A screen is placed `50 cm` from a single slit, which is illuminated with `6000 Å` light. If the distance between the first and third minima in the diffraction pattern is `3.00 mm`, what is the width of the slit ?

Options

A. `1 xx 10^(-4)m`
B. `2 xx 10^(-4)m`
C. `0.5 xx 10^(-4)m`
D. `4 xx 10^(-4)m`

Explanation

To solve the problem, we need to find the width of the slit using the given information about the diffraction pattern created by a single slit.

### Step-by-Step Solution:

1. **Identify Given Data:**
- Distance from the slit to the screen (D) = 50 cm = 0.50 m
- Wavelength of light (λ) = 6000 Å = 6000 × 10^(-10) m = 6 × 10^(-7) m
- Distance between the first and third minima (Δx) = 3.00 mm = 3 × 10^(-3) m

2. **Understand the Formula for Minima:**
The position of the minima in a single slit diffraction pattern is given by:
\[ x_n = \frac{n D \lambda}{a} \]
where \( n \) is the order of the minima, \( D \) is the distance from the slit to the screen, \( \lambda \) is the wavelength of light, and \( a \) is the width of the slit.

3. **Calculate Positions of the First and Third Minima:**
- For the first minima (n = 1):
\[ x_1 = \frac{1 \cdot D \cdot \lambda}{a} = \frac{D \lambda}{a} \]
- For the third minima (n = 3):
\[ x_3 = \frac{3 \cdot D \cdot \lambda}{a} \]

4. **Find the Difference Between the Positions:**
The distance between the first and third minima is given by:
\[ x_3 - x_1 = \frac{3D\lambda}{a} - \frac{D\lambda}{a} = \frac{(3 - 1)D\lambda}{a} = \frac{2D\lambda}{a} \]
This is equal to the given distance between the minima:
\[ \Delta x = x_3 - x_1 = \frac{2D\lambda}{a} \]

5. **Rearranging to Find the Slit Width (a):**
Rearranging the equation gives:
\[ a = \frac{2D\lambda}{\Delta x} \]

6. **Substituting Values:**
Now, substituting the known values into the equation:
\[ a = \frac{2 \cdot (0.50 \, \text{m}) \cdot (6 \times 10^{-7} \, \text{m})}{3 \times 10^{-3} \, \text{m}} \]

7. **Calculating the Width of the Slit:**
- Calculate the numerator:
\[ 2 \cdot 0.50 \cdot 6 \times 10^{-7} = 6 \times 10^{-7} \, \text{m} \]
- Now divide by the distance between the minima:
\[ a = \frac{6 \times 10^{-7}}{3 \times 10^{-3}} = 2 \times 10^{-4} \, \text{m} = 0.2 \, \text{mm} \]

### Final Answer:
The width of the slit (a) is **0.2 mm**.