A beam of light of wavelength 600 nm from a distant source
falls on a single slit 1.0 mm wide and the resulting diffraction pattern is
observed on a screen 2m away. What is the distance between the first dark
fringe on either side of the central bright fringe?

Options

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

Answer

To find 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: Understand the Problem
We need to find the distance between the first dark fringe on either side of the central bright fringe when light of wavelength 600 nm passes through a single slit of width 1.0 mm and is observed on a screen 2 m away.

### Step 2: Identify the Relevant Formula
For a single slit diffraction pattern, the condition for dark fringes is given by:
\[ d \sin \theta = n \lambda \]
where:
- \( d \) = width of the slit
- \( \theta \) = angle of the dark fringe
- \( n \) = order of the dark fringe (for the first dark fringe, \( n = 1 \))
- \( \lambda \) = wavelength of light

### Step 3: Small Angle Approximation
For small angles, we can use the approximation:
\[ \sin \theta \approx \tan \theta \approx \frac{y}{D} \]
where:
- \( y \) = distance from the central maximum to the first dark fringe
- \( D \) = distance from the slit to the screen

### Step 4: Substitute and Rearrange the Formula
Substituting the small angle approximation into the dark fringe condition:
\[ d \frac{y}{D} = n \lambda \]
Rearranging gives:
\[ y = \frac{n \lambda D}{d} \]

### Step 5: Plug in the Values
Given:
- \( \lambda = 600 \, \text{nm} = 600 \times 10^{-9} \, \text{m} \)
- \( D = 2 \, \text{m} \)
- \( d = 1.0 \, \text{mm} = 1.0 \times 10^{-3} \, \text{m} \)
- For the first dark fringe, \( n = 1 \)

Now substituting the values:
\[ y = \frac{1 \times (600 \times 10^{-9}) \times 2}{1.0 \times 10^{-3}} \]

### Step 6: Calculate \( y \)
Calculating \( y \):
\[ y = \frac{600 \times 10^{-9} \times 2}{1.0 \times 10^{-3}} \]
\[ y = \frac{1200 \times 10^{-9}}{1.0 \times 10^{-3}} \]
\[ y = 1.2 \times 10^{-3} \, \text{m} = 1.2 \, \text{mm} \]

### Step 7: Find the Total Distance Between the Dark Fringes
Since we need the distance between the first dark fringe on either side of the central maximum, we multiply \( y \) by 2:
\[ \text{Total distance} = 2y = 2 \times 1.2 \, \text{mm} = 2.4 \, \text{mm} \]

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

---