The separation between the. consecutive dark fringes in a Young's double slit experiment is 1 mm. he is placed at a distance of 2.5 m from the slits screen s and separation between the slits is 1.0 mm. Calculate the he wavelength of light used for the experiment.

Explanation

To solve the problem, we will use the formula for the separation between consecutive dark fringes in a Young's double slit experiment. The formula is given by:

\[ \beta = \frac{\lambda D}{d} \]

where:
- \(\beta\) is the separation between consecutive dark fringes,
- \(\lambda\) is the wavelength of light,
- \(D\) is the distance from the slits to the screen,
- \(d\) is the separation between the slits.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Separation between dark fringes, \(\beta = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m}\)
- Distance from slits to screen, \(D = 2.5 \, \text{m}\)
- Separation between the slits, \(d = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m}\)

2. **Write the Formula:**
\[ \beta = \frac{\lambda D}{d} \]

3. **Rearrange the Formula to Solve for Wavelength (\(\lambda\)):**
\[ \lambda = \frac{\beta d}{D} \]

4. **Substitute the Known Values into the Formula:**
\[ \lambda = \frac{(1 \times 10^{-3} \, \text{m}) \times (1 \times 10^{-3} \, \text{m})}{2.5 \, \text{m}} \]

5. **Calculate the Wavelength:**
\[ \lambda = \frac{1 \times 10^{-6} \, \text{m}^2}{2.5 \, \text{m}} = 4 \times 10^{-7} \, \text{m} \]

6. **Convert Wavelength to Nanometers:**
\[ \lambda = 4 \times 10^{-7} \, \text{m} = 400 \, \text{nm} \]

### Final Answer:
The wavelength of light used in the experiment is \(400 \, \text{nm}\).