Among two interfering sources, let `S_1` be ahead of the phase by `90^@` relative to `S_2` . If an observation point P is such that `PS_1 - PS_2 = 1.5 lambda` , the phase difference between the waves from `S_1 and S_2` reaching P is

Options

A. `(a)3pi`
B. `(b)(5pi)/2`
C. `(c)(7pi)/2`
D. `(d)4pi`

Explanation

To solve the problem, we need to determine the phase difference between the waves from two sources \( S_1 \) and \( S_2 \) at an observation point \( P \), given that \( S_1 \) is ahead in phase by \( 90^\circ \) (or \( \frac{\pi}{2} \) radians) relative to \( S_2 \) and that the path difference \( PS_1 - PS_2 = 1.5 \lambda \).

### Step-by-Step Solution:

1. **Identify the phase of the sources:**
- The phase of source \( S_1 \) is given as \( \phi_1 = \frac{\pi}{2} \) (90 degrees).
- The phase of source \( S_2 \) is \( \phi_2 \), which we need to calculate based on the path difference.

2. **Calculate the phase difference due to path difference:**
- The phase difference \( \Delta \phi \) due to the path difference \( \Delta x = PS_1 - PS_2 \) can be calculated using the formula:
\[ \Delta \phi = \frac{2\pi}{\lambda} \Delta x \]
- Substituting \( \Delta x = 1.5 \lambda \):
\[ \Delta \phi = \frac{2\pi}{\lambda} \times 1.5 \lambda = 3\pi \]

3. **Determine the total phase of \( S_2 \):**
- Since \( S_1 \) is ahead by \( \frac{\pi}{2} \), the phase of \( S_2 \) can be expressed as:
\[ \phi_2 = \Delta \phi + \phi_1 = 3\pi + \frac{\pi}{2} \]

4. **Combine the phases:**
- To combine the phases, convert \( 3\pi \) into a fraction with a common denominator:
\[ 3\pi = \frac{6\pi}{2} \]
- Now, adding \( \phi_1 \):
\[ \phi_2 = \frac{6\pi}{2} + \frac{\pi}{2} = \frac{7\pi}{2} \]

5. **Calculate the net phase difference:**
- The net phase difference \( \Delta \Phi \) between the waves from \( S_1 \) and \( S_2 \) at point \( P \) is:
\[ \Delta \Phi = \phi_2 - \phi_1 = \frac{7\pi}{2} - \frac{\pi}{2} = \frac{6\pi}{2} = 3\pi \]

6. **Final Result:**
- The phase difference between the waves from \( S_1 \) and \( S_2 \) reaching point \( P \) is \( 3\pi \).

### Conclusion:
The phase difference between the waves from \( S_1 \) and \( S_2 \) reaching point \( P \) is \( 3\pi \).