If the ratio of amounts of scattering of two light waves is 1:4, the ratio of their wavelength is

Options

A. `1:2`
B. `sqrt(2):1`
C. `1:sqrt(2)`
D. `1:1`

Explanation

To solve the problem of finding the ratio of wavelengths given the ratio of scattering amounts of two light waves, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between scattering and wavelength**:
The amount of scattering of light is inversely proportional to the fourth power of its wavelength. This relationship can be expressed mathematically as:
\[ S \propto \frac{1}{\lambda^4} \]
where \( S \) is the amount of scattering and \( \lambda \) is the wavelength.

2. **Set up the ratio of scattering**:
Given that the ratio of the amounts of scattering for two light waves is \( S_1 : S_2 = 1 : 4 \), we can express this as:
\[ \frac{S_1}{S_2} = \frac{1}{4} \]

3. **Relate scattering to wavelength**:
Using the relationship established in step 1, we can write:
\[ \frac{S_1}{S_2} = \frac{\lambda_2^4}{\lambda_1^4} \]

4. **Substitute the known ratio into the equation**:
Substituting the ratio of scattering into the equation gives us:
\[ \frac{1}{4} = \frac{\lambda_2^4}{\lambda_1^4} \]

5. **Cross-multiply to find a relationship between wavelengths**:
Rearranging the equation, we get:
\[ \lambda_2^4 = \frac{1}{4} \lambda_1^4 \]

6. **Take the fourth root of both sides**:
To find the ratio of the wavelengths, we take the fourth root:
\[ \frac{\lambda_2}{\lambda_1} = \left(\frac{1}{4}\right)^{1/4} \]

7. **Simplify the expression**:
We know that \( 4^{1/4} = 2^{1/2} = \sqrt{2} \), thus:
\[ \frac{\lambda_2}{\lambda_1} = \frac{1}{\sqrt{2}} \]

8. **Express the ratio of wavelengths**:
Inverting the ratio gives us:
\[ \frac{\lambda_1}{\lambda_2} = \sqrt{2} \]

### Final Result:
The ratio of the wavelengths is:
\[ \lambda_1 : \lambda_2 = \sqrt{2} : 1 \]