The path difference equivalent to a phase difference of `180^(@)` (given wavelength of wave = `lambda`) is

Options

A. Zero
B. `lambda//2`
C. `3lambda//4`
D. `lambda`

Answer

To find the path difference equivalent to a phase difference of \(180^\circ\) (or \(\pi\) radians), we can use the relationship between phase difference and path difference. Here’s the step-by-step solution:

### Step 1: Understand the relationship between phase difference and path difference
The phase difference (\(\Delta \phi\)) is related to the path difference (\(\Delta x\)) by the formula:
\[ \Delta \phi = \frac{2\pi}{\lambda} \Delta x \]
where \(\lambda\) is the wavelength of the wave.

### Step 2: Set the phase difference to \(180^\circ\)
We want to find the path difference that corresponds to a phase difference of \(180^\circ\). We convert \(180^\circ\) to radians:
\[ 180^\circ = \pi \text{ radians} \]

### Step 3: Substitute the phase difference into the formula
Now, we substitute \(\Delta \phi = \pi\) into the relationship:
\[ \pi = \frac{2\pi}{\lambda} \Delta x \]

### Step 4: Solve for the path difference (\(\Delta x\))
To isolate \(\Delta x\), we can rearrange the equation:
\[ \Delta x = \frac{\pi \lambda}{2\pi} \]
This simplifies to:
\[ \Delta x = \frac{\lambda}{2} \]

### Conclusion
Thus, the path difference equivalent to a phase difference of \(180^\circ\) is:
\[ \Delta x = \frac{\lambda}{2} \]