What will be the ratio of temperatures of sun and moon if the wavelengths of their maximum emission radiations rates are `140A^(@)` and `4200A^(@)` respectively .

Options

A. (a)`1:30`
B. (b)`30:1`
C. (c)`42:14`
D. (d)`14:12`

Answer

To find the ratio of temperatures of the Sun and the Moon based on their maximum emission wavelengths, we can use Wien's Law. According to Wien's Law, the product of the wavelength of maximum emission (λ_max) and the temperature (T) of a black body is a constant.

### Step-by-Step Solution:

1. **Understand Wien's Law**:
\[ \lambda_{\text{max}} \cdot T = b \]
where \( b \) is Wien's displacement constant.

2. **Define Variables**:
- Let \( T_1 \) be the temperature of the Sun.
- Let \( T_2 \) be the temperature of the Moon.
- The maximum wavelength for the Sun, \( \lambda_1 = 140 \, \text{Å} \).
- The maximum wavelength for the Moon, \( \lambda_2 = 4200 \, \text{Å} \).

3. **Apply Wien's Law for the Sun and Moon**:
- For the Sun:
\[ \lambda_1 \cdot T_1 = b \quad \text{(1)} \]
- For the Moon:
\[ \lambda_2 \cdot T_2 = b \quad \text{(2)} \]

4. **Set Equations Equal**:
Since both equations equal the same constant \( b \), we can set them equal to each other:
\[ \lambda_1 \cdot T_1 = \lambda_2 \cdot T_2 \]

5. **Rearrange for Temperature Ratio**:
\[ \frac{T_1}{T_2} = \frac{\lambda_2}{\lambda_1} \]

6. **Substitute the Wavelengths**:
- Substitute \( \lambda_1 = 140 \, \text{Å} \) and \( \lambda_2 = 4200 \, \text{Å} \):
\[ \frac{T_1}{T_2} = \frac{4200}{140} \]

7. **Calculate the Ratio**:
\[ \frac{T_1}{T_2} = \frac{4200}{140} = 30 \]

8. **Final Ratio**:
Therefore, the ratio of the temperatures of the Sun to the Moon is:
\[ T_1 : T_2 = 30 : 1 \]

### Conclusion:
The ratio of the temperatures of the Sun and Moon is \( 30 : 1 \).

---