What is the width of a single slit if the first minimum is observed at an angle `2^@` with a light of wavelength 9680 Å ?

Options

A. 0.2 mm
B. `2 xx 10^(-2) mm`
C. `2 xx 10^5 mm`
D. `2 mm`

Answer

To find the width of a single slit given the angle of the first minimum and the wavelength of light, we can use the formula for single-slit diffraction:

\[ a \sin \theta = n \lambda \]

where:
- \( a \) is the width of the slit,
- \( \theta \) is the angle at which the first minimum occurs,
- \( n \) is the order of the minimum (for the first minimum, \( n = 1 \)),
- \( \lambda \) is the wavelength of the light.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Wavelength, \( \lambda = 9680 \, \text{Å} = 9680 \times 10^{-10} \, \text{m} \)
- Angle for the first minimum, \( \theta = 2^\circ \)

2. **Convert the Angle to Radians (if necessary)**:
- In this case, we can directly use the sine function in degrees.

3. **Use the Formula for the First Minimum**:
- Since we are looking for the first minimum, we set \( n = 1 \).
- The formula becomes:
\[ a \sin(2^\circ) = 1 \times \lambda \]

4. **Rearranging the Formula to Solve for \( a \)**:
\[ a = \frac{\lambda}{\sin(2^\circ)} \]

5. **Calculate \( \sin(2^\circ) \)**:
- Using a calculator, find \( \sin(2^\circ) \):
\[ \sin(2^\circ) \approx 0.0349 \]

6. **Substituting the Values**:
\[ a = \frac{9680 \times 10^{-10}}{0.0349} \]

7. **Perform the Calculation**:
- Calculate \( a \):
\[ a \approx \frac{9680 \times 10^{-10}}{0.0349} \approx 2.776 \times 10^{-5} \, \text{m} \]

8. **Convert to Millimeters**:
- Since \( 1 \, \text{m} = 1000 \, \text{mm} \):
\[ a \approx 2.776 \times 10^{-5} \, \text{m} \times 1000 \approx 2.776 \, \text{mm} \]

9. **Final Answer**:
- The width of the single slit is approximately \( 2.776 \, \text{mm} \).