If a thin mica sheet of thickness 't' and refractive index `mu` is placed in the path of one of the waves producing interference , then the whole interference pattern shifts towards the side of the sheet by a distance

Options

A. `d/D(mu-1)t`
B. `D/d(mu-1)t`
C. `Dd(mu-1)t`
D. `(mu-1)t`

Answer

To solve the problem of how the interference pattern shifts when a thin mica sheet of thickness 't' and refractive index 'µ' is placed in the path of one of the waves, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two slits (S1 and S2) that produce an interference pattern on a screen.
- A thin mica sheet is placed in front of one of the slits (let's say S1).

2. **Path Difference Introduction**:
- The mica sheet introduces a path difference between the two waves coming from the slits.
- The wave passing through the mica sheet travels a longer effective distance compared to the wave passing through air.

3. **Calculating the Effective Path Length**:
- The effective optical path length for the wave traveling through the mica sheet is given by:
\[ \text{Effective Path Length} = \mu t \]
- The wave traveling through air has an effective path length of:
\[ \text{Path Length in Air} = t \]
- Therefore, the path difference (Δ) introduced by the mica sheet is:
\[ \Delta = \mu t - t = (\mu - 1)t \]

4. **Relating Path Difference to Shift in Interference Pattern**:
- The shift in the interference pattern can be related to the path difference using the formula:
\[ \Delta = d \sin \theta \]
- For small angles (which is often the case in interference patterns), we can use the approximation:
\[ \sin \theta \approx \tan \theta \approx \frac{y}{D} \]
- Here, \(d\) is the distance between the slits, \(y\) is the shift in the interference pattern, and \(D\) is the distance from the slits to the screen.

5. **Finding the Shift (y)**:
- Substituting the path difference into the equation gives:
\[ (\mu - 1)t = d \frac{y}{D} \]
- Rearranging to find \(y\):
\[ y = \frac{(\mu - 1)tD}{d} \]

6. **Conclusion**:
- The whole interference pattern shifts towards the side of the mica sheet by a distance \(y\) given by:
\[ y = \frac{(\mu - 1)tD}{d} \]