In a double slit experiment, the two slits are `1mm` apart and the screen is placed `1m` away. A monochromatic light of wavelength `500nm` is used. What will be the width of each slit for obtaining ten maxima of double slit within the central maxima of single-slit pattern?

Options

A. (a) `0.2mm`
B. (b) `0.1mm`
C. (c) `0.5mm`
D. (d) `0.02mm`

Explanation

To solve the problem, we need to determine the width of each slit (denoted as \( a \)) in a double slit experiment such that there are 10 maxima of the double slit pattern within the central maximum of the single slit pattern.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Distance between the slits, \( d = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m} \)
- Distance from the slits to the screen, \( D = 1 \, \text{m} \)
- Wavelength of light, \( \lambda = 500 \, \text{nm} = 500 \times 10^{-9} \, \text{m} \)

2. **Calculate the Width of the Central Maximum in Single Slit:**
The width of the central maximum in a single slit pattern is given by:
\[ \beta = \frac{2\lambda D}{a} \]

3. **Calculate the Fringe Width in Double Slit Experiment:**
The fringe width (distance between adjacent maxima) in a double slit experiment is given by:
\[ \beta' = \frac{\lambda D}{d} \]

4. **Set Up the Condition for 10 Maxima:**
For there to be 10 maxima of the double slit pattern within the central maximum of the single slit pattern, we have:
\[ 10 \beta' = \beta \]

5. **Substituting the Expressions:**
Substituting the expressions for \( \beta \) and \( \beta' \):
\[ 10 \left(\frac{\lambda D}{d}\right) = \frac{2\lambda D}{a} \]

6. **Simplifying the Equation:**
Cancel \( \lambda D \) from both sides (since they are non-zero):
\[ 10 \left(\frac{1}{d}\right) = \frac{2}{a} \]

7. **Rearranging to Solve for \( a \):**
Rearranging gives:
\[ a = \frac{2d}{10} = \frac{d}{5} \]

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

### Final Answer:
The width of each slit \( a \) is \( 0.2 \, \text{mm} \).

---