A single slit of width 0.1 mm is illuminated by a parallel beam of light of wavelength 6000Å and diffraction bands are observed on a screen 0.5 m from the slit. The distance of the third dark band from the central bright band is _______ mm.

Explanation

To solve the problem of finding the distance of the third dark band from the central bright band in a single slit diffraction pattern, we can follow these steps:

### Step 1: Understand the given data
- Width of the slit (d) = 0.1 mm = \(0.1 \times 10^{-3}\) m = \(1 \times 10^{-4}\) m
- Wavelength of light (\(\lambda\)) = 6000 Å = \(6000 \times 10^{-10}\) m = \(6 \times 10^{-7}\) m
- Distance from the slit to the screen (D) = 0.5 m

### Step 2: Use the formula for dark fringes in single slit diffraction
The condition for dark fringes in single slit diffraction is given by:
\[ d \sin \theta = n \lambda \]
where \(n\) is the order of the dark fringe (n = 1, 2, 3, ...).

### Step 3: Relate \(\sin \theta\) to the position on the screen
For small angles, \(\sin \theta \approx \tan \theta \approx \frac{x}{D}\), where \(x\) is the distance of the dark fringe from the central maximum. Thus, we can rewrite the equation as:
\[ d \frac{x}{D} = n \lambda \]

### Step 4: Rearrange the equation to solve for \(x\)
Rearranging gives:
\[ x = \frac{n \lambda D}{d} \]

### Step 5: Substitute the values into the equation
For the third dark band, \(n = 3\):
\[ x = \frac{3 \times (6 \times 10^{-7} \text{ m}) \times (0.5 \text{ m})}{1 \times 10^{-4} \text{ m}} \]

### Step 6: Calculate the value of \(x\)
Calculating the above expression:
\[ x = \frac{3 \times 6 \times 10^{-7} \times 0.5}{1 \times 10^{-4}} = \frac{9 \times 10^{-7}}{1 \times 10^{-4}} = 9 \times 10^{-3} \text{ m} \]

### Step 7: Convert \(x\) to mm
To convert meters to millimeters:
\[ x = 9 \times 10^{-3} \text{ m} = 9 \text{ mm} \]

### Final Answer
The distance of the third dark band from the central bright band is **9 mm**.