In the double-slit experiment, the distance of the second dark fringe from the central line are `3 mm`. The distance of the fourth bright fringe from the central line is

Options

A. `6 mm`
B. `8 mm`
C. `12 mm`
D. `16 mm`

Explanation

To solve the problem, we need to find the distance of the fourth bright fringe from the central line in a double-slit experiment, given that the distance of the second dark fringe from the central line is 3 mm.

### Step-by-Step Solution:

1. **Understand the relationship for dark fringes**:
The formula for the position of the nth dark fringe in a double-slit experiment is given by:
\[ x_n = \left(2n + 1\right) \frac{d \lambda}{2d} \]
where \( n \) is the order of the dark fringe, \( d \) is the distance between the slits, \( \lambda \) is the wavelength of light, and \( D \) is the distance from the slits to the screen.

2. **Identify the parameters for the second dark fringe**:
For the second dark fringe, \( n = 2 \). We know from the problem that:
\[ x_2 = 3 \text{ mm} \]
Therefore, substituting \( n = 2 \) into the formula:
\[ x_2 = (2 \cdot 2 + 1) \frac{d \lambda}{2d} = 5 \frac{d \lambda}{2d} \]
Setting this equal to 3 mm:
\[ 5 \frac{d \lambda}{2d} = 3 \text{ mm} \]

3. **Solve for \( \frac{d \lambda}{d} \)**:
Rearranging the equation gives:
\[ \frac{d \lambda}{d} = \frac{3 \text{ mm} \cdot 2}{5} = \frac{6}{5} \text{ mm} = 1.2 \text{ mm} \]

4. **Use the formula for the fourth bright fringe**:
The formula for the position of the nth bright fringe is:
\[ x_n = n \frac{d \lambda}{d} \]
For the fourth bright fringe, \( n = 4 \):
\[ x_4 = 4 \frac{d \lambda}{d} = 4 \cdot 1.2 \text{ mm} = 4.8 \text{ mm} \]

5. **Final Calculation**:
Therefore, the distance of the fourth bright fringe from the central line is:
\[ x_4 = 4.8 \text{ mm} \]

### Final Answer:
The distance of the fourth bright fringe from the central line is **4.8 mm**.