When a mica plate of thickness `0.1mm` is introduced in one of the interfering beams, the central fringe is displaced by a distance equal to 10 fringes. If the wavelength of the light is `6000 A^(0)`, the refractive index of the mica is

Options

A. `1.06`
B. `1.6`
C. `2.4`
D. `1.2`

Answer

To find the refractive index of the mica plate, we can follow these steps:

### Step 1: Understand the relationship between fringe shift and refractive index
When a mica plate is introduced in one of the interfering beams, it causes a path length change that results in a fringe shift. The formula for the fringe shift (Δx) when a mica plate of thickness \( t \) and refractive index \( \mu \) is introduced is given by:

\[ \Delta x = \frac{\beta}{\lambda} \cdot ( \mu - 1 ) \cdot t \]

Where:
- \( \Delta x \) = fringe shift (in terms of fringe width)
- \( \beta \) = fringe width
- \( \lambda \) = wavelength of light
- \( t \) = thickness of the mica plate
- \( \mu \) = refractive index of the mica

### Step 2: Relate fringe shift to the number of fringes
It is given that the central fringe is displaced by a distance equal to 10 fringes. Therefore, we can express the fringe shift as:

\[ \Delta x = 10 \cdot \beta \]

### Step 3: Substitute the expression for fringe shift into the formula
Substituting \( \Delta x = 10 \cdot \beta \) into the original formula gives us:

\[ 10 \cdot \beta = \frac{\beta}{\lambda} \cdot ( \mu - 1 ) \cdot t \]

### Step 4: Cancel out \( \beta \)
Assuming \( \beta \neq 0 \), we can cancel \( \beta \) from both sides:

\[ 10 = \frac{1}{\lambda} \cdot ( \mu - 1 ) \cdot t \]

### Step 5: Rearrange the equation to solve for \( \mu - 1 \)
Rearranging the equation gives:

\[ \mu - 1 = 10 \cdot \frac{\lambda}{t} \]

### Step 6: Substitute the known values
Now, substitute the values for \( \lambda \) and \( t \):
- \( \lambda = 6000 \, \text{Å} = 6000 \times 10^{-10} \, \text{m} \)
- \( t = 0.1 \, \text{mm} = 0.1 \times 10^{-3} \, \text{m} = 10^{-4} \, \text{m} \)

Substituting these values into the equation:

\[ \mu - 1 = 10 \cdot \frac{6000 \times 10^{-10}}{0.1 \times 10^{-3}} \]

### Step 7: Calculate \( \mu - 1 \)
Calculating the right side:

\[ \mu - 1 = 10 \cdot \frac{6000 \times 10^{-10}}{10^{-4}} = 10 \cdot 6000 \times 10^{-6} = 60 \times 10^{-6} = 0.06 \]

### Step 8: Solve for \( \mu \)
Now, adding 1 to both sides gives:

\[ \mu = 1 + 0.06 = 1.06 \]

### Conclusion
The refractive index of the mica is:

\[ \mu = 1.06 \]