Two coherent sources are 0.3 mm apart. They are 0.9m away from the screen. The second dark fringe is at a distance of 0.3cm from the centre. Find the distance of fourth bright fringe from the centre. Also, find the wavelength of light used.

Explanation

To solve the given problem step by step, we will first find the distance of the fourth bright fringe from the center and then calculate the wavelength of light used.

### Step 1: Identify the given data
- Distance between the two coherent sources (d) = 0.3 mm = 0.3 × 10^(-3) m
- Distance from the coherent sources to the screen (D) = 0.9 m
- Distance of the second dark fringe from the center (y_2) = 0.3 cm = 0.3 × 10^(-2) m

### Step 2: Use the formula for the position of dark fringes
The position of the nth dark fringe in a double-slit interference pattern is given by:
\[ y_n = \frac{(2n - 1) \lambda D}{2d} \]
For the second dark fringe (n = 2):
\[ y_2 = \frac{(2 \cdot 2 - 1) \lambda D}{2d} = \frac{3 \lambda D}{2d} \]
Setting \(y_2 = 0.3 \times 10^{-2}\) m, we have:
\[ 0.3 \times 10^{-2} = \frac{3 \lambda (0.9)}{2(0.3 \times 10^{-3})} \]

### Step 3: Solve for the wavelength (λ)
Rearranging the equation:
\[ \lambda = \frac{0.3 \times 10^{-2} \cdot 2(0.3 \times 10^{-3})}{3 \cdot 0.9} \]
Calculating the right-hand side:
\[ \lambda = \frac{0.3 \times 10^{-2} \cdot 0.6 \times 10^{-3}}{2.7} = \frac{0.18 \times 10^{-5}}{2.7} \approx 6.67 \times 10^{-7} \text{ m} \]

### Step 4: Find the distance of the fourth bright fringe from the center
The position of the nth bright fringe is given by:
\[ y_n = \frac{n \lambda D}{d} \]
For the fourth bright fringe (n = 4):
\[ y_4 = \frac{4 \lambda D}{d} \]
Substituting the values of λ, D, and d:
\[ y_4 = \frac{4 \cdot (6.67 \times 10^{-7}) \cdot 0.9}{0.3 \times 10^{-3}} \]
Calculating:
\[ y_4 = \frac{4 \cdot 6.67 \cdot 0.9 \times 10^{-7}}{0.3 \times 10^{-3}} = \frac{24.0 \times 10^{-7}}{0.3 \times 10^{-3}} = 0.8 \times 10^{-2} \text{ m} = 0.8 \text{ cm} \]

### Final Answers:
- Distance of the fourth bright fringe from the center = **0.8 cm**
- Wavelength of light used = **6.67 × 10^(-7) m**