If the wavelength corresponding to maximum energy radiated from the moon is 14 micron, and wien's constant is `2.8xx10^(-3)mK`, then temperature of moon is

Options

A. 207 K
B. 270 K
C. `207^(@)`C
D. `270^(@)`C

Explanation

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

### Step 1: Understand the relationship
Wien's Law states that the wavelength corresponding to the maximum energy radiated by a black body is inversely proportional to its temperature. The formula is given by:

\[ \lambda_{m} = \frac{b}{T} \]

where:
- \(\lambda_{m}\) is the wavelength corresponding to maximum energy,
- \(b\) is Wien's displacement constant,
- \(T\) is the temperature in Kelvin.

### Step 2: Convert the given wavelength
The problem states that the wavelength \(\lambda_{m}\) is 14 microns. We need to convert this into meters for consistency with the units of Wien's constant.

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

### Step 3: Identify Wien's constant
Wien's constant \(b\) is given as \(2.8 \times 10^{-3} \text{ m K}\).

### Step 4: Rearrange the formula to find temperature
From the formula \(\lambda_{m} = \frac{b}{T}\), we can rearrange it to solve for \(T\):

\[ T = \frac{b}{\lambda_{m}} \]

### Step 5: Substitute the values into the equation
Now we can substitute the values of \(b\) and \(\lambda_{m}\) into the equation:

\[ T = \frac{2.8 \times 10^{-3} \text{ m K}}{14 \times 10^{-6} \text{ m}} \]

### Step 6: Perform the calculation
Calculating the above expression:

\[ T = \frac{2.8 \times 10^{-3}}{14 \times 10^{-6}} = \frac{2.8}{14} \times 10^{3} \]

Calculating \(\frac{2.8}{14}\):

\[ \frac{2.8}{14} = 0.2 \]

Thus,

\[ T = 0.2 \times 10^{3} = 200 \text{ K} \]

### Conclusion
The temperature of the moon is approximately \(200 \text{ K}\).

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