In Young's double-slit experiment, the slit are 0.5 mm apart and the interference is observed on a screen at a distance of 100 cm from the slits, It is found that the ninth bright fringe is at a distance of 7.5 mm from the second dark fringe from the center of the fringe pattern. The wavelength of the light used is

Options

A. `5000 Å`
B. `(5000)/(7) Å`
C. `2500 Å`
D. `(2500)/(7) Å`

Explanation

To solve the problem, we will use the formula for the positions of bright and dark fringes in Young's double-slit experiment.

### Step-by-Step Solution:

1. **Identify the given data:**
- Distance between the slits, \( d = 0.5 \, \text{mm} = 0.5 \times 10^{-3} \, \text{m} \)
- Distance from the slits to the screen, \( D = 100 \, \text{cm} = 1 \, \text{m} \)
- Distance between the 9th bright fringe and the 2nd dark fringe, \( y = 7.5 \, \text{mm} = 7.5 \times 10^{-3} \, \text{m} \)

2. **Understand the positions of the fringes:**
- The position of the \( n \)-th bright fringe is given by:
\[ y_n = \frac{n \lambda D}{d} \]
- The position of the \( m \)-th dark fringe is given by:
\[ y_m = \frac{(m + \frac{1}{2}) \lambda D}{d} \]

3. **Set up the equation for the distance between the 9th bright fringe and the 2nd dark fringe:**
- Let \( n = 9 \) (for the 9th bright fringe) and \( m = 2 \) (for the 2nd dark fringe).
- The position of the 9th bright fringe:
\[ y_9 = \frac{9 \lambda D}{d} \]
- The position of the 2nd dark fringe:
\[ y_2 = \frac{(2 + \frac{1}{2}) \lambda D}{d} = \frac{(2.5) \lambda D}{d} \]

4. **Calculate the distance between the two fringes:**
- The distance \( y \) between the 9th bright fringe and the 2nd dark fringe is:
\[ y = y_9 - y_2 = \frac{9 \lambda D}{d} - \frac{2.5 \lambda D}{d} \]
- Simplifying this gives:
\[ y = \frac{(9 - 2.5) \lambda D}{d} = \frac{6.5 \lambda D}{d} \]

5. **Substituting the known values:**
- We know \( y = 7.5 \times 10^{-3} \, \text{m} \), \( D = 1 \, \text{m} \), and \( d = 0.5 \times 10^{-3} \, \text{m} \).
- Therefore:
\[ 7.5 \times 10^{-3} = \frac{6.5 \lambda (1)}{0.5 \times 10^{-3}} \]
- Rearranging gives:
\[ \lambda = \frac{7.5 \times 10^{-3} \times 0.5 \times 10^{-3}}{6.5} \]

6. **Calculating \( \lambda \):**
- Calculate \( \lambda \):
\[ \lambda = \frac{7.5 \times 0.5}{6.5} \times 10^{-6} = \frac{3.75}{6.5} \times 10^{-6} \approx 0.5769 \times 10^{-6} \, \text{m} = 576.9 \, \text{nm} \]

### Final Answer:
The wavelength of the light used is approximately \( 577 \, \text{nm} \).