In a Young's experiment, one of the slits is covered with a transparent sheet of thickness `3.6xx10^(-3)cm` due to which position of central fringe shifts to a position originally occupied by 30th fringe. If `lambda=6000 Å`, then find the refractive index of the sheet.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
In Young's double-slit experiment, a transparent sheet is placed over one of the slits, causing the central fringe to shift to the position of the 30th fringe. We need to find the refractive index of the sheet.

### Step 2: Identify given values
- Thickness of the sheet, \( T = 3.6 \times 10^{-3} \, \text{cm} = 3.6 \times 10^{-5} \, \text{m} \)
- Wavelength, \( \lambda = 6000 \, \text{Å} = 6000 \times 10^{-10} \, \text{m} = 6 \times 10^{-7} \, \text{m} \)
- Fringe shift corresponds to the 30th fringe.

### Step 3: Write the formula for fringe shift
The shift in the fringe position due to the introduction of the sheet is given by:
\[ \text{Shift} = (n - 1) \cdot T \cdot \frac{D}{d} \]
where \( n \) is the refractive index of the sheet, \( T \) is the thickness of the sheet, \( D \) is the distance from the slits to the screen, and \( d \) is the distance between the slits.

### Step 4: Relate the shift to the fringe number
Since the central fringe has shifted to the position of the 30th fringe, we can equate the shift to the position of the 30th fringe:
\[ (n - 1) \cdot T \cdot \frac{D}{d} = 30 \cdot \lambda \]

### Step 5: Simplify the equation
From the above equation, we can express \( n - 1 \):
\[ n - 1 = \frac{30 \cdot \lambda \cdot d}{T \cdot D} \]

### Step 6: Substitute the values
We know:
- \( \lambda = 6 \times 10^{-7} \, \text{m} \)
- \( T = 3.6 \times 10^{-5} \, \text{m} \)

Now, substituting the values into the equation:
\[ n - 1 = \frac{30 \cdot (6 \times 10^{-7})}{3.6 \times 10^{-5}} \]

### Step 7: Calculate \( n - 1 \)
Calculating the right side:
\[ n - 1 = \frac{180 \times 10^{-7}}{3.6 \times 10^{-5}} = \frac{180}{3.6} \times 10^{-2} \]
\[ n - 1 = 50 \times 10^{-2} = 0.5 \]

### Step 8: Find the refractive index \( n \)
Now, adding 1 to both sides:
\[ n = 1 + 0.5 = 1.5 \]

### Final Answer
The refractive index of the sheet is \( n = 1.5 \).

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