In a Young's double slit experiment, the fringe width is found to be `0.4mm`. If the whole apparatus is immersed in water of refractive index `4//3` without disturbing the geometrical arrangement, the new fringe width will be

Options

A. (a) `0.30mm`
B. (b) `0.40mm`
C. (c) `0.53mm`
D. (d) `450micron`

Explanation

To solve the problem of finding the new fringe width when the Young's double slit experiment apparatus is immersed in water, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for fringe width**:
The fringe width (β) in a Young's double slit experiment is given by the formula:
\[ \beta = \frac{D \lambda}{d} \]
where:
- \(D\) = distance from the slits to the screen (remains constant),
- \(\lambda\) = wavelength of light,
- \(d\) = distance between the slits (also remains constant).

2. **Identify the initial fringe width**:
The initial fringe width is given as:
\[ \beta_1 = 0.4 \, \text{mm} \]

3. **Determine the effect of immersion in water**:
When the apparatus is immersed in water with a refractive index \( \mu = \frac{4}{3} \), the wavelength of light changes. The new wavelength (\(\lambda'\)) in the medium is given by:
\[ \lambda' = \frac{\lambda}{\mu} = \frac{\lambda}{\frac{4}{3}} = \frac{3}{4} \lambda \]

4. **Relate the new fringe width to the old fringe width**:
Since the fringe width is directly proportional to the wavelength, we can express the new fringe width (\(\beta_2\)) in terms of the old fringe width:
\[ \frac{\beta_1}{\beta_2} = \frac{\lambda_1}{\lambda_2} \]
Substituting \(\lambda_2 = \frac{3}{4} \lambda_1\):
\[ \frac{\beta_1}{\beta_2} = \frac{\lambda_1}{\frac{3}{4} \lambda_1} = \frac{4}{3} \]

5. **Calculate the new fringe width**:
Rearranging the equation gives:
\[ \beta_2 = \frac{3}{4} \beta_1 \]
Substituting the value of \(\beta_1\):
\[ \beta_2 = \frac{3}{4} \times 0.4 \, \text{mm} = 0.3 \, \text{mm} \]

6. **Conclusion**:
The new fringe width when the apparatus is immersed in water is:
\[ \beta_2 = 0.3 \, \text{mm} \]

### Final Answer:
The new fringe width is \(0.3 \, \text{mm}\).

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