The distance between the first and the sixth minima in the diffraction pattern of a single slit is 0.5 mm. The screen is 0.5 m away from the slit. If the wavelength of light used is 5000 Å. Then the slit width will be

Options

A. 5 mm
B. 2.5 mm
C. 1.25 mm
D. 1.0 mm

Explanation

To find the slit width (A) in the given problem, we will use the formula for the position of minima in a single slit diffraction pattern. The position of the minima on the screen is given by:

\[ y_m = \frac{m \lambda D}{A} \]

where:
- \( y_m \) is the position of the m-th minima,
- \( m \) is the order of the minima (1 for the first minima, 6 for the sixth minima),
- \( \lambda \) is the wavelength of light,
- \( D \) is the distance from the slit to the screen,
- \( A \) is the width of the slit.

### Step 1: Identify the given values
- Distance between the first and sixth minima, \( y_6 - y_1 = 0.5 \, \text{mm} = 0.5 \times 10^{-3} \, \text{m} \)
- Distance from the slit to the screen, \( D = 0.5 \, \text{m} \)
- Wavelength of light, \( \lambda = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m} = 5 \times 10^{-7} \, \text{m} \)

### Step 2: Write the equations for the positions of the first and sixth minima
Using the formula for minima:
- For the first minima (\( m = 1 \)):
\[ y_1 = \frac{1 \cdot \lambda D}{A} = \frac{\lambda D}{A} \]

- For the sixth minima (\( m = 6 \)):
\[ y_6 = \frac{6 \cdot \lambda D}{A} = \frac{6 \lambda D}{A} \]

### Step 3: Find the difference between the sixth and first minima
The distance between the first and sixth minima is:
\[ y_6 - y_1 = \frac{6 \lambda D}{A} - \frac{\lambda D}{A} = \frac{(6 - 1) \lambda D}{A} = \frac{5 \lambda D}{A} \]

### Step 4: Set up the equation using the known distance
We know that:
\[ y_6 - y_1 = 0.5 \times 10^{-3} \, \text{m} \]
So we can write:
\[ \frac{5 \lambda D}{A} = 0.5 \times 10^{-3} \]

### Step 5: Rearranging to find the slit width (A)
Rearranging the equation gives:
\[ A = \frac{5 \lambda D}{0.5 \times 10^{-3}} \]

### Step 6: Substitute the known values
Now substitute the values of \( \lambda \) and \( D \):
\[ A = \frac{5 \cdot (5 \times 10^{-7}) \cdot (0.5)}{0.5 \times 10^{-3}} \]

### Step 7: Calculate A
Calculating the above expression:
\[ A = \frac{5 \cdot 5 \times 10^{-7} \cdot 0.5}{0.5 \times 10^{-3}} \]
\[ A = \frac{25 \times 10^{-7}}{0.5 \times 10^{-3}} \]
\[ A = \frac{25 \times 10^{-7}}{0.5} \times 10^{3} \]
\[ A = 50 \times 10^{-4} \]
\[ A = 5 \times 10^{-3} \, \text{m} \]
\[ A = 5 \, \text{mm} \]

### Final Answer
The slit width \( A \) is \( 5 \, \text{mm} \).