Light of wavelength `lambda=5000Å` falls normally on a narrow slit. A screen is placed at a distance of `1m` from the slit and perpendicular to the direction of light. The first minima of the diffraction pattern is situated at `5mm` from the centre of central maximum. The width of the slit is

Options

A. 0.1 mm
B. 1.0 mm
C. 0.5 mm
D. 0.2 mm

Answer

To solve the problem, we need to find the width of the slit (denoted as \( d \)) using the information provided about the diffraction pattern created by light of a specific wavelength.

### Step-by-step Solution:

1. **Identify Given Values**:
- Wavelength of light, \( \lambda = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m} \)
- Distance from the slit to the screen, \( D = 1 \, \text{m} \)
- Position of the first minima from the center, \( x_1 = 5 \, \text{mm} = 5 \times 10^{-3} \, \text{m} \)
- Order of the minima, \( n = 1 \)

2. **Use the Formula for Minima in Single-Slit Diffraction**:
The position of the \( n \)-th minima in a single-slit diffraction pattern is given by the formula:
\[ x_n = \frac{n \lambda D}{d} \]
For the first minima (\( n = 1 \)):
\[ x_1 = \frac{\lambda D}{d} \]

3. **Rearranging the Formula to Solve for \( d \)**:
We can rearrange the equation to find the width of the slit \( d \):
\[ d = \frac{\lambda D}{x_1} \]

4. **Substituting the Known Values**:
Now, substitute the known values into the equation:
\[ d = \frac{(5000 \times 10^{-10} \, \text{m})(1 \, \text{m})}{5 \times 10^{-3} \, \text{m}} \]

5. **Calculating the Width of the Slit**:
\[ d = \frac{5000 \times 10^{-10}}{5 \times 10^{-3}} = \frac{5000}{5} \times 10^{-10 + 3} = 1000 \times 10^{-7} \, \text{m} = 10^{-4} \, \text{m} \]
Converting \( d \) to millimeters:
\[ d = 10^{-4} \, \text{m} = 0.1 \, \text{mm} \]

### Final Answer:
The width of the slit is \( d = 0.1 \, \text{mm} \).

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