If wavelength of maximum intensity of radiation emitted by sun and moon are `0.5xx10^(-6)` m and `10^(-4)` m respectively. Calculate the ratio of their temperatures

Explanation

To solve the problem of finding the ratio of the temperatures of the sun and the moon based on their respective wavelengths of maximum intensity, we will use 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 corresponding to the maximum intensity of radiation emitted by a black body is inversely proportional to its temperature. Mathematically, it can be expressed as:
\[ \lambda_{max} \cdot T = b \]
where \( \lambda_{max} \) is the wavelength of maximum intensity, \( T \) is the temperature, and \( b \) is a constant.

### Step 2: Set Up the Equation
For the sun and the moon, we can express the relationship as:
\[ \lambda_{sun} \cdot T_{sun} = \lambda_{moon} \cdot T_{moon} \]

### Step 3: Rearranging for Temperature Ratio
We want to find the ratio of the temperatures:
\[ \frac{T_{sun}}{T_{moon}} = \frac{\lambda_{moon}}{\lambda_{sun}} \]

### Step 4: Substitute the Given Values
From the problem, we have:
- Wavelength of maximum intensity for the sun, \( \lambda_{sun} = 0.5 \times 10^{-6} \) m
- Wavelength of maximum intensity for the moon, \( \lambda_{moon} = 10^{-4} \) m

Substituting these values into the equation gives:
\[ \frac{T_{sun}}{T_{moon}} = \frac{10^{-4}}{0.5 \times 10^{-6}} \]

### Step 5: Simplifying the Ratio
Now, simplify the right-hand side:
1. Rewrite \( 0.5 \times 10^{-6} \) as \( \frac{0.5}{1} \times 10^{-6} \).
2. The ratio becomes:
\[ \frac{T_{sun}}{T_{moon}} = \frac{10^{-4}}{0.5 \times 10^{-6}} = \frac{10^{-4}}{0.5} \times \frac{1}{10^{-6}} \]
3. This simplifies to:
\[ \frac{T_{sun}}{T_{moon}} = \frac{10^{-4} \times 10^{6}}{0.5} = \frac{10^{2}}{0.5} = 200 \]

### Step 6: Final Result
Thus, the ratio of the temperatures of the sun to the moon is:
\[ \frac{T_{sun}}{T_{moon}} = 200:1 \]