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 in nm is

Options

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

Answer

To solve the problem, we need to find the wavelength of light used in Young's double-slit experiment based on the given parameters.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Distance between slits (b) = 0.5 mm = \(0.5 \times 10^{-3}\) m
- Distance from slits to screen (D) = 100 cm = 1 m
- Distance from the center to the ninth bright fringe (y) = Distance from the center to the second dark fringe (x) + 7.5 mm
- Distance from the center to the second dark fringe (x) = 2nd dark fringe position

2. **Understanding Fringe Positions:**
- The position of the nth bright fringe is given by:
\[ y_n = \frac{n \lambda D}{b} \]
- The position of the mth dark fringe is given by:
\[ y_m = \frac{(m + 0.5) \lambda D}{b} \]
- For the second dark fringe (m=2):
\[ y_2 = \frac{(2 + 0.5) \lambda D}{b} = \frac{2.5 \lambda D}{b} \]

3. **Setting Up the Equation:**
- We know that the distance from the ninth bright fringe to the second dark fringe is given as 7.5 mm:
\[ y_9 - y_2 = 7.5 \text{ mm} = 7.5 \times 10^{-3} \text{ m} \]
- Substitute the expressions for \(y_9\) and \(y_2\):
\[ \frac{9 \lambda D}{b} - \frac{2.5 \lambda D}{b} = 7.5 \times 10^{-3} \]
- Simplifying this gives:
\[ \frac{(9 - 2.5) \lambda D}{b} = 7.5 \times 10^{-3} \]
\[ \frac{6.5 \lambda D}{b} = 7.5 \times 10^{-3} \]

4. **Solving for Wavelength (λ):**
- Rearranging the equation:
\[ \lambda = \frac{7.5 \times 10^{-3} \cdot b}{6.5 \cdot D} \]
- Substitute the values of \(b\) and \(D\):
\[ \lambda = \frac{7.5 \times 10^{-3} \cdot (0.5 \times 10^{-3})}{6.5 \cdot 1} \]
\[ \lambda = \frac{3.75 \times 10^{-6}}{6.5} \]
\[ \lambda \approx 5.769 \times 10^{-7} \text{ m} \]

5. **Convert to Nanometers:**
- To convert meters to nanometers:
\[ \lambda \approx 5.769 \times 10^{-7} \text{ m} = 576.9 \text{ nm} \]

### Final Answer:
The wavelength of the light used is approximately **577 nm**.