The central fringe of the interference pattern produced by light of wavelength 6000A is found to shift to the position of 4th bright fringe after a glass plate of refraction index 1.5 is introduced in front of one slit in young's experiment. The thickness of the glass plate will be

Options

A. `4.8 mu m`
B. `8.23 mu m`
C. `14.98 mu m`
D. `3.78 mu m`

Answer

To solve the problem, we need to find the thickness of the glass plate introduced in front of one slit in Young's double-slit experiment. The introduction of the glass plate causes a shift in the interference pattern, specifically moving the central fringe to the position of the 4th bright fringe.

### Step-by-Step Solution:

1. **Understanding the Shift in Fringe Position**:
- The central fringe (0th order) shifts to the position of the 4th bright fringe. This indicates that the path difference introduced by the glass plate is equivalent to the path difference for 4 bright fringes.

2. **Path Difference Calculation**:
- The path difference for the m-th bright fringe in Young's experiment is given by:
\[ \text{Path Difference} = m \lambda \]
- For the 4th bright fringe (m = 4), the path difference is:
\[ \text{Path Difference} = 4 \lambda \]

3. **Substituting the Wavelength**:
- Given that the wavelength \( \lambda = 6000 \, \text{Å} = 6000 \times 10^{-10} \, \text{m} = 6 \times 10^{-7} \, \text{m} \).
- Therefore, the path difference becomes:
\[ \text{Path Difference} = 4 \times 6 \times 10^{-7} \, \text{m} = 24 \times 10^{-7} \, \text{m} = 2.4 \times 10^{-6} \, \text{m} \]

4. **Using the Refractive Index**:
- The effective path difference introduced by the glass plate is given by:
\[ \Delta = (n - 1) t \]
- Where \( n \) is the refractive index of the glass plate (1.5) and \( t \) is the thickness of the glass plate.

5. **Setting Up the Equation**:
- From the previous steps, we have:
\[ (n - 1) t = 2.4 \times 10^{-6} \, \text{m} \]
- Substituting \( n = 1.5 \):
\[ (1.5 - 1) t = 2.4 \times 10^{-6} \, \text{m} \]
\[ 0.5 t = 2.4 \times 10^{-6} \, \text{m} \]

6. **Solving for Thickness \( t \)**:
- Rearranging the equation gives:
\[ t = \frac{2.4 \times 10^{-6} \, \text{m}}{0.5} = 4.8 \times 10^{-6} \, \text{m} \]
- Converting this to micrometers:
\[ t = 4.8 \, \mu m \]

### Final Answer:
The thickness of the glass plate is \( 4.8 \, \mu m \).