A thin plastic of refractive index 1.6 is used to cover one of the slits of a double slit arrangement. The central point omn the screen is now occupied by what would have been the 7th bright fringe before the plastic was used. If the wavelength of light is 600 nm, what is the thickness ( in `mu`m ) of the plastic?

Options

A. 7
B. 4
C. 8
D. 6

Answer

To solve the problem, we need to determine the thickness of the plastic that causes the central maximum to shift to the position of the 7th bright fringe in a double slit interference pattern.

### Step-by-Step Solution:

1. **Understanding Path Difference**:
When a thin plastic sheet of refractive index \( \mu \) is placed in front of one of the slits, it changes the optical path length for the light passing through that slit. The optical path length through the plastic is given by \( \mu t \), where \( t \) is the thickness of the plastic.

2. **Calculating the Path Difference**:
The path difference \( \Delta \) introduced by the plastic is:
\[ \Delta = \mu t - t = (\mu - 1) t \]
Given that \( \mu = 1.6 \), we can substitute this value:
\[ \Delta = (1.6 - 1) t = 0.6 t \]

3. **Condition for Bright Fringe**:
The central maximum now coincides with the position of the 7th bright fringe. The path difference for the 7th bright fringe is given by:
\[ \Delta = 7 \lambda \]
where \( \lambda \) is the wavelength of light. Given \( \lambda = 600 \, \text{nm} = 600 \times 10^{-9} \, \text{m} \).

4. **Setting the Path Differences Equal**:
To find the thickness \( t \), we set the path difference due to the plastic equal to the path difference for the 7th fringe:
\[ 0.6 t = 7 \lambda \]
Substituting the value of \( \lambda \):
\[ 0.6 t = 7 \times 600 \times 10^{-9} \]

5. **Solving for Thickness \( t \)**:
Rearranging the equation to solve for \( t \):
\[ t = \frac{7 \times 600 \times 10^{-9}}{0.6} \]
Calculating the right-hand side:
\[ t = \frac{4200 \times 10^{-9}}{0.6} = 7000 \times 10^{-9} \, \text{m} = 7 \times 10^{-6} \, \text{m} \]
Converting to micrometers:
\[ t = 7 \, \mu\text{m} \]

### Final Answer:
The thickness of the plastic is \( \boxed{7} \, \mu\text{m} \).