Two light waves having the same wavelength `lambda` in vacuum are in phase initially. Then the first ray travels a path of length `L_(1)` through a medium of refractive index `mu_(1)`. Then second ray travels a path of length `L_(2)` throug a medium of refractive index `mu_(2)`. The two waves are then combined to observed interference effects. The phase difference between the two, when they interfere, is

Options

A. `2pi/lambda*(L_2-L_1)`
B. `2pi/lambda*(n_1L_1-n_2L_2)`
C. `2pi/lambda*(n_2L_1-n_1L_2)`
D. `2pi/lambda*(L_1/n_1-L_2/n_2)`

Answer

To find the phase difference between two light waves after they have traveled through different media, we can follow these steps:

### Step 1: Understand the Optical Path Length
The optical path length (OPL) for a light wave traveling through a medium is given by the product of the refractive index of the medium and the physical path length traveled by the light in that medium.

For the first wave traveling through a medium with refractive index \( \mu_1 \) and path length \( L_1 \):
\[ \text{OPL}_1 = \mu_1 L_1 \]

For the second wave traveling through a medium with refractive index \( \mu_2 \) and path length \( L_2 \):
\[ \text{OPL}_2 = \mu_2 L_2 \]

### Step 2: Calculate the Path Difference
The path difference \( \Delta x \) between the two waves when they are combined is given by the difference in their optical path lengths:
\[ \Delta x = \text{OPL}_1 - \text{OPL}_2 = \mu_1 L_1 - \mu_2 L_2 \]

### Step 3: Relate Path Difference to Phase Difference
The phase difference \( \Delta \phi \) corresponding to a path difference can be calculated using the formula:
\[ \Delta \phi = \frac{2\pi}{\lambda} \Delta x \]
Substituting the expression for path difference:
\[ \Delta \phi = \frac{2\pi}{\lambda} (\mu_1 L_1 - \mu_2 L_2) \]

### Final Result
Thus, the phase difference between the two waves when they interfere is given by:
\[ \Delta \phi = \frac{2\pi}{\lambda} (\mu_1 L_1 - \mu_2 L_2) \]