A black body at `1373 ^@C` emits maximum energy corresponding to a wavelength of 1.78 microns. The temperature of the moon for which `lamda_m=14` micron wood be

Options

A. `62.6 ^@C`
B. `-58.9^@C`
C. `-63.7 ^@C`
D. `64.2 ^@C`

Answer

To solve the problem, we will use Wien's Displacement Law, which states that the product of the maximum wavelength of radiation emitted by a black body and its absolute temperature is a constant.

### Step-by-step Solution:

1. **Understanding the Given Information**:
- The temperature of the black body (T_B) is given as \(1373 \, ^\circ C\).
- The maximum wavelength emitted by the black body (λ_B) is \(1.78 \, \text{microns}\).
- We need to find the temperature of the moon (T_M) for which the maximum wavelength (λ_M) is \(14 \, \text{microns}\).

2. **Convert Temperature to Kelvin**:
- Temperature in Kelvin (T_B) is calculated as:
\[ T_B = 1373 + 273 = 1646 \, \text{K} \]

3. **Applying Wien's Displacement Law**:
- According to Wien's Displacement Law:
\[ \lambda_B \cdot T_B = \lambda_M \cdot T_M \]
- Rearranging the formula to find T_M:
\[ T_M = \frac{\lambda_B \cdot T_B}{\lambda_M} \]

4. **Substituting the Values**:
- Substitute the known values into the equation:
\[ T_M = \frac{1.78 \, \text{microns} \cdot 1646 \, \text{K}}{14 \, \text{microns}} \]

5. **Calculating T_M**:
- Performing the multiplication and division:
\[ T_M = \frac{2925.68}{14} \approx 208.98 \, \text{K} \]

6. **Convert Temperature to Celsius**:
- To convert from Kelvin to Celsius:
\[ T_M (\text{in } ^\circ C) = 208.98 - 273 \approx -64.02 \, ^\circ C \]

### Final Answer:
The temperature of the moon is approximately \(-64.02 \, ^\circ C\).