In Young’s experiment, monochromatic light through a single slit S is used to illuminate the two slits `S_1`and `S_2`. Interference fringes are obtained on a screen. The fringe width is found to be w. Now a thin sheet of mica (thickness t and refractive index `mu`) is placed near and in front of one of the two slits. Now the fringe width is found to be w’, then :

Options

A. `w'=w//mu`
B. `w'=w mu`
C. `w'=(mu-1) tw`
D. `w'=w`

Answer

To solve the problem, we need to analyze the effect of placing a thin mica sheet in front of one of the slits in Young's double-slit experiment. We will derive the relationship between the original fringe width \( w \) and the new fringe width \( w' \) after inserting the mica sheet.

### Step-by-Step Solution:

1. **Understanding the Original Fringe Width**:
The fringe width \( w \) in Young's double-slit experiment is given by the formula:
\[ w = \frac{\lambda D}{d} \]
where:
- \( \lambda \) = wavelength of the light used,
- \( D \) = distance from the slits to the screen,
- \( d \) = distance between the two slits.

2. **Introducing the Mica Sheet**:
When a thin sheet of mica of thickness \( t \) and refractive index \( \mu \) is placed in front of one of the slits (let's say slit \( S_1 \)), it introduces an additional optical path difference due to the change in speed of light in the mica.

3. **Calculating the Extra Path Difference**:
The extra path difference \( \Delta x' \) introduced by the mica sheet can be calculated as:
\[ \Delta x' = (\mu - 1) t \]
This is because light travels slower in the mica than in air, leading to a longer effective path.

4. **Total Path Difference**:
The total path difference \( \Delta x \) for constructive interference at the position of the bright fringe will now be:
\[ \Delta x = \Delta x - \Delta x' = \Delta x - (\mu - 1)t \]
For constructive interference, this path difference must be equal to \( n\lambda \) (where \( n \) is an integer).

5. **Finding the New Fringe Position**:
The position of the bright fringe on the screen is given by:
\[ y_n = \frac{n \lambda D}{d} + \text{(additional term due to mica)} \]
The additional term due to the mica will be \( \frac{(\mu - 1)t D}{d} \).

6. **Calculating New Fringe Width**:
The new fringe width \( w' \) can be derived from the difference in positions of two consecutive bright fringes (for \( n \) and \( n+1 \)):
\[ w' = y_{n+1} - y_n = \left(\frac{(n+1)\lambda D}{d} + \frac{(\mu - 1)t D}{d}\right) - \left(\frac{n\lambda D}{d} + \frac{(\mu - 1)t D}{d}\right) \]
Simplifying this gives:
\[ w' = \frac{\lambda D}{d} \]
This shows that the additional term cancels out, and thus:
\[ w' = w \]

### Conclusion:
The new fringe width \( w' \) remains equal to the original fringe width \( w \) even after placing the mica sheet in front of one of the slits.