Assuming the sun to be a spherical body (e = 1) of radius R at a temperature of T K, evaluate the total radiant power, incident on Earth having radius `r_0`, at a distance `r` from the sun, where `r_0` is the radius of the earth and `sigma` is Stefan's constant.

Options

A. `(pir_(0)^(2)R^(2)sigmaT^(4))/r^(2)`
B. `(r_(0)^(2)R^(2)sigmaT^(4))/(4pir^(2))`
C. `(R^(2)sigmaT^(4))/r^(2)`
D. `(4pir_0^(2)R^(2)sigmaT^(4))/r^(2)`

Answer

To evaluate 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 power radiated by a black body is given by the Stefan-Boltzmann law:
\[ P_{\text{sun}} = e \cdot A \cdot \sigma \cdot T^4 \]
where:
- \( e \) is the emissivity (given as 1 for a black body),
- \( A \) is the surface area of the Sun, which can be calculated as \( 4\pi R^2 \) (where \( R \) is the radius of the Sun),
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( T \) is the temperature of the Sun in Kelvin.

Substituting these values, we have:
\[ P_{\text{sun}} = 1 \cdot (4\pi R^2) \cdot \sigma \cdot T^4 = 4\pi R^2 \sigma T^4 \]

### Step 2: Determine the Intensity of Radiation at Distance \( r \)
The intensity \( I \) at a distance \( r \) from the Sun is the total power radiated divided by the surface area of a sphere with radius \( r \):
\[ I = \frac{P_{\text{sun}}}{A'} = \frac{4\pi R^2 \sigma T^4}{4\pi r^2} = \frac{R^2 \sigma T^4}{r^2} \]

### Step 3: Calculate the Projected Area of the Earth
The projected area \( A_0 \) of the Earth, which is a circle, can be calculated as:
\[ A_0 = \pi R_0^2 \]
where \( R_0 \) is the radius of the Earth.

### Step 4: Calculate the Total Power Received by the Earth
The total power \( P_{\text{Earth}} \) incident on the Earth is given by the intensity at distance \( r \) multiplied by the projected area of the Earth:
\[ P_{\text{Earth}} = I \cdot A_0 = \left(\frac{R^2 \sigma T^4}{r^2}\right) \cdot \left(\pi R_0^2\right) \]
Substituting the values, we get:
\[ P_{\text{Earth}} = \frac{R^2 \sigma T^4}{r^2} \cdot \pi R_0^2 \]

### Step 5: Simplify the Expression
Thus, the total radiant power incident on the Earth can be expressed as:
\[ P_{\text{Earth}} = \frac{\pi R_0^2 R^2 \sigma T^4}{r^2} \]

### Final Answer
The total radiant power incident on Earth at a distance \( r \) from the Sun is:
\[ P_{\text{Earth}} = \frac{\pi R_0^2 R^2 \sigma T^4}{r^2} \]

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