A parallel beam of light of wavelength 500 nm falls on a narrow slit and the resulting diffraction pattern is observe on screen 1 m away. It is observed that the first minimum is at a distance of `2.5 mm` from the centre of the screen. Find the width of the slit.

Options

A. 0.2 mm
B. 1 mm
C. 2 mm
D. 1.5 mm

Explanation

To solve the problem step by step, we will use the formula for the position of the first minimum in a single-slit diffraction pattern.

### Step 1: Understand the diffraction pattern
In a single-slit diffraction experiment, the first minimum occurs at a distance \( x \) from the center of the screen, which can be described by the formula:
\[ x = \frac{n \lambda D}{d} \]
where:
- \( n \) is the order of the minimum (for the first minimum, \( n = 1 \)),
- \( \lambda \) is the wavelength of light,
- \( D \) is the distance from the slit to the screen,
- \( d \) is the width of the slit.

### Step 2: Rearrange the formula to find the width of the slit
We need to find the width of the slit \( d \). Rearranging the formula gives:
\[ d = \frac{n \lambda D}{x} \]

### Step 3: Substitute the known values
Given:
- Wavelength \( \lambda = 500 \, \text{nm} = 500 \times 10^{-9} \, \text{m} \)
- Distance to the screen \( D = 1 \, \text{m} \)
- Distance of the first minimum from the center \( x = 2.5 \, \text{mm} = 2.5 \times 10^{-3} \, \text{m} \)
- Order of the minimum \( n = 1 \)

Substituting these values into the rearranged formula:
\[ d = \frac{1 \times (500 \times 10^{-9}) \times 1}{2.5 \times 10^{-3}} \]

### Step 4: Calculate the width of the slit
Now, calculate \( d \):
\[ d = \frac{500 \times 10^{-9}}{2.5 \times 10^{-3}} = \frac{500}{2.5} \times 10^{-6} = 200 \times 10^{-6} \, \text{m} = 2 \times 10^{-4} \, \text{m} \]

### Step 5: Convert to millimeters
To express the width in millimeters:
\[ d = 2 \times 10^{-4} \, \text{m} = 0.2 \, \text{mm} \]

### Conclusion
The width of the slit is \( 0.2 \, \text{mm} \).