The path difference between two wavefronts emitted by coherent sources of wavelength 5460 Å is 2.1 micron . The phase difference between the wavefronts at that point is –

Options

A. `7.692`
B. `7.692 pi`
C. `(7.692)/(pi)`
D. `(7.692)/(3pi)`

Answer

To find the phase difference between two wavefronts emitted by coherent sources, we can use the relationship between path difference and phase difference.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Wavelength (\( \lambda \)) = 5460 Å = \( 5460 \times 10^{-10} \) meters = \( 5.46 \times 10^{-7} \) meters.
- Path difference (\( \Delta x \)) = 2.1 microns = \( 2.1 \times 10^{-6} \) meters.

2. **Use the Formula for Phase Difference:**
The phase difference (\( \phi \)) can be calculated using the formula:
\[ \phi = \frac{2\pi}{\lambda} \Delta x \]

3. **Substitute the Values into the Formula:**
\[ \phi = \frac{2\pi}{5.46 \times 10^{-7}} \times (2.1 \times 10^{-6}) \]

4. **Calculate the Value:**
First, calculate \( \frac{2\pi}{5.46 \times 10^{-7}} \):
\[ \frac{2\pi}{5.46 \times 10^{-7}} \approx 1.152 \times 10^{7} \text{ radians/meter} \]

Now, multiply this by the path difference:
\[ \phi \approx 1.152 \times 10^{7} \times 2.1 \times 10^{-6} \approx 24.192 \text{ radians} \]

5. **Convert Radians to a Multiple of \( \pi \):**
To express the phase difference in terms of \( \pi \):
\[ \phi \approx \frac{24.192}{\pi} \approx 7.692 \pi \]

### Final Answer:
The phase difference between the wavefronts at that point is approximately \( 7.692 \pi \) radians.

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