Interference fringes were produced using light in a doulbe-slit experiment. When a mica sheet of uniform thickness and refractive index 1.6 (relative to air) is placed in the path of light from one of the slits, the central fringe moves through some distance. This distance is equal to the width of 30 interference bands if light of wavelength 4800 is used. The thickness (in `mu m`) of mica is

Options

A. 90
B. 12
C. 14
D. 24

Explanation

To solve the problem, we need to find the thickness of the mica sheet placed in the path of light in a double-slit experiment. The key points from the question are:

- The refractive index of mica (ν) = 1.6
- The wavelength of light (λ) = 4800 Å (or 4800 x 10^-10 m)
- The central fringe moves through a distance equal to the width of 30 interference bands.

### Step-by-Step Solution:

1. **Calculate the Shift in the Central Fringe:**
The distance moved by the central fringe (Y) is equal to the width of 30 interference bands.
\[ Y = \beta \times 30 \]
where \(\beta\) is the fringe width.

2. **Calculate the Fringe Width (β):**
The fringe width (β) is given by the formula:
\[ \beta = \frac{D \lambda}{d} \]
where:
- \(D\) = distance from the slits to the screen
- \(d\) = distance between the slits

Thus, substituting this into the equation for Y:
\[ Y = \frac{D \lambda}{d} \times 30 \]

3. **Calculate the Path Difference Due to Mica:**
The path difference (ΔX) introduced by the mica sheet is given by:
\[ \Delta X = (ν - 1) \times T \]
where \(T\) is the thickness of the mica sheet.

Given \(ν = 1.6\):
\[ \Delta X = (1.6 - 1) \times T = 0.6T \]

4. **Equate the Path Difference to the Shift:**
The path difference (ΔX) is also equal to the shift Y calculated earlier:
\[ 0.6T = Y \]
Substituting for Y:
\[ 0.6T = \frac{D \lambda}{d} \times 30 \]

5. **Rearranging the Equation:**
Rearranging for thickness \(T\):
\[ T = \frac{30D\lambda}{0.6d} \]

6. **Substituting the Wavelength:**
Substitute \(\lambda = 4800 \times 10^{-10} \text{ m}\):
\[ T = \frac{30D(4800 \times 10^{-10})}{0.6d} \]

7. **Simplifying the Equation:**
Simplifying the equation:
\[ T = \frac{30 \times 4800 \times 10^{-10}}{0.6} \times \frac{D}{d} \]

Since \(D/d\) cancels out, we can focus on the numerical part:
\[ T = 240000 \times 10^{-10} \text{ m} \]

8. **Convert to Micrometers:**
Convert \(T\) to micrometers:
\[ T = 240000 \times 10^{-10} \text{ m} = 24 \text{ micrometers} \]

### Final Answer:
The thickness of the mica sheet is **24 micrometers (μm)**.