In Young's experiment, what will be the phase difference and the path difference between the light waves reaching (i) third bright fringe and (ii) third dark fringe from the central fringe. Take `lambda = 5000 Å`.

Explanation

To solve the problem, we need to find the phase difference and path difference for both the third bright fringe and the third dark fringe in Young's double-slit experiment. Given the wavelength \( \lambda = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m} \).

### Step 1: Calculate the Path Difference for the Third Bright Fringe

The path difference \( \Delta x \) for the \( n \)-th bright fringe is given by:
\[ \Delta x = n \lambda \]
For the third bright fringe (\( n = 3 \)):
\[ \Delta x = 3 \lambda = 3 \times (5000 \times 10^{-10}) \, \text{m} \]
\[ \Delta x = 1.5 \times 10^{-6} \, \text{m} \]

### Step 2: Calculate the Phase Difference for the Third Bright Fringe

The phase difference \( \Delta \phi \) is related to the path difference by the formula:
\[ \Delta \phi = \frac{2\pi}{\lambda} \Delta x \]
Substituting \( \Delta x \) for the third bright fringe:
\[ \Delta \phi = \frac{2\pi}{\lambda} (3 \lambda) = 6\pi \, \text{radians} \]

### Step 3: Calculate the Path Difference for the Third Dark Fringe

The path difference for the \( n \)-th dark fringe is given by:
\[ \Delta x = \left( n - \frac{1}{2} \right) \lambda \]
For the third dark fringe (\( n = 3 \)):
\[ \Delta x = \left( 3 - \frac{1}{2} \right) \lambda = \frac{5}{2} \lambda \]
Calculating this:
\[ \Delta x = \frac{5}{2} \times (5000 \times 10^{-10}) \, \text{m} = 1.25 \times 10^{-6} \, \text{m} \]

### Step 4: Calculate the Phase Difference for the Third Dark Fringe

Using the same formula for phase difference:
\[ \Delta \phi = \frac{2\pi}{\lambda} \Delta x \]
Substituting \( \Delta x \) for the third dark fringe:
\[ \Delta \phi = \frac{2\pi}{\lambda} \left( \frac{5}{2} \lambda \right) = 5\pi \, \text{radians} \]

### Summary of Results

- For the third bright fringe:
- Path Difference: \( 1.5 \times 10^{-6} \, \text{m} \)
- Phase Difference: \( 6\pi \, \text{radians} \)

- For the third dark fringe:
- Path Difference: \( 1.25 \times 10^{-6} \, \text{m} \)
- Phase Difference: \( 5\pi \, \text{radians} \)