A young's double slit apporatus is immersed in a liquid of refractive index 1.33.It has slit separation of 1 mm and interference pattern is observed on the screen at a distance 1.33 m from plane of slits.The wavelength in air is `6300 Å`
One of the slits of the apparatus is covered by a thin glass sheet of refractive index 1.53. Find the fringe width

Options

A. 0.63 mm
B. 1.26 mm
C. 1.67 mm
D. 2.2 mm

Explanation

To solve the problem of finding the fringe width in a Young's double slit apparatus immersed in a liquid, follow these steps:

### Step 1: Understand the given data
- Refractive index of the liquid (μ_liquid) = 1.33
- Refractive index of the glass (μ_glass) = 1.53
- Wavelength in air (λ) = 6300 Å = 6300 × 10⁻¹⁰ m
- Slit separation (d) = 1 mm = 1 × 10⁻³ m
- Distance from slits to screen (D) = 1.33 m

### Step 2: Calculate the effective wavelength in the liquid
The effective wavelength (λ') in the liquid can be calculated using the formula:
\[ \lambda' = \frac{\lambda}{\mu_{\text{liquid}}} \]
Substituting the values:
\[ \lambda' = \frac{6300 \times 10^{-10}}{1.33} \]

### Step 3: Calculate λ'
Calculating λ':
\[ \lambda' = \frac{6300 \times 10^{-10}}{1.33} \approx 4736.84 \times 10^{-10} \text{ m} \approx 4.73684 \times 10^{-7} \text{ m} \]

### Step 4: Use the formula for fringe width
The formula for fringe width (β) is given by:
\[ \beta = \frac{\lambda' D}{d} \]
Substituting the values:
\[ \beta = \frac{(4.73684 \times 10^{-7}) \times 1.33}{1 \times 10^{-3}} \]

### Step 5: Calculate β
Calculating β:
\[ \beta = \frac{(4.73684 \times 10^{-7}) \times 1.33}{1 \times 10^{-3}} \approx 0.629 \text{ mm} \]

### Step 6: Final result
Thus, the fringe width is approximately:
\[ \beta \approx 0.629 \text{ mm} \]