Statement I: In Young's double-slit experiment, the two slits are at distance d apart. Interference pattern is observed on a screen at distance D from the slits. At a point on the screen which is directly opposite to one of the slits, a dark fringe is observed. Then, the wavelength of wave is proportional to the squar of distance between two slits.
Statement II: For a dark fringe, intensity is zero

Options

A. Statement I is True, statement II is True, Statement II is a correct explanation for Statement I.
B. Statement I is Ture, Statement II is Ture, Statement II is NOT a correct explanation for Statement I.
C. Statement I is True, Statement II is False.
D. Statement I is False, Statement II is True.

Explanation

To solve the question, we need to analyze both statements provided and determine their validity and relationship.

### Step-by-Step Solution:

1. **Understanding Young's Double-Slit Experiment**:
- In Young's double-slit experiment, two slits are separated by a distance \( d \). When coherent light passes through these slits, an interference pattern of bright and dark fringes is formed on a screen placed at a distance \( D \) from the slits.

2. **Condition for Dark Fringe**:
- A dark fringe occurs at a point on the screen when the path difference between the light waves coming from the two slits is an odd multiple of half the wavelength (\( \lambda \)). This can be mathematically expressed as:
\[ \Delta x = \frac{(2n + 1)\lambda}{2} \]
- Here, \( n \) is an integer (0, 1, 2, ...).

3. **Position of Dark Fringe**:
- The position \( Y \) of the dark fringe on the screen can be calculated using the formula:
\[ Y = \frac{D \cdot (2n + 1) \cdot \lambda}{2d} \]
- For the first dark fringe (\( n = 0 \)), this simplifies to:
\[ Y = \frac{D \cdot \lambda}{2d} \]

4. **Analyzing Statement I**:
- The first statement claims that if a dark fringe is observed directly opposite one of the slits, then the wavelength \( \lambda \) is proportional to the square of the distance between the two slits.
- From the formula derived, we can rearrange it to show:
\[ \lambda = \frac{2dY}{D} \]
- This indicates that the wavelength \( \lambda \) is indeed proportional to the distance \( d \) between the slits, but not to the square of \( d \). Therefore, **Statement I is false**.

5. **Analyzing Statement II**:
- The second statement asserts that for a dark fringe, the intensity is zero. This is true because at the dark fringe, the waves from the two slits interfere destructively, resulting in zero intensity.
- Thus, **Statement II is true**.

6. **Conclusion**:
- Statement I is false, and Statement II is true. Therefore, the correct option is that Statement I is false and Statement II is true.

### Final Answer:
- **Statement I is false, and Statement II is true.**