Frequencies of light incident on a system of scattering particles are in the ratio of 1 : 2. Then, the intensity of scattered light in a particular direction is …

Options

A. `1:4`
B. `1:2`
C. `1:8`
D. `1: 16`

Explanation

To solve the problem, we need to determine the intensity of scattered light in a particular direction when the frequencies of light incident on a system of scattering particles are in the ratio of 1:2.

### Step-by-Step Solution:

1. **Understand the Given Ratios**:
We are given that the frequencies of light \( f_1 \) and \( f_2 \) are in the ratio:
\[ f_1 : f_2 = 1 : 2 \]
This means we can express \( f_1 \) and \( f_2 \) as:
\[ f_1 = k \quad \text{and} \quad f_2 = 2k \]
where \( k \) is a constant.

2. **Use Rayleigh Scattering Formula**:
According to Rayleigh scattering, the intensity \( I \) of scattered light is inversely proportional to the fourth power of the wavelength \( \lambda \):
\[ I \propto \frac{1}{\lambda^4} \]
Since frequency \( f \) and wavelength \( \lambda \) are related by the equation \( c = f \lambda \) (where \( c \) is the speed of light), we can express \( \lambda \) in terms of \( f \):
\[ \lambda = \frac{c}{f} \]
Therefore, we can rewrite the intensity in terms of frequency:
\[ I \propto \frac{1}{\left(\frac{c}{f}\right)^4} = \frac{f^4}{c^4} \]

3. **Calculate the Intensity Ratio**:
Now, we can find the ratio of the intensities \( I_1 \) and \( I_2 \) corresponding to frequencies \( f_1 \) and \( f_2 \):
\[ \frac{I_1}{I_2} = \frac{f_1^4}{f_2^4} \]
Substituting \( f_1 = k \) and \( f_2 = 2k \):
\[ \frac{I_1}{I_2} = \frac{k^4}{(2k)^4} = \frac{k^4}{16k^4} = \frac{1}{16} \]

4. **Conclusion**:
Thus, the intensity of scattered light in a particular direction when the frequencies are in the ratio of 1:2 is:
\[ I_1 : I_2 = 1 : 16 \]

### Final Answer:
The intensity of scattered light in a particular direction is in the ratio of **1:16**.

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