In Young.s double slit experiment, the 10th bright fringe is at a distance x from the central fringe. Then
a) the 10th dark fringe ia at a distance of `19x"/"20` from the central fringe
b) the 10th dark fringe is at a distance of `21x"/"20` from the central fringe.
c) the 5th dark fringe is at a distance of `x"/"2` from the central fringe. d) the 5th dark fringe is at a distance of `9x"/"20` from the central fringe.

Options

A. a, b, c only
B. b, c, d only
C. a, d only
D. a, b, c, d only

Answer

To solve the problem, we need to analyze the Young's double slit experiment and derive the positions of the bright and dark fringes based on the given information.

### Step-by-Step Solution:

1. **Understanding the Bright Fringe Position**:
- The position of the nth bright fringe in Young's double slit experiment is given by the formula:
\[ y_n = \frac{n \lambda D}{d} \]
- Here, \( n \) is the fringe order, \( \lambda \) is the wavelength, \( D \) is the distance from the slits to the screen, and \( d \) is the distance between the slits.
- For the 10th bright fringe (\( n = 10 \)):
\[ y_{10} = \frac{10 \lambda D}{d} \]
- According to the problem, this distance is given as \( x \):
\[ x = \frac{10 \lambda D}{d} \]

2. **Finding the Position of the 10th Dark Fringe**:
- The position of the nth dark fringe is given by the formula:
\[ y_n = \frac{(2n - 1) \lambda D}{2d} \]
- For the 10th dark fringe (\( n = 10 \)):
\[ y_{10} = \frac{(2 \cdot 10 - 1) \lambda D}{2d} = \frac{19 \lambda D}{2d} \]
- Now substituting \( \frac{10 \lambda D}{d} = x \):
\[ y_{10} = \frac{19}{2} \cdot \frac{\lambda D}{d} = \frac{19}{20} \cdot x \]
- Therefore, the position of the 10th dark fringe is:
\[ y_{10} = \frac{19x}{20} \]

3. **Finding the Position of the 5th Dark Fringe**:
- For the 5th dark fringe (\( n = 5 \)):
\[ y_5 = \frac{(2 \cdot 5 - 1) \lambda D}{2d} = \frac{9 \lambda D}{2d} \]
- Again substituting \( \frac{10 \lambda D}{d} = x \):
\[ y_5 = \frac{9}{2} \cdot \frac{\lambda D}{d} = \frac{9}{20} \cdot x \]
- Therefore, the position of the 5th dark fringe is:
\[ y_5 = \frac{9x}{20} \]

### Summary of Results:
- The 10th dark fringe is at a distance of \( \frac{19x}{20} \) from the central fringe (Option A).
- The 5th dark fringe is at a distance of \( \frac{9x}{20} \) from the central fringe (Option D).

### Conclusion:
- The correct options are:
- a) The 10th dark fringe is at a distance of \( \frac{19x}{20} \) from the central fringe.
- d) The 5th dark fringe is at a distance of \( \frac{9x}{20} \) from the central fringe.