If λm for the moon is 14.5 micron ,then find its temperature.

Explanation

To find the temperature of the moon using the given wavelength (λm = 14.5 microns), we can apply Wien's Displacement Law. Here's a step-by-step solution:

### Step 1: Understand Wien's Displacement Law
Wien's Displacement Law states that the wavelength at which the emission of a black body spectrum is maximized (λm) is inversely proportional to the temperature (T) of the black body. The relationship is given by the formula:
\[ \lambda_m \cdot T = b \]
where \( b \) is a constant approximately equal to \( 2.89 \times 10^{-3} \, \text{m} \cdot \text{K} \).

### Step 2: Convert Wavelength to Meters
The wavelength provided is in microns. We need to convert this to meters for consistency in units.
\[ \lambda_m = 14.5 \, \text{microns} = 14.5 \times 10^{-6} \, \text{m} \]

### Step 3: Rearrange the Formula to Solve for Temperature
From the formula \( \lambda_m \cdot T = b \), we can rearrange it to solve for temperature (T):
\[ T = \frac{b}{\lambda_m} \]

### Step 4: Substitute Known Values
Now, we can substitute the known values into the equation:
- \( b = 2.89 \times 10^{-3} \, \text{m} \cdot \text{K} \)
- \( \lambda_m = 14.5 \times 10^{-6} \, \text{m} \)

Substituting these values gives:
\[ T = \frac{2.89 \times 10^{-3}}{14.5 \times 10^{-6}} \]

### Step 5: Perform the Calculation
Now, we perform the division:
\[ T = \frac{2.89 \times 10^{-3}}{14.5 \times 10^{-6}} = 199.31 \, \text{K} \]

### Conclusion
Thus, the temperature of the moon is approximately:
\[ T \approx 199.3 \, \text{K} \]