The solar energy received by the Earth persquare metre per minute is `8.315 xx 10^(4)Jm^(-2)min^(-1)`. If the radius of the Sun is` 7.5 xx 10^(5)` km and the distance of the Earth from the Sun is `1.5 xx10^(8)` km, calculate the surface temperature of Sun. Assume the Sun as a perfect black body. Given that Stefen constanta ` sigma = 5.7 xx 10^(-8)Wm^(-2)K^(-4).`

Answer

To calculate the surface temperature of the Sun, we will use Stefan's Law, which relates the power radiated by a black body to its temperature. Here are the steps to solve the problem:

### Step 1: Convert the given solar energy received by the Earth to per second
The solar energy received by the Earth is given as \( 8.315 \times 10^4 \, \text{J/m}^2/\text{min} \). We need to convert this to Joules per second (Watts).

\[ \text{Energy per second} = \frac{8.315 \times 10^4 \, \text{J/m}^2}{60 \, \text{s}} = 1385.25 \, \text{J/m}^2/\text{s} \]

### Step 2: Write down the formula for the power radiated by the Sun
According to Stefan's Law, the power radiated by the Sun can be expressed as:

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

Where:
- \( P_R \) is the total power radiated by the Sun,
- \( \sigma \) is the Stefan-Boltzmann constant (\( 5.7 \times 10^{-8} \, \text{W/m}^2\text{K}^4 \)),
- \( A \) is the surface area of the Sun,
- \( T \) is the surface temperature of the Sun.

### Step 3: Calculate the surface area of the Sun
The surface area \( A \) of the Sun can be calculated using the formula for the surface area of a sphere:

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

Where \( R \) is the radius of the Sun. Given that the radius of the Sun is \( 7.5 \times 10^5 \, \text{km} = 7.5 \times 10^8 \, \text{m} \):

\[ A = 4 \pi (7.5 \times 10^8)^2 \]

Calculating this gives:

\[ A \approx 4 \pi (5.625 \times 10^{17}) \approx 7.06858 \times 10^{18} \, \text{m}^2 \]

### Step 4: Set up the equation for the power received by the Earth
The power received by the Earth per unit area is given by:

\[ \frac{P_R}{4 \pi D^2} = \text{Energy received per unit area} \]

Where \( D \) is the distance from the Sun to the Earth, given as \( 1.5 \times 10^8 \, \text{km} = 1.5 \times 10^{11} \, \text{m} \).

### Step 5: Substitute the values into the equation
Now, substituting the values into the equation:

\[ \frac{\sigma A T^4}{4 \pi D^2} = 1385.25 \]

Substituting \( A \) and rearranging gives:

\[ \sigma (4 \pi (7.5 \times 10^8)^2) T^4 = 1385.25 \times 4 \pi (1.5 \times 10^{11})^2 \]

### Step 6: Solve for \( T^4 \)
Now we can solve for \( T^4 \):

\[ T^4 = \frac{1385.25 \times 4 \pi (1.5 \times 10^{11})^2}{\sigma (4 \pi (7.5 \times 10^8)^2)} \]

The \( 4 \pi \) cancels out:

\[ T^4 = \frac{1385.25 \times (1.5 \times 10^{11})^2}{\sigma (7.5 \times 10^8)^2} \]

### Step 7: Calculate \( T \)
Substituting \( \sigma = 5.7 \times 10^{-8} \):

\[ T^4 = \frac{1385.25 \times (2.25 \times 10^{22})}{5.7 \times 10^{-8} \times (5.625 \times 10^{17})} \]

Calculating this gives:

\[ T^4 \approx 972.515 \times 10^{12} \]

Taking the fourth root:

\[ T \approx (972.515 \times 10^{12})^{1/4} \approx 5584 \, \text{K} \]

### Final Answer
The surface temperature of the Sun is approximately \( 5584 \, \text{K} \).

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