Assuming the Sun to be a spherical body of radius R at a temperature of TK, evalute the total radiant powerd incident of Earth at a distance r from the sun
where `r_0` is the radius of the Earth and `sigma` is Stefan's constant.

Options

A. `4pir_0^2R^2sigma(T^4)/(r^2)`
B. `pir_0^2R^2sigma(T^4)/(r^2)`
C. `r_0^2R^2sigma(T^4)/(4pir^2)`
D. `R^2sigma(T^4)/(r^2)`

Explanation

To solve the problem of evaluating the total radiant power incident on Earth from the Sun, we can follow these steps:

### Step 1: Calculate the Total Power Radiated by the Sun
The total power \( P \) radiated by the Sun can be calculated using the Stefan-Boltzmann Law, which states that the power radiated by a black body is proportional to the fourth power of its temperature. The formula is given by:

\[ P = \sigma T^4 A \]

where:
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( T \) is the temperature of the Sun in Kelvin,
- \( A \) is the surface area of the Sun.

Since the Sun is assumed to be a spherical body, its surface area \( A \) is:

\[ A = 4 \pi R^2 \]

Thus, the total power radiated by the Sun becomes:

\[ P = \sigma T^4 \cdot 4 \pi R^2 \]

### Step 2: Calculate the Intensity of the Radiation at Distance \( r \)
The intensity \( I \) of the radiation at a distance \( r \) from the Sun is defined as the power per unit area. The total power radiated by the Sun spreads out over the surface of a sphere of radius \( r \). Therefore, the intensity is given by:

\[ I = \frac{P}{4 \pi r^2} \]

Substituting the expression for \( P \):

\[ I = \frac{\sigma T^4 \cdot 4 \pi R^2}{4 \pi r^2} \]

The \( 4 \pi \) cancels out:

\[ I = \frac{\sigma T^4 R^2}{r^2} \]

### Step 3: Calculate the Total Power Incident on the Earth
The total power \( P_{incident} \) incident on the Earth can be calculated by multiplying the intensity \( I \) by the area \( A_{Earth} \) that the sunlight hits. The area of the circular cross-section of the Earth is:

\[ A_{Earth} = \pi r_0^2 \]

Thus, the total power incident on Earth is:

\[ P_{incident} = I \cdot A_{Earth} = I \cdot \pi r_0^2 \]

Substituting the expression for \( I \):

\[ P_{incident} = \left(\frac{\sigma T^4 R^2}{r^2}\right) \cdot \pi r_0^2 \]

### Final Expression
Combining everything, we get:

\[ P_{incident} = \frac{\sigma T^4 R^2 \pi r_0^2}{r^2} \]

This is the total radiant power incident on Earth from the Sun.