For the moon, the wavelength corresponding to the maximum spectral emissive power is 14 microns. If the Wien's constant `b=2.884 xx10^(-3)mK`, then the temperature of the moon is close to

Options

A. 206 K
B. 112 K
C. 314 K
D. 156 K

Answer

To find the temperature of the moon using Wien's Law, we follow these steps:

### Step 1: Understand Wien's Law
Wien's Law states that the wavelength corresponding to the maximum spectral emissive power (λ_max) is inversely proportional to the temperature (T) of the black body. The relationship is given by the formula:

\[ \lambda_{\text{max}} \cdot T = b \]

where \( b \) is Wien's constant.

### Step 2: Convert Wavelength to Meters
The given wavelength is 14 microns. We need to convert this to meters for our calculations:

\[ \lambda_{\text{max}} = 14 \, \text{microns} = 14 \times 10^{-6} \, \text{meters} \]

### Step 3: Substitute Values into Wien's Law
We know the value of Wien's constant \( b = 2.884 \times 10^{-3} \, \text{mK} \). Now, we can rearrange the formula to solve for temperature \( T \):

\[ T = \frac{b}{\lambda_{\text{max}}} \]

### Step 4: Plug in the Values
Substituting the values into the equation:

\[ T = \frac{2.884 \times 10^{-3} \, \text{mK}}{14 \times 10^{-6} \, \text{m}} \]

### Step 5: Calculate the Temperature
Now, we perform the calculation:

\[ T = \frac{2.884 \times 10^{-3}}{14 \times 10^{-6}} = \frac{2.884}{14} \times 10^{3} \]

Calculating \( \frac{2.884}{14} \):

\[ \frac{2.884}{14} \approx 0.2057 \]

Now multiplying by \( 10^{3} \):

\[ T \approx 0.2057 \times 10^{3} \approx 205.7 \, \text{K} \]

### Step 6: Round the Temperature
Rounding to the nearest whole number, we find:

\[ T \approx 206 \, \text{K} \]

### Conclusion
The temperature of the moon is close to **206 K**.

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