In YDSE, a glass slab of refractive index, `mu= 1.5` and thickness 'l' is introduced in one of the interfening beams of wavelength `lambda = 5000 A`. If on introducing the slab, the central fringe shift by 2 mm, then thickness of stab would be (Fringe width, `beta = 0.2 mm)`

Options

A. `3*10^-3 mm`
B. `1*10^-4 mm`
C. `1*10^-2 mm`
D. `2*10^-3 mm`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given data
We have the following information:
- Refractive index of the glass slab, \( \mu = 1.5 \)
- Wavelength of light, \( \lambda = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m} \)
- Fringe shift, \( \Delta y = 2 \, \text{mm} = 2 \times 10^{-3} \, \text{m} \)
- Fringe width, \( \beta = 0.2 \, \text{mm} = 0.2 \times 10^{-3} \, \text{m} \)

### Step 2: Calculate the order of fringe shift (n)
The fringe shift can be expressed as:
\[ \Delta y = n \cdot \beta \]
Rearranging this gives:
\[ n = \frac{\Delta y}{\beta} \]
Substituting the values:
\[ n = \frac{2 \times 10^{-3}}{0.2 \times 10^{-3}} = 10 \]

### Step 3: Determine the path difference due to the glass slab
The path difference caused by the introduction of the glass slab is given by:
\[ \text{Path difference} = t (\mu - 1) \]
Where \( t \) is the thickness of the slab.

### Step 4: Relate path difference to the order of the fringe
The condition for maxima in Young's Double Slit Experiment is:
\[ \text{Path difference} = n \lambda \]
Substituting the values we have:
\[ t (\mu - 1) = n \lambda \]
Substituting \( n = 10 \), \( \mu = 1.5 \), and \( \lambda = 5000 \times 10^{-10} \):
\[ t (1.5 - 1) = 10 \cdot (5000 \times 10^{-10}) \]
This simplifies to:
\[ t (0.5) = 10 \cdot (5000 \times 10^{-10}) \]

### Step 5: Solve for the thickness \( t \)
Rearranging gives:
\[ t = \frac{10 \cdot (5000 \times 10^{-10})}{0.5} \]
Calculating this:
\[ t = \frac{50000 \times 10^{-10}}{0.5} = 100000 \times 10^{-10} = 10^{-5} \, \text{m} \]

### Step 6: Convert thickness to millimeters
To convert meters to millimeters:
\[ t = 10^{-5} \, \text{m} = 10^{-5} \times 10^3 \, \text{mm} = 10^{-2} \, \text{mm} \]

### Final Answer
The thickness of the glass slab is:
\[ \boxed{0.01 \, \text{mm}} \]