On introducing a thin sheet of mica (thickness `12 xx 10^(-7)` cm) in path of one of the interfering beams (`lambda_("vacuum") = 600 nm`) in Young's double-slit experiment, the central fringe is shifted through a distance equal to the spacing between successive bright fringes. Calculate the refractive index of mica.

Options

A. 1.33
B. 1.4
C. 1.5
D. 2.5

Answer

To solve the problem, we need to calculate the refractive index of mica when a thin sheet is introduced in the path of one of the interfering beams in Young's double-slit experiment. Here are the steps to arrive at the solution:

### Step 1: Understand the problem
In Young's double-slit experiment, introducing a thin sheet of mica causes a shift in the interference pattern. The central fringe is shifted by a distance equal to the spacing between successive bright fringes.

### Step 2: Define the parameters
- Thickness of mica, \( t = 12 \times 10^{-7} \) cm \( = 12 \times 10^{-9} \) m (since \( 1 \) cm = \( 10^{-2} \) m)
- Wavelength of light in vacuum, \( \lambda = 600 \) nm \( = 600 \times 10^{-9} \) m

### Step 3: Calculate the extra optical path length introduced by the mica
When the mica sheet is introduced, the optical path length increases by \( (n - 1) t \), where \( n \) is the refractive index of mica. Therefore, the extra path length is given by:
\[ \Delta x = (n - 1) t \]

### Step 4: Relate the shift in fringe position to the wavelength
The shift in the central fringe is equal to the spacing between successive bright fringes, which is given by:
\[ \text{Fringe spacing} = \frac{\lambda}{d} \]
where \( d \) is the distance between the slits. However, since we are told that the shift is equal to \( \lambda \), we can equate:
\[ (n - 1) t = \lambda \]

### Step 5: Rearranging to find the refractive index
From the equation above, we can express the refractive index \( n \) as:
\[ n - 1 = \frac{\lambda}{t} \]
\[ n = 1 + \frac{\lambda}{t} \]

### Step 6: Substitute the values
Now, substituting the values of \( \lambda \) and \( t \):
\[ n = 1 + \frac{600 \times 10^{-9}}{12 \times 10^{-9}} \]
\[ n = 1 + \frac{600}{12} \]
\[ n = 1 + 50 \]
\[ n = 51 \]

### Final Answer
The refractive index of mica is \( n = 51 \).