A beam of light `(lamda=600 nm)` from a distant source falls on a single slit 1 mm wide and the resulting diffraction pattern is observed on a screen 2 m away. The distance between the first dark fringes on either side of the central bright fringe is

Options

A. 1.2 cm
B. 1.2 cm
C. 2.4 cm
D. 2.4 mm

Answer

To solve the problem of finding the distance between the first dark fringes on either side of the central bright fringe in a single-slit diffraction pattern, we can follow these steps:

### Step-by-Step Solution:

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

2. **Understand the Formula for Dark Fringes:**
The position of the dark fringes in a single-slit diffraction pattern is given by the formula:
\[ x_n = \frac{n \lambda D}{a} \]
where \( n \) is the order of the dark fringe (1 for the first dark fringe).

3. **Calculate the Position of the First Dark Fringe:**
For the first dark fringe, \( n = 1 \):
\[ x_1 = \frac{1 \cdot \lambda \cdot D}{a} = \frac{1 \cdot (600 \times 10^{-9}) \cdot 2}{1 \times 10^{-3}} \]

4. **Perform the Calculation:**
\[ x_1 = \frac{600 \times 10^{-9} \times 2}{1 \times 10^{-3}} = \frac{1200 \times 10^{-9}}{10^{-3}} = 1200 \times 10^{-6} \, \text{m} = 1.2 \times 10^{-3} \, \text{m} = 1.2 \, \text{mm} \]

5. **Find the Total Distance Between the First Dark Fringes on Both Sides:**
The distance between the first dark fringes on either side of the central bright fringe is:
\[ \text{Total distance} = 2x_1 = 2 \cdot 1.2 \, \text{mm} = 2.4 \, \text{mm} \]

### Final Answer:
The distance between the first dark fringes on either side of the central bright fringe is **2.4 mm**.

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