The distace 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 wavelength of the light used is `5000 A^(0)`, then the slit width will be

Options

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

Answer

To find the slit width \( A \) given the distance between the first and sixth minima in the diffraction pattern of a single slit, we can follow these steps:

### Step 1: Understand the given data
- 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: \( L = 0.5 \, \text{m} \)
- Wavelength of light: \( \lambda = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m} \)

### Step 2: Write the formula for the positions of minima
For a single slit diffraction pattern, the position of the minima can be given by:
- For the first minima:
\[ y_1 = \frac{L \lambda}{A} \]
- For the sixth minima:
\[ y_6 = \frac{6L \lambda}{A} \]

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

### Step 4: Set up the equation
From the given data, we have:
\[ y_6 - y_1 = 0.5 \times 10^{-3} \, \text{m} \]
Thus, we can write:
\[ \frac{5L \lambda}{A} = 0.5 \times 10^{-3} \]

### Step 5: Substitute the known values
Substituting \( L = 0.5 \, \text{m} \) and \( \lambda = 5000 \times 10^{-10} \, \text{m} \) into the equation:
\[ \frac{5 \times 0.5 \times (5000 \times 10^{-10})}{A} = 0.5 \times 10^{-3} \]

### Step 6: Simplify the equation
Calculating the left-hand side:
\[ \frac{5 \times 0.5 \times 5000 \times 10^{-10}}{A} = \frac{12500 \times 10^{-10}}{A} \]
Now, we can equate it to the right-hand side:
\[ \frac{12500 \times 10^{-10}}{A} = 0.5 \times 10^{-3} \]

### Step 7: Solve for \( A \)
Rearranging gives:
\[ A = \frac{12500 \times 10^{-10}}{0.5 \times 10^{-3}} = \frac{12500 \times 10^{-10}}{0.5 \times 10^{-3}} = \frac{12500}{0.5} \times 10^{-7} \]
Calculating this:
\[ A = 25000 \times 10^{-7} = 2.5 \times 10^{-3} \, \text{m} = 2.5 \, \text{mm} \]

### Final Answer
The slit width \( A \) is \( 2.5 \, \text{mm} \).

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