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

Options

A. (a) `1.2mm`
B. (b) `1.2cm`
C. (c) `2.4cm`
D. (d) `2.4mm`

Explanation

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

### Step 2: Understand the condition for dark fringes
In single slit diffraction, the positions of the dark fringes are given by the formula:
\[ y_n = \frac{n \lambda D}{d} \]
where \( n \) is the order of the dark fringe (1 for the first dark fringe, 2 for the second, etc.).

### Step 3: Calculate the position of the first dark fringe
For the first dark fringe (\( n = 1 \)):
\[ y_1 = \frac{1 \cdot \lambda D}{d} \]
Substituting the known values:
\[ y_1 = \frac{1 \cdot (600 \times 10^{-9} \, \text{m}) \cdot (2 \, \text{m})}{1 \times 10^{-3} \, \text{m}} \]
\[ y_1 = \frac{600 \times 10^{-9} \cdot 2}{1 \times 10^{-3}} \]
\[ y_1 = \frac{1200 \times 10^{-9}}{10^{-3}} = 1200 \times 10^{-6} \, \text{m} = 1.2 \, \text{mm} \]

### Step 4: Calculate the distance between the first dark fringes on either side of the central maximum
The distance between the first dark fringe on one side and the first dark fringe on the other side of the central maximum is:
\[ \text{Distance between dark fringes} = 2y_1 = 2 \cdot 1.2 \, \text{mm} = 2.4 \, \text{mm} \]

### Final Answer:
The distance between the first dark fringes on either side of the central bright fringe is \( 2.4 \, \text{mm} \).

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