Light of wavelength `500 nm` falls from a distant source on a slit `0.5 mm` wide. Find the distance between the two dark bands on either side of central maximum, if diffraction pattern is observed on a screen at `2 m` from the slit.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the given data
- Wavelength of light, \( \lambda = 500 \, \text{nm} = 500 \times 10^{-9} \, \text{m} \)
- Width of the slit, \( d = 0.5 \, \text{mm} = 0.5 \times 10^{-3} \, \text{m} \)
- Distance from the slit to the screen, \( D = 2 \, \text{m} \)

### Step 2: Use the formula for the distance between dark bands
The distance between the dark bands on either side of the central maximum in a single-slit diffraction pattern can be calculated using the formula for fringe width:
\[ x = \frac{\lambda D}{d} \]
Where:
- \( x \) is the distance from the central maximum to the first dark band.

### Step 3: Calculate \( x \)
Substituting the known values into the formula:
\[ x = \frac{500 \times 10^{-9} \, \text{m} \times 2 \, \text{m}}{0.5 \times 10^{-3} \, \text{m}} \]

### Step 4: Simplify the calculation
Calculating the numerator:
\[ 500 \times 10^{-9} \times 2 = 1000 \times 10^{-9} = 1 \times 10^{-6} \, \text{m} \]
Now, substituting back into the equation:
\[ x = \frac{1 \times 10^{-6}}{0.5 \times 10^{-3}} = \frac{1 \times 10^{-6}}{0.5 \times 10^{-3}} = \frac{1}{0.5} \times 10^{-3} = 2 \times 10^{-3} \, \text{m} \]

### Step 5: Calculate the total distance between the two dark bands
Since \( x \) is the distance from the central maximum to the first dark band, the total distance between the two dark bands on either side of the central maximum is:
\[ \text{Total distance} = 2x = 2 \times (2 \times 10^{-3}) = 4 \times 10^{-3} \, \text{m} \]

### Step 6: Convert to millimeters
Converting meters to millimeters:
\[ 4 \times 10^{-3} \, \text{m} = 4 \, \text{mm} \]

### Final Answer
The distance between the two dark bands on either side of the central maximum is \( 4 \, \text{mm} \).

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