In YDSE, let A and B be two slits. Films of thickness `t_(A)` and `t_(B)` and refractive `mu_(A)` and `mu_(B)` are placed in front of A and B, respectively. If `mu_(A) t_(A) = mu_(A) t_(B)`, then the central maxima will

Options

A. not shift
B. shift toward A
C. shift toward B
D. (b) if `t_(B) lt t_(A)` and (C ) if `t_(B) lt t_(A)`

Explanation

To solve the problem, we need to analyze the situation in the Young's Double Slit Experiment (YDSE) with the introduction of films of different thicknesses and refractive indices in front of the slits.

### Step 1: Understand the Phase Change Due to the Films
In YDSE, when light passes through a medium with a refractive index, it experiences a phase change. The phase change (or optical path difference) introduced by a film of thickness \( t \) and refractive index \( \mu \) is given by:
\[ \Delta x = (\mu - 1) \cdot t \]
This means that the phase change depends on both the thickness of the film and its refractive index.

### Step 2: Calculate the Optical Path Difference
For slits A and B, the optical path differences due to the films can be expressed as:
- For slit A:
\[ \Delta x_A = (\mu_A - 1) \cdot t_A \]
- For slit B:
\[ \Delta x_B = (\mu_B - 1) \cdot t_B \]

### Step 3: Determine the Net Optical Path Difference
The net optical path difference (\( \Delta x_{net} \)) between the two slits is given by:
\[ \Delta x_{net} = \Delta x_A - \Delta x_B = (\mu_A - 1) \cdot t_A - (\mu_B - 1) \cdot t_B \]

### Step 4: Use the Given Condition
We are given that:
\[ \mu_A \cdot t_A = \mu_B \cdot t_B \]
From this, we can rearrange to find:
\[ \mu_A \cdot t_A - \mu_B \cdot t_B = 0 \]
This implies that:
\[ \Delta x_A = \Delta x_B \]
Thus, substituting this into our equation for \( \Delta x_{net} \):
\[ \Delta x_{net} = (\mu_A - 1) \cdot t_A - (\mu_B - 1) \cdot t_B \]

### Step 5: Analyze the Result
Since we know that \( \mu_A \cdot t_A = \mu_B \cdot t_B \), we can conclude that the contributions to the optical path difference from both slits are equal. Therefore:
\[ \Delta x_{net} = 0 \]
This means that the central maxima will not shift.

### Conclusion
The central maxima will **not shift**.