In a single slit diffraction patternm, the distance between the first minimum on the left and the first minimum on the right is 5 mm. The screen on which the diffraction pattern is displayed is at a distance of 80 cm friom the silt. The wavelength is 6000 Å. The slit width ( in mm) is about .

Options

A. 0.576
B. 0.348
C. 0.192
D. 0.096

Answer

To solve the problem of finding the slit width in a single slit diffraction pattern, we can follow these steps:

### Step 1: Understand the given data
- Distance between the first minimum on the left and the first minimum on the right = 5 mm
- Distance from the slit to the screen (D) = 80 cm = 0.8 m
- Wavelength (λ) = 6000 Å = 6000 × 10^(-10) m = 6 × 10^(-7) m

### Step 2: Determine the distance from the central maximum to the first minimum
Since the distance between the first minimum on the left and the first minimum on the right is 5 mm, the distance from the central maximum to the first minimum (y) is half of this distance:
\[ y = \frac{5 \text{ mm}}{2} = 2.5 \text{ mm} = 2.5 \times 10^{-3} \text{ m} \]

### Step 3: Use the formula for single slit diffraction
The formula for the position of the minima in a single slit diffraction pattern is given by:
\[ y = \frac{m \lambda D}{W} \]
where:
- \( m \) = order of the minimum (for the first minimum, \( m = 1 \))
- \( \lambda \) = wavelength of light
- \( D \) = distance from the slit to the screen
- \( W \) = width of the slit

### Step 4: Rearranging the formula to find W
Rearranging the formula to solve for W gives:
\[ W = \frac{m \lambda D}{y} \]

### Step 5: Substitute the known values
Substituting \( m = 1 \), \( \lambda = 6 \times 10^{-7} \text{ m} \), \( D = 0.8 \text{ m} \), and \( y = 2.5 \times 10^{-3} \text{ m} \):
\[ W = \frac{1 \times (6 \times 10^{-7}) \times (0.8)}{2.5 \times 10^{-3}} \]

### Step 6: Calculate W
Calculating the numerator:
\[ 6 \times 10^{-7} \times 0.8 = 4.8 \times 10^{-7} \]
Now, substituting into the equation for W:
\[ W = \frac{4.8 \times 10^{-7}}{2.5 \times 10^{-3}} \]

Calculating this gives:
\[ W = \frac{4.8}{2.5} \times 10^{-4} = 1.92 \times 10^{-4} \text{ m} \]

### Step 7: Convert W to mm
To convert from meters to millimeters:
\[ W = 1.92 \times 10^{-4} \text{ m} \times 1000 = 0.192 \text{ mm} \]

### Final Answer
The slit width \( W \) is approximately **0.192 mm**.

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