Light waves of wavelength 5460 A, emitted by two coherent sources, meet at a point after travelling different paths. The path difference between the two wave trains at that point is `2.1 mum`. then phase difference will be ?

Options

A. `7.69pi`
B. `7pi`
C. `(pi)/(4)`
D. `(pi)/(2)`

Answer

To find the phase difference between two coherent light waves that have traveled different paths, we can use the formula that relates path difference to phase difference.

### Step-by-Step Solution:

1. **Identify the given values:**
- Wavelength of light, \( \lambda = 5460 \, \text{Å} = 5460 \times 10^{-10} \, \text{m} \)
- Path difference, \( \Delta x = 2.1 \, \mu m = 2.1 \times 10^{-6} \, \text{m} \)

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

3. **Substitute the values into the formula:**
\[ \Delta \phi = \frac{2\pi}{5460 \times 10^{-10}} \times (2.1 \times 10^{-6}) \]

4. **Calculate the wavelength in meters:**
\[ \lambda = 5460 \times 10^{-10} \, \text{m} = 5.46 \times 10^{-7} \, \text{m} \]

5. **Plug in the values:**
\[ \Delta \phi = \frac{2\pi}{5.46 \times 10^{-7}} \times (2.1 \times 10^{-6}) \]

6. **Perform the calculations:**
- First, calculate \( \frac{2\pi}{5.46 \times 10^{-7}} \):
\[ \frac{2\pi}{5.46 \times 10^{-7}} \approx 1.152 \times 10^{7} \, \text{rad/m} \]
- Now multiply by \( 2.1 \times 10^{-6} \):
\[ \Delta \phi \approx 1.152 \times 10^{7} \times 2.1 \times 10^{-6} \approx 24.192 \, \text{radians} \]

7. **Convert radians to a more manageable form:**
Since phase difference is often expressed in terms of multiples of \( 2\pi \):
\[ \Delta \phi = 24.192 \, \text{radians} \approx \frac{24.192}{2\pi} \approx 3.85 \, \text{cycles} \]
This means the phase difference is approximately \( 3.85 \times 2\pi \).

### Final Result:
The phase difference \( \Delta \phi \) is approximately \( 24.192 \, \text{radians} \).