Ratio waves originating from sources `S_(1)` and `S_(2)` having zero phase difference and common wavelength `lambda` will show completely destructive interference at a point `P` is `S_(1)P-S_(2)P` is

Options

A. `5 lambda`
B. `2 lambda`
C. `(3lambda)/4`
D. `(11lambda)/2`

Answer

To solve the problem, we need to determine the path difference between the two waves originating from sources \( S_1 \) and \( S_2 \) at point \( P \) that results in completely destructive interference.

### Step-by-Step Solution:

1. **Understanding Destructive Interference**:
- Destructive interference occurs when two waves meet in such a way that their crests and troughs cancel each other out. This happens when the phase difference between the two waves is an odd multiple of \( \pi \) (180 degrees).
- Mathematically, for two waves to interfere destructively, the path difference must be given by:
\[ \Delta d = S_1P - S_2P = \left( n + \frac{1}{2} \right) \lambda \]
where \( n \) is an integer (0, 1, 2, ...).

2. **Common Wavelength**:
- Given that both sources have a common wavelength \( \lambda \), we can use this in our calculations.

3. **Condition for Complete Destructive Interference**:
- For complete destructive interference, we need the path difference to be equal to an odd multiple of half the wavelength:
\[ S_1P - S_2P = \left( n + \frac{1}{2} \right) \lambda \]
- The simplest case (for \( n = 0 \)) gives:
\[ S_1P - S_2P = \frac{1}{2} \lambda \]

4. **Conclusion**:
- Therefore, the path difference \( S_1P - S_2P \) must be equal to \( \frac{1}{2} \lambda \) for the waves from sources \( S_1 \) and \( S_2 \) to show completely destructive interference at point \( P \).

### Final Answer:
The path difference \( S_1P - S_2P \) for completely destructive interference is:
\[ S_1P - S_2P = \frac{1}{2} \lambda \]