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 8 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

Explanation

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 (y) = 5 mm = 5 × 10^-3 m
- Distance from the slit to the screen (D) = 8 cm = 8 × 10^-2 m
- Wavelength (λ) = 6000 Å = 6000 × 10^-10 m = 6 × 10^-7 m

### Step 2: Use the formula for single slit diffraction
The position of the first minimum in a single slit diffraction pattern is given by the formula:
\[ y = \frac{\lambda D}{a} \]
where:
- \( y \) is the distance from the central maximum to the first minimum,
- \( \lambda \) is the wavelength,
- \( D \) is the distance from the slit to the screen,
- \( a \) is the slit width.

Since the distance between the first minimum on the left and the first minimum on the right is given as 5 mm, we can express this as:
\[ 2y = 5 \text{ mm} \]
Thus,
\[ y = \frac{5}{2} \text{ mm} = 2.5 \text{ mm} = 2.5 \times 10^{-3} \text{ m} \]

### Step 3: Rearranging the formula to find slit width (a)
We can rearrange the formula to solve for \( a \):
\[ a = \frac{\lambda D}{y} \]

### Step 4: Substitute the values into the formula
Now we can substitute the known values into the rearranged formula:
\[ a = \frac{(6 \times 10^{-7} \text{ m}) (8 \times 10^{-2} \text{ m})}{2.5 \times 10^{-3} \text{ m}} \]

### Step 5: Calculate the slit width
Calculating the above expression:
\[ a = \frac{(6 \times 10^{-7}) (8 \times 10^{-2})}{2.5 \times 10^{-3}} \]
\[ a = \frac{4.8 \times 10^{-8}}{2.5 \times 10^{-3}} \]
\[ a = 1.92 \times 10^{-5} \text{ m} \]

### Step 6: Convert to mm
To convert the slit width from meters to millimeters:
\[ a = 1.92 \times 10^{-5} \text{ m} \times 1000 \text{ mm/m} = 0.192 \text{ mm} \]

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

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