In Young's double slit experiment, the slits are `0.2 mm` apart and the screen is `1.5 m` away. It is observed that the distance between the central bright fringe and fourth dark fringe is `1.8 cm`. Calculate wavelength of light used.

Explanation

To solve the problem, we will use the formula for the position of dark fringes in Young's double slit experiment. The distance between the central bright fringe and the nth dark fringe is given by:

\[ y_n = \frac{(2n - 1) \lambda D}{2d} \]

where:
- \( y_n \) is the distance from the central maximum to the nth dark fringe,
- \( n \) is the order of the dark fringe (in this case, \( n = 4 \)),
- \( \lambda \) is the wavelength of light,
- \( D \) is the distance from the slits to the screen,
- \( d \) is the distance between the slits.

### Step-by-Step Solution:

1. **Identify the given values:**
- Distance between the slits, \( d = 0.2 \, \text{mm} = 0.2 \times 10^{-3} \, \text{m} = 2 \times 10^{-4} \, \text{m} \)
- Distance from the slits to the screen, \( D = 1.5 \, \text{m} \)
- Distance from the central bright fringe to the fourth dark fringe, \( y_4 = 1.8 \, \text{cm} = 1.8 \times 10^{-2} \, \text{m} \)

2. **Use the formula for the position of the nth dark fringe:**
- For the fourth dark fringe (\( n = 4 \)):
\[ y_4 = \frac{(2 \times 4 - 1) \lambda D}{2d} \]
This simplifies to:
\[ y_4 = \frac{7 \lambda D}{2d} \]

3. **Substitute the known values into the equation:**
\[ 1.8 \times 10^{-2} = \frac{7 \lambda (1.5)}{2(2 \times 10^{-4})} \]

4. **Rearranging the equation to solve for \( \lambda \):**
\[ 1.8 \times 10^{-2} = \frac{7 \lambda (1.5)}{4 \times 10^{-4}} \]
\[ 1.8 \times 10^{-2} \cdot 4 \times 10^{-4} = 7 \lambda (1.5) \]
\[ 7 \lambda (1.5) = 7.2 \times 10^{-6} \]
\[ \lambda = \frac{7.2 \times 10^{-6}}{10.5} \]

5. **Calculate \( \lambda \):**
\[ \lambda = \frac{7.2 \times 10^{-6}}{10.5} \approx 6.857 \times 10^{-7} \, \text{m} \]
\[ \lambda \approx 6.86 \times 10^{-7} \, \text{m} \text{ or } 686 \, \text{nm} \]

### Final Answer:
The wavelength of light used is approximately \( 6.86 \times 10^{-7} \, \text{m} \) or \( 686 \, \text{nm} \).