Which of the following is the path difference for destructive interference ?

Options

A. `n(lambda+1)`
B. `(2n+1)(lambda)/(2)`
C. `n lambda`
D. `(n+1)(lambda)/(2)`

Explanation

To determine the path difference for destructive interference in wave optics, we need to understand the conditions that lead to constructive and destructive interference.

### Step-by-Step Solution:

1. **Understanding Interference**:
- Interference occurs when two waves meet. Depending on their phase relationship, they can interfere constructively (amplifying each other) or destructively (cancelling each other).

2. **Path Difference for Constructive Interference**:
- For constructive interference, the path difference (Δx) must be an even multiple of the wavelength (λ). This can be expressed mathematically as:
\[ \Delta x = n\lambda \quad (n = 0, 1, 2, \ldots) \]

3. **Path Difference for Destructive Interference**:
- For destructive interference, the path difference must be an odd multiple of half the wavelength (λ/2). This can be expressed as:
\[ \Delta x = (2n - 1)\frac{\lambda}{2} \quad (n = 1, 2, 3, \ldots) \]
- This means that the possible values for the path difference for destructive interference are:
- λ/2 (for n=1)
- 3λ/2 (for n=2)
- 5λ/2 (for n=3)
- and so on.

4. **Identifying the Correct Option**:
- Given the options, we need to identify which one corresponds to the above conditions for destructive interference. The correct answer will be any of the odd multiples of λ/2.

5. **Conclusion**:
- After analyzing the options, we conclude that the correct path difference for destructive interference is represented by the expression \( \Delta x = (2n - 1)\frac{\lambda}{2} \).