\[ I = \frac{\sigma r^2 (T + 273)^4}{R^2} \]