For destructive interference to take place between two monochronic light waves of wavelength the path difference should be

Options

A. `((2n-1)lambda)/2`
B. `2nlambda`
C. `((2n+1)lambda)/4`
D. `nlambda`

Answer

To determine the path difference required for destructive interference between two monochromatic light waves, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Destructive Interference**:
- Destructive interference occurs when two waves are out of phase, resulting in a reduced amplitude or cancellation of the waves.

2. **Amplitude of Resulting Wave**:
- The amplitude of the resulting wave from two interfering waves can be expressed as:
\[ A_{net}^2 = A_1^2 + A_2^2 + 2A_1A_2 \cos(\phi) \]
- Here, \(A_1\) and \(A_2\) are the amplitudes of the two waves, and \(\phi\) is the phase difference between them.

3. **Condition for Minimum Amplitude**:
- For destructive interference, the resulting amplitude should be minimum. This occurs when \(\cos(\phi)\) is at its minimum value, which is \(-1\).

4. **Finding Phase Difference**:
- Setting \(\cos(\phi) = -1\), we find:
\[ \phi = (2n - 1) \pi \]
- Here, \(n\) is an integer (0, 1, 2, ...).

5. **Relating Phase Difference to Path Difference**:
- The phase difference \(\phi\) is also related to the path difference \(\Delta x\) by the equation:
\[ \phi = k \Delta x \]
- Where \(k\) is the wave number given by:
\[ k = \frac{2\pi}{\lambda} \]
- Substituting for \(k\):
\[ \phi = \frac{2\pi}{\lambda} \Delta x \]

6. **Equating the Two Expressions for Phase Difference**:
- We set the two expressions for \(\phi\) equal to each other:
\[ \frac{2\pi}{\lambda} \Delta x = (2n - 1) \pi \]

7. **Solving for Path Difference**:
- Dividing both sides by \(\pi\):
\[ \frac{2}{\lambda} \Delta x = 2n - 1 \]
- Rearranging gives:
\[ \Delta x = \frac{(2n - 1) \lambda}{2} \]

8. **Conclusion**:
- The path difference for destructive interference is given by:
\[ \Delta x = \frac{(2n - 1) \lambda}{2} \]

### Final Answer:
For destructive interference to take place between two monochromatic light waves of wavelength \(\lambda\), the path difference should be:
\[ \Delta x = \frac{(2n - 1) \lambda}{2} \]