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

Answer

To solve the problem, we need to find the minimum thickness \( t \) of a glass plate with a refractive index \( \mu = 1.5 \) such that the intensity at the position of the central maximum in a double-slit experiment remains unchanged after the introduction of the glass plate.

### Step-by-Step Solution:

1. **Understanding the Shift Due to the Glass Plate**:
When a glass plate is introduced in one of the paths of the interfering beams, it causes a phase shift. This phase shift can be quantified as an extra path length, denoted as \( \Delta x \).

2. **Formula for Extra Shift**:
The extra shift \( \Delta x \) due to the glass plate can be expressed as:
\[ \Delta x = (\mu - 1) t \]
where \( \mu \) is the refractive index of the glass plate and \( t \) is its thickness.

3. **Condition for Unchanged Intensity**:
For the intensity at the position of the central maximum to remain unchanged, the extra shift must be an integer multiple of the wavelength \( \lambda \):
\[ \Delta x = n \lambda \]
where \( n \) is an integer.

4. **Setting Up the Equation**:
By substituting the expression for \( \Delta x \) into the condition for unchanged intensity, we get:
\[ (\mu - 1) t = n \lambda \]

5. **Finding Minimum Thickness**:
To find the minimum thickness \( t \), we set \( n = 1 \) (the smallest integer value):
\[ (\mu - 1) t = \lambda \]
Rearranging this gives:
\[ t = \frac{\lambda}{\mu - 1} \]

6. **Substituting the Values**:
Now, substituting \( \mu = 1.5 \):
\[ t = \frac{\lambda}{1.5 - 1} = \frac{\lambda}{0.5} = 2\lambda \]

7. **Final Result**:
Therefore, the minimum thickness \( t \) of the glass plate is:
\[ t = 2\lambda \]

### Conclusion:
The minimum thickness of the glass plate is \( 2\lambda \).