The wavelength `lambda_(m)` of maximum intensity of emission of solar radiation is `lambda_(m) = 4753 Å` and from moon is `lambda_(m) = 14 mum` The surface temperature of sun and moon are (given `b = 2.898 xx 10^(-3)` meter/Kelvin)

Options

A. 6097 K, 207 K
B. 8097 K, 307 K
C. 10,000 K, 400 K
D. 3000 K, 100 K

Answer

To find the surface temperatures of the Sun and the Moon using Wien's displacement law, we can follow these steps:

### Step 1: Understand Wien's Displacement Law
Wien's displacement law states that the wavelength of maximum intensity of emission (λm) is inversely proportional to the temperature (T) of the black body. The formula is given by:

\[ \lambda_m \cdot T = b \]

where \( b = 2.898 \times 10^{-3} \) m·K (Wien's constant).

### Step 2: Convert Wavelengths to Meters
We need to convert the given wavelengths from Angstroms and micrometers to meters.

- For the Sun:
\[ \lambda_{m, \text{sun}} = 4753 \, \text{Å} = 4753 \times 10^{-10} \, \text{m} = 4.753 \times 10^{-7} \, \text{m} \]

- For the Moon:
\[ \lambda_{m, \text{moon}} = 14 \, \mu m = 14 \times 10^{-6} \, \text{m} \]

### Step 3: Calculate Temperature of the Sun
Using the formula from Wien's law for the Sun:

\[ T_{\text{sun}} = \frac{b}{\lambda_{m, \text{sun}}} \]

Substituting the values:

\[ T_{\text{sun}} = \frac{2.898 \times 10^{-3}}{4.753 \times 10^{-7}} \]

Calculating this gives:

\[ T_{\text{sun}} = \frac{2.898 \times 10^{-3}}{4.753 \times 10^{-7}} \approx 6097 \, \text{K} \]

### Step 4: Calculate Temperature of the Moon
Using the formula from Wien's law for the Moon:

\[ T_{\text{moon}} = \frac{b}{\lambda_{m, \text{moon}}} \]

Substituting the values:

\[ T_{\text{moon}} = \frac{2.898 \times 10^{-3}}{14 \times 10^{-6}} \]

Calculating this gives:

\[ T_{\text{moon}} = \frac{2.898 \times 10^{-3}}{14 \times 10^{-6}} \approx 207 \, \text{K} \]

### Final Results
- Temperature of the Sun: \( T_{\text{sun}} \approx 6097 \, \text{K} \)
- Temperature of the Moon: \( T_{\text{moon}} \approx 207 \, \text{K} \)

### Summary
Thus, the surface temperatures are:
- Sun: 6097 K
- Moon: 207 K