Light from a sources emitting two wavelengths `lambda_(1)` and `lambda_(2)` is allowed to fall on Young's double-slit apparatus after filtering one of the wavelengths. The position of interference maxima is noted. When the filter is removed both the wavelengths are incident and it is found that maximum intensity is produced where the fourth maxima occured previously. If the other wavelength if filtered, at the same location the third maxima is found. What is the raito of wavelength?

Options

A. `(2)/(3)`
B. `(3)/(2)`
C. `(3)/(4)`
D. `(4)/(3)`

Explanation

To solve the problem, we need to analyze the information given about the interference maxima produced by two wavelengths in Young's double-slit experiment.

### Step-by-Step Solution:

1. **Understanding the Setup:**
- We have two wavelengths, \( \lambda_1 \) and \( \lambda_2 \).
- When only \( \lambda_1 \) is present, the position of the fourth maxima is noted.
- When both wavelengths are present, the maximum intensity occurs at the position of the fourth maxima of \( \lambda_1 \).
- When \( \lambda_2 \) is filtered out, the third maxima of \( \lambda_2 \) is found at the same position.

2. **Using the Formula for Maxima Position:**
- The position of the maxima in Young's double-slit experiment is given by the formula:
\[ x_n = \frac{n \lambda D}{d} \]
where \( n \) is the order of the maxima, \( \lambda \) is the wavelength, \( D \) is the distance from the slits to the screen, and \( d \) is the distance between the slits.

3. **Setting Up the Equations:**
- For \( \lambda_1 \) at the fourth maxima:
\[ x_4 = \frac{4 \lambda_1 D}{d} \]
- For \( \lambda_2 \) at the third maxima:
\[ x_3 = \frac{3 \lambda_2 D}{d} \]

4. **Equating the Positions:**
- Since both maxima occur at the same position when both wavelengths are present, we can equate the two expressions:
\[ \frac{4 \lambda_1 D}{d} = \frac{3 \lambda_2 D}{d} \]

5. **Canceling Common Terms:**
- The terms \( D \) and \( d \) can be canceled from both sides:
\[ 4 \lambda_1 = 3 \lambda_2 \]

6. **Finding the Ratio of Wavelengths:**
- Rearranging the equation gives:
\[ \frac{\lambda_1}{\lambda_2} = \frac{3}{4} \]

### Final Answer:
The ratio of the wavelengths \( \lambda_1 : \lambda_2 \) is \( 3 : 4 \).