In Fraunhofer diffraction pattern, slit width is `0.2 mm` and screen is at 2 m away from the lens. If wavelength of light used is `5000 Å`, then the distance between the first minimum on either side of the central maximum is (`theta` is small and measured in radian)

Options

A. `10^(-1) m`
B. `10^(-2)m`
C. `2xx10^(-2)m`
D. `2xx10^(-1)m`

Answer

To solve the problem of finding the distance between the first minima on either side of the central maximum in a Fraunhofer diffraction pattern, we can follow these steps:

### Step 1: Identify the given values
- Slit width (d) = 0.2 mm = \(0.2 \times 10^{-3}\) m
- Distance to the screen (L) = 2 m
- Wavelength of light (λ) = 5000 Å = \(5000 \times 10^{-10}\) m = \(5 \times 10^{-7}\) m

### Step 2: Use the formula for minima in single-slit diffraction
The condition for the minima in a single-slit diffraction pattern is given by:
\[ d \sin \theta = n \lambda \]
For the first minima (n = 1):
\[ d \sin \theta = \lambda \]

### Step 3: Approximate sin θ for small angles
Since θ is small, we can use the small angle approximation:
\[ \sin \theta \approx \theta \]
Thus, we can rewrite the equation as:
\[ d \theta = \lambda \]
From this, we can find θ:
\[ \theta = \frac{\lambda}{d} \]

### Step 4: Calculate θ
Substituting the values of λ and d:
\[ \theta = \frac{5 \times 10^{-7}}{0.2 \times 10^{-3}} \]
\[ \theta = \frac{5 \times 10^{-7}}{2 \times 10^{-4}} = \frac{5}{2} \times 10^{-3} = 2.5 \times 10^{-3} \text{ radians} \]

### Step 5: Calculate the distance between the first minima
The distance between the first minima on either side of the central maximum can be calculated as:
\[ y = L \cdot 2\theta \]
Where L is the distance from the slit to the screen. Thus:
\[ y = 2 \cdot 2 \cdot 2.5 \times 10^{-3} \]
\[ y = 4 \cdot 2.5 \times 10^{-3} = 10 \times 10^{-3} = 0.01 \text{ m} \]

### Final Answer
The distance between the first minima on either side of the central maximum is:
\[ \text{Distance} = 0.01 \text{ m} = 10 \text{ mm} \]