light of wavelength 600 nm is incident on a single slit. The first minimum of athe diffraction pattern is obtained at a distance of 4 m m from the centre. The distance between the screen and the slit is 2 m . What is the width of the slit ?

Options

A. 0 . 1 m m
B. 0 . 3 m m
C. 0 . 5 m
D. 0 . 6m m

Answer

To solve the problem, we need to find the width of the slit (denoted as \( d \)) using the information provided about the diffraction pattern. The first minimum in a single-slit diffraction pattern occurs at a specific distance from the center, which can be calculated using the formula for the position of minima.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Wavelength of light, \( \lambda = 600 \, \text{nm} = 600 \times 10^{-9} \, \text{m} \)
- Distance from the screen to the slit, \( D = 2 \, \text{m} \)
- Distance from the center to the first minimum, \( Y = 4 \, \text{mm} = 4 \times 10^{-3} \, \text{m} \)

2. **Use the Formula for the Position of the First Minimum:**
The position of the first minimum in a single-slit diffraction pattern is given by:
\[ Y = \frac{N \lambda D}{d} \]
where \( N \) is the order of the minimum (for the first minimum, \( N = 1 \)).

3. **Substituting Known Values:**
For the first minimum (\( N = 1 \)):
\[ Y = \frac{1 \cdot \lambda \cdot D}{d} \]
Substituting the known values:
\[ 4 \times 10^{-3} = \frac{1 \cdot (600 \times 10^{-9}) \cdot 2}{d} \]

4. **Rearranging the Equation to Solve for \( d \):**
Rearranging gives:
\[ d = \frac{(600 \times 10^{-9}) \cdot 2}{4 \times 10^{-3}} \]

5. **Calculating \( d \):**
\[ d = \frac{1200 \times 10^{-9}}{4 \times 10^{-3}} = \frac{1200}{4} \times 10^{-6} = 300 \times 10^{-6} \, \text{m} \]
Converting to millimeters:
\[ d = 300 \times 10^{-6} \, \text{m} = 0.3 \, \text{mm} \]

### Final Answer:
The width of the slit \( d \) is **0.3 mm**.

---