A narrow slit of width 2 mm is illuminated by monochromatic light fo wavelength 500nm. The distance between the first minima on either side on a screen at a distance of 1 m is

Options

A. 5 mm
B. 0.5 mm
C. 1 mm
D. 10 mm

Explanation

To solve the problem, we need to find the distance between the first minima on either side of a narrow slit illuminated by monochromatic light. Here's the step-by-step solution:

### Step 1: Identify the given quantities
- Width of the slit (a) = 2 mm = \(2 \times 10^{-3}\) m
- Wavelength of the light (\(\lambda\)) = 500 nm = \(500 \times 10^{-9}\) m
- Distance from the slit to the screen (D) = 1 m

### Step 2: Understand the formula for the position of minima
For a single slit diffraction, the position of the first minima on either side of the central maximum is given by the formula:
\[ y = \frac{m \lambda D}{a} \]
where:
- \(y\) is the distance from the central maximum to the m-th minima,
- \(m\) is the order of the minima (for the first minima, \(m = 1\)),
- \(\lambda\) is the wavelength of the light,
- \(D\) is the distance from the slit to the screen,
- \(a\) is the width of the slit.

### Step 3: Calculate the distance to the first minima
For the first minima (\(m = 1\)):
\[ y = \frac{1 \cdot \lambda \cdot D}{a} \]
Substituting the known values:
\[ y = \frac{1 \cdot (500 \times 10^{-9} \text{ m}) \cdot (1 \text{ m})}{2 \times 10^{-3} \text{ m}} \]

### Step 4: Simplify the equation
Calculating the above expression:
\[ y = \frac{500 \times 10^{-9}}{2 \times 10^{-3}} = \frac{500}{2} \times 10^{-9 + 3} = 250 \times 10^{-6} \text{ m} \]

### Step 5: Convert to more convenient units
Convert \(250 \times 10^{-6}\) m to millimeters:
\[ 250 \times 10^{-6} \text{ m} = 0.25 \text{ mm} \]

### Step 6: Calculate the total distance between the first minima on either side
Since we need the distance between the first minima on either side, we multiply by 2:
\[ \text{Total distance} = 2y = 2 \times 0.25 \text{ mm} = 0.5 \text{ mm} \]

### Final Answer
The distance between the first minima on either side is \(0.5 \text{ mm}\).

---