Distance between sun and earth is `2 xx 10^(8)` km, temperature of sun 6000 K, radius of sun `7 xx 10^(5)` km, if emmisivity of earth is 0.6 , then find out temperature of earth in thermal equilibrium.

Options

A. 400 K
B. 300 K
C. 500 K
D. 600 K

Explanation

To find the temperature of the Earth in thermal equilibrium with the Sun, we can use the concept of radiative heat transfer and the Stefan-Boltzmann law. Here’s a step-by-step solution:

### Step 1: Understand the Power Received by the Earth
The power emitted by the Sun can be calculated using the Stefan-Boltzmann law:
\[ P_{sun} = \sigma A_{sun} T_{sun}^4 \]
where:
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( A_{sun} = 4 \pi R_{sun}^2 \) is the surface area of the Sun,
- \( T_{sun} = 6000 \, K \) is the temperature of the Sun.

### Step 2: Calculate the Power Emitted by the Sun
Substituting the values:
- \( R_{sun} = 7 \times 10^5 \, km = 7 \times 10^8 \, m \),
- \( P_{sun} = \sigma (4 \pi (7 \times 10^8)^2) (6000)^4 \).

### Step 3: Calculate the Power Received by the Earth
The power received by the Earth from the Sun at a distance \( d \) is given by:
\[ P_{received} = \frac{P_{sun}}{4 \pi d^2} A_{projected} \]
where \( A_{projected} = \pi R_{earth}^2 \) (the projected area of the Earth).

### Step 4: Set Up the Equation for Thermal Equilibrium
At thermal equilibrium, the power received by the Earth equals the power emitted by the Earth:
\[ P_{received} = P_{emitted} \]
Using the emissivity \( e \) of the Earth:
\[ P_{emitted} = \sigma e A_{earth} T_{earth}^4 \]
where \( A_{earth} = 4 \pi R_{earth}^2 \).

### Step 5: Substitute and Simplify
Equating the two powers:
\[ \frac{\sigma (4 \pi (7 \times 10^8)^2) (6000)^4}{4 \pi (2 \times 10^8)^2} \cdot \pi R_{earth}^2 = \sigma e (4 \pi R_{earth}^2) T_{earth}^4 \]

### Step 6: Cancel and Rearrange
Cancel common terms and rearrange to solve for \( T_{earth} \):
\[ \frac{(6000)^4 (7 \times 10^8)^2}{(2 \times 10^8)^2} = e T_{earth}^4 \]

### Step 7: Solve for \( T_{earth} \)
Substituting \( e = 0.6 \):
\[ T_{earth}^4 = \frac{(6000)^4 (7 \times 10^8)^2}{(2 \times 10^8)^2 \cdot 0.6} \]

### Step 8: Calculate \( T_{earth} \)
Calculate \( T_{earth} \):
1. Calculate \( (6000)^4 \).
2. Calculate \( (7 \times 10^8)^2 \) and \( (2 \times 10^8)^2 \).
3. Substitute these values into the equation and solve for \( T_{earth} \).

After performing the calculations, you will find:
\[ T_{earth} \approx 300 \, K \]

### Final Answer
The temperature of the Earth in thermal equilibrium is approximately **300 K**.

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