In Young's double slit experiment the two slits are d distance apart. Interference pattern is observed on a screen at a distance D from the slits. A dark fringe is observed on the screen directly opposite to one of the slits. The wavelength of light is

Options

A. `(D^(2))/(2d)`
B. `(d^(2))/(2D)`
C. `(D^(2))/(d)`
D. `(d^(2))/(D)`

Explanation

To solve the problem, we will use the principles of Young's double slit experiment and the conditions for dark fringes (minima) in the interference pattern.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two slits separated by a distance \( d \).
- The screen is located at a distance \( D \) from the slits.
- A dark fringe is observed directly opposite to one of the slits.

2. **Condition for Dark Fringe**:
- The condition for dark fringes (minima) in Young's double slit experiment is given by:
\[ d \sin \theta = (2n - 1) \frac{\lambda}{2} \]
where \( n \) is the order of the dark fringe (1, 2, 3,...).

3. **Small Angle Approximation**:
- For small angles, we can approximate \( \sin \theta \) as:
\[ \sin \theta \approx \tan \theta \approx \frac{y}{D} \]
where \( y \) is the distance from the central maximum to the dark fringe on the screen.

4. **Setting Up the Equation**:
- Substituting the small angle approximation into the dark fringe condition gives:
\[ d \cdot \frac{y}{D} = (2n - 1) \frac{\lambda}{2} \]
- Rearranging this, we find:
\[ y = \frac{(2n - 1) \lambda D}{2d} \]

5. **Finding the Position of the Dark Fringe**:
- Since a dark fringe is observed directly opposite one of the slits, we can set \( y = \frac{D}{2} \) (the position of the first dark fringe).
- Thus, we have:
\[ \frac{D}{2} = \frac{(2n - 1) \lambda D}{2d} \]

6. **Solving for Wavelength \( \lambda \)**:
- Canceling \( D \) from both sides (assuming \( D \neq 0 \)):
\[ 1 = \frac{(2n - 1) \lambda}{2d} \]
- Rearranging gives:
\[ \lambda = \frac{2d}{2n - 1} \]

7. **Considering the First Minimum**:
- For the first minimum, we set \( n = 1 \):
\[ \lambda = \frac{2d}{2 \cdot 1 - 1} = 2d \]

### Final Result:
The wavelength of light \( \lambda \) is:
\[ \lambda = \frac{2d}{1} = 2d \]