Sun and moon emit maximum radiant energy at wavelength 5000 Å and `15mu` respectively. Iff surface temperature of sun is 6000 K, then value of surface temperature of moon is

Options

A. 100 K
B. 450 K
C. 200 K
D. 150 K

Explanation

To find the surface temperature of the Moon given the maximum radiant energy wavelengths for both the Sun and the Moon, we can use Wien's Displacement Law, which states that the product of the wavelength of maximum emission (λ) and the absolute temperature (T) of a black body is a constant.

### Step-by-Step Solution:

1. **Identify the given values:**
- Wavelength of maximum radiant energy for the Sun (λ_sun) = 5000 Å = \(5000 \times 10^{-10}\) m
- Wavelength of maximum radiant energy for the Moon (λ_moon) = 15 µm = \(15 \times 10^{-6}\) m
- Surface temperature of the Sun (T_sun) = 6000 K

2. **Write down Wien's Displacement Law:**
\[ \lambda \cdot T = b \]
where \(b\) is a constant.

3. **Set up the equation for the Sun and Moon:**
\[ \lambda_{sun} \cdot T_{sun} = \lambda_{moon} \cdot T_{moon} \]

4. **Rearranging the equation to find the temperature of the Moon:**
\[ T_{moon} = \frac{\lambda_{sun} \cdot T_{sun}}{\lambda_{moon}} \]

5. **Substituting the known values into the equation:**
\[ T_{moon} = \frac{(5000 \times 10^{-10} \text{ m}) \cdot (6000 \text{ K})}{(15 \times 10^{-6} \text{ m})} \]

6. **Calculating the numerator:**
\[ 5000 \times 10^{-10} \text{ m} \cdot 6000 \text{ K} = 30000000 \times 10^{-10} \text{ K m} = 3 \times 10^{6} \text{ K m} \]

7. **Calculating the denominator:**
\[ 15 \times 10^{-6} \text{ m} \]

8. **Now, divide the numerator by the denominator:**
\[ T_{moon} = \frac{3 \times 10^{6} \text{ K m}}{15 \times 10^{-6} \text{ m}} = \frac{3 \times 10^{6}}{15} \times 10^{6} \text{ K} \]

9. **Simplifying the division:**
\[ T_{moon} = 0.2 \times 10^{12} \text{ K} = 200 \text{ K} \]

10. **Final Result:**
The surface temperature of the Moon is \(200 \text{ K}\).