In YDSE , ten maxima of two pattern (Interference ) is formed in central maxima of single slit pattern (diffraction ) If separtion between two slits is 1mm, then what is the width of each slit ?

Options

A. `0.2` mm
B. `0.4` mm
C. `0.1 `mm
D. `0.02` mm

Answer

To solve the problem, we need to find the width of each slit in a Young's Double Slit Experiment (YDSE) where ten maxima of the interference pattern are formed within the central maximum of a single slit diffraction pattern. The separation between the two slits is given as 1 mm.

### Step-by-Step Solution:

1. **Identify Given Data:**
- Separation between the two slits (d) = 1 mm = 1 × 10^-3 m
- Number of maxima (n) = 10

2. **Formula for Linear Separation of Maxima:**
The linear position of the n-th maximum in the interference pattern is given by:
\[ x = \frac{n \lambda D}{d} \]
where:
- \( \lambda \) = wavelength of light,
- \( D \) = distance from the slits to the screen.

3. **Angular Separation of Maxima:**
The angular separation corresponding to the n-th maximum is:
\[ \theta_1 = \frac{x}{D} = \frac{n \lambda}{d} \]
For the 10th maximum, we have:
\[ \theta_{10} = \frac{10 \lambda}{d} \]

4. **Angular Width of Central Maximum in Single Slit Diffraction:**
The angular width of the central maximum in the single slit diffraction pattern is given by:
\[ \theta_2 = \frac{2 \lambda}{D_1} \]
where \( D_1 \) is the width of each slit.

5. **Equating Angular Separations:**
Since the 10 maxima fit within the central maximum of the single slit pattern, we can equate the two angles:
\[ \theta_1 = \theta_2 \]
Therefore,
\[ \frac{10 \lambda}{d} = \frac{2 \lambda}{D_1} \]

6. **Solving for Width of Each Slit (D1):**
Rearranging the equation gives:
\[ 10 \lambda D_1 = 2 \lambda d \]
Dividing both sides by \( \lambda \) (assuming \( \lambda \neq 0 \)):
\[ 10 D_1 = 2 d \]
Thus,
\[ D_1 = \frac{2 d}{10} = \frac{d}{5} \]

7. **Substituting the Value of d:**
Now substituting \( d = 1 \text{ mm} = 1 \times 10^{-3} \text{ m} \):
\[ D_1 = \frac{1 \times 10^{-3}}{5} = 0.2 \times 10^{-3} \text{ m} = 0.2 \text{ mm} \]

8. **Final Answer:**
The width of each slit \( D_1 \) is 0.2 mm.

### Conclusion:
The width of each slit is **0.2 mm**.