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 much 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 coherent light waves producing an interference pattern.
- When a thin mica sheet is introduced in the path of one of the waves, it alters the effective path length of that wave.

2. **Path Length Change**:
- The introduction of the mica sheet causes a change in the optical path length. The optical path length is given by the product of the physical thickness of the medium and its refractive index.
- The change in optical path length (Δx) due to the mica sheet is given by:
\[ \Delta x = (μ - 1) \cdot t \]
- Here, 'μ' is the refractive index of the mica, and 't' is the thickness of the mica sheet.

3. **Relating Path Difference to Fringe Shift**:
- The path difference also relates to the fringe shift on the screen. The path difference (Δx) can also be expressed in terms of the fringe shift (y) and the geometry of the setup.
- For small angles, we can use the approximation:
\[ \Delta x = d \cdot \tan(\theta) \]
- For small angles, \(\tan(\theta) \approx \frac{y}{D}\), where 'y' is the fringe shift and 'D' is the distance from the slits to the screen.

4. **Setting Up the Equation**:
- We can equate the two expressions for the path difference:
\[ \frac{d \cdot y}{D} = (μ - 1) \cdot t \]

5. **Solving for Fringe Shift (y)**:
- Rearranging the equation to solve for 'y':
\[ y = \frac{D}{d} \cdot (μ - 1) \cdot t \]
- This gives us the distance by which the interference pattern shifts.

6. **Final Result**:
- The final expression for the shift in the interference pattern is:
\[ y = \frac{D}{d} \cdot (μ - 1) \cdot t \]