If wavelengths of maximum intensity of radiations emitted by the sun and the moon are `0.5xx10^(-6)m " and " 10^(-4)`m respectively, the ratio of their temperature is ……………

Options

A. 2000
B. 1000
C. 100
D. 200

Explanation

To solve the problem, we will use Wien's Displacement Law, which states that the wavelength of maximum intensity of radiation emitted by a black body is inversely proportional to its temperature. The formula can be expressed as:

\[ \lambda_{\text{max}} \propto \frac{1}{T} \]

Where:
- \(\lambda_{\text{max}}\) is the wavelength of maximum intensity,
- \(T\) is the absolute temperature in Kelvin.

### Step 1: Write the relationship using Wien's Law

From Wien's Law, we can write the relationship between the wavelengths and temperatures of the sun and the moon as follows:

\[ \frac{\lambda_s}{\lambda_m} = \frac{T_m}{T_s} \]

Where:
- \(\lambda_s\) is the wavelength of the sun,
- \(\lambda_m\) is the wavelength of the moon,
- \(T_s\) is the temperature of the sun,
- \(T_m\) is the temperature of the moon.

### Step 2: Substitute the given values

The problem provides the following wavelengths:
- \(\lambda_s = 0.5 \times 10^{-6} \, \text{m}\)
- \(\lambda_m = 10^{-4} \, \text{m}\)

Now we can substitute these values into the equation:

\[ \frac{0.5 \times 10^{-6}}{10^{-4}} = \frac{T_m}{T_s} \]

### Step 3: Simplify the equation

To simplify the left side:

\[ \frac{0.5 \times 10^{-6}}{10^{-4}} = 0.5 \times 10^{-6} \times 10^{4} = 0.5 \times 10^{-2} \]

So we have:

\[ \frac{T_m}{T_s} = 0.5 \times 10^{-2} \]

### Step 4: Find the ratio of the temperatures

To find the ratio of the temperatures, we can rearrange the equation:

\[ \frac{T_s}{T_m} = \frac{1}{0.5 \times 10^{-2}} = \frac{1}{0.005} = 200 \]

### Conclusion

Thus, the ratio of the temperature of the sun to the temperature of the moon is:

\[ \frac{T_s}{T_m} = 200 \]

### Final Answer

The ratio of their temperatures is **200**.

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