In the ideal double-slit experiment, when a glass-plate (refractive index 1.5) of thickness t is introduced in the path of one of the interfering beams (wavelength `lambda`), the intensity at the position where the central maximum occurred previously remains unchanged. The minimum thickness of the glass-plate is

Options

A. `2lambda`
B. `(2lambda)/3`
C. `lambda/3`
D. `lambda`

Explanation

To solve the problem, we need to determine the minimum thickness \( t \) of the glass plate that, when introduced in the path of one of the beams in a double-slit experiment, keeps the intensity at the position of the central maximum unchanged.

### Step-by-Step Solution:

1. **Understanding the Setup**:
In a double-slit experiment, when a glass plate is introduced in one of the paths, it changes the optical path length of that beam due to its refractive index. This can shift the position of the interference pattern.

2. **Optical Path Length Change**:
The optical path length \( OP \) is given by the product of the refractive index \( \mu \) and the physical thickness \( t \) of the glass plate. The change in optical path length when the glass plate is introduced is:
\[ \Delta OP = (\mu - 1) t \]
where \( \mu \) is the refractive index of the glass plate.

3. **Condition for Unchanged Intensity**:
For the intensity at the central maximum to remain unchanged, the additional optical path length introduced by the glass plate must equal an integer multiple of the wavelength \( \lambda \):
\[ \Delta OP = n\lambda \]
where \( n \) is an integer (typically \( n = 1 \) for the minimum thickness).

4. **Substituting Values**:
Given that the refractive index \( \mu = 1.5 \), we can substitute this into the equation:
\[ (\mu - 1) t = n\lambda \]
This becomes:
\[ (1.5 - 1) t = n\lambda \]
Simplifying gives:
\[ 0.5 t = n\lambda \]

5. **Finding Minimum Thickness**:
For the minimum thickness, we can take \( n = 1 \):
\[ 0.5 t = \lambda \]
Rearranging gives:
\[ t = 2\lambda \]

6. **Conclusion**:
Thus, the minimum thickness \( t \) of the glass plate required to keep the intensity at the central maximum unchanged is:
\[ t = 2\lambda \]

### Final Answer:
The minimum thickness of the glass plate is \( t = 2\lambda \).

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