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 light waves that have traveled different paths, we can use the formula:

\[ \text{Phase Difference} (\Delta \phi) = \frac{2\pi}{\lambda} \times \text{Path Difference} (\Delta x) \]

Where:
- \(\lambda\) is the wavelength of the light,
- \(\Delta x\) is the path difference.

### Step 1: Convert Wavelength to Meters
The given wavelength is \(5460 \, \text{Å}\) (angstroms). We need to convert this to meters:

\[ \lambda = 5460 \, \text{Å} = 5460 \times 10^{-10} \, \text{m} = 5.46 \times 10^{-7} \, \text{m} \]

### Step 2: Convert Path Difference to Meters
The path difference is given as \(2.1 \, \mu m\) (micrometers). We need to convert this to meters:

\[ \Delta x = 2.1 \, \mu m = 2.1 \times 10^{-6} \, \text{m} \]

### Step 3: Substitute Values into the Phase Difference Formula
Now we can substitute the values into the phase difference formula:

\[ \Delta \phi = \frac{2\pi}{\lambda} \times \Delta x \]

Substituting the values we calculated:

\[ \Delta \phi = \frac{2\pi}{5.46 \times 10^{-7}} \times 2.1 \times 10^{-6} \]

### Step 4: Simplify the Expression
Calculating the fraction:

\[ \Delta \phi = \frac{2\pi \times 2.1 \times 10^{-6}}{5.46 \times 10^{-7}} \]

Calculating the denominator:

\[ \Delta \phi = \frac{2\pi \times 2.1}{5.46} \times 10^{1} \quad (\text{since } 10^{-6} / 10^{-7} = 10^{1}) \]

### Step 5: Calculate the Numerical Value
Now, calculate the numerical value:

\[ \Delta \phi \approx \frac{2 \times 3.14 \times 2.1}{5.46} \times 10 \]

Calculating \(2 \times 3.14 \times 2.1 \approx 13.188\):

\[ \Delta \phi \approx \frac{13.188}{5.46} \times 10 \approx 2.41 \times 10 \approx 7.69 \, \text{radians} \]

### Final Result
Thus, the phase difference is approximately:

\[ \Delta \phi \approx 7.69 \, \text{radians} \]