The radiant energy from the Sun incident normally at the surface of earth is `20kcal//m^(2)` min What would have been the radiant energy incident normally on the earth if the sun had a temperature twice of the present one ? .

Options

A. `160 kcal//m^(2)` min
B. `40 kcal//m^(2)` min
C. `320 kcal//m^(2)` min
D. `80 kcal//m^(2)` min

Answer

To solve the problem, we will use Stefan's Law, which states that the radiant energy emitted by a black body is directly proportional to the fourth power of its absolute temperature.

### Step-by-Step Solution:

1. **Identify Given Values**:
- The radiant energy incident normally on the Earth from the Sun, \( E_1 = 20 \, \text{kcal/m}^2 \, \text{min} \).
- Present temperature of the Sun, \( T_1 = T \).
- New temperature of the Sun, \( T_2 = 2T \).

2. **Apply Stefan's Law**:
According to Stefan's Law, the relationship between the energies and temperatures is given by:
\[ \frac{E_1}{E_2} = \left( \frac{T_1}{T_2} \right)^4 \]

3. **Substitute Known Values**:
Substitute \( E_1 \), \( T_1 \), and \( T_2 \) into the equation:
\[ \frac{20 \, \text{kcal/m}^2 \, \text{min}}{E_2} = \left( \frac{T}{2T} \right)^4 \]

4. **Simplify the Temperature Ratio**:
Simplifying the temperature ratio:
\[ \frac{T}{2T} = \frac{1}{2} \]
Therefore,
\[ \left( \frac{1}{2} \right)^4 = \frac{1}{16} \]

5. **Rewrite the Equation**:
Now, substituting back into the equation gives:
\[ \frac{20}{E_2} = \frac{1}{16} \]

6. **Cross-Multiply to Solve for \( E_2 \)**:
Cross-multiplying gives:
\[ 20 = \frac{E_2}{16} \]
Therefore,
\[ E_2 = 20 \times 16 \]

7. **Calculate \( E_2 \)**:
Finally, calculate \( E_2 \):
\[ E_2 = 320 \, \text{kcal/m}^2 \, \text{min} \]

### Final Answer:
The radiant energy incident normally on the Earth if the Sun had a temperature twice the present one would be \( E_2 = 320 \, \text{kcal/m}^2 \, \text{min} \).

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