A slit of width `'a'` is illuminated by red light of wavelenght `6500 Å`. For what value of `'a'` will the (i) first minimum fall at an angle of diffraction of `30^(@)` ? (ii) first maximum fall at an angle of diffraction of `30^(@)` ?

Answer

To solve the problem, we will use the principles of single-slit diffraction.

### Given:
- Wavelength of red light, \( \lambda = 6500 \, \text{Å} = 6500 \times 10^{-10} \, \text{m} \)
- Angle of diffraction, \( \theta = 30^\circ \)

### Part (i): Finding the slit width \( a \) for the first minimum at \( 30^\circ \)

1. **Understanding the condition for the first minimum**:
The condition for the first minimum in single-slit diffraction is given by:
\[ a \sin \theta = n \lambda \]
where \( n = 1 \) for the first minimum.

2. **Substituting the known values**:
For the first minimum, we have:
\[ a \sin(30^\circ) = 1 \cdot \lambda \]

3. **Calculating \( \sin(30^\circ) \)**:
\[ \sin(30^\circ) = \frac{1}{2} \]

4. **Setting up the equation**:
\[ a \cdot \frac{1}{2} = 6500 \times 10^{-10} \]

5. **Solving for \( a \)**:
\[ a = 6500 \times 10^{-10} \cdot 2 \]
\[ a = 13000 \times 10^{-10} \, \text{m} = 1.3 \times 10^{-6} \, \text{m} \]

### Part (ii): Finding the slit width \( a \) for the first maximum at \( 30^\circ \)

1. **Understanding the condition for the first maximum**:
The condition for the first maximum in single-slit diffraction is given by:
\[ a \sin \theta = (n + \frac{1}{2}) \lambda \]
where \( n = 0 \) for the first maximum.

2. **Substituting the known values**:
For the first maximum, we have:
\[ a \sin(30^\circ) = \left(0 + \frac{1}{2}\right) \lambda \]

3. **Using \( \sin(30^\circ) \)**:
\[ a \cdot \frac{1}{2} = \frac{1}{2} \cdot 6500 \times 10^{-10} \]

4. **Setting up the equation**:
\[ a \cdot \frac{1}{2} = \frac{6500 \times 10^{-10}}{2} \]

5. **Solving for \( a \)**:
\[ a = 6500 \times 10^{-10} \]
\[ a = 6500 \times 10^{-10} \cdot 2 \]
\[ a = 19500 \times 10^{-10} \, \text{m} = 1.95 \times 10^{-6} \, \text{m} \]

### Final Answers:
- (i) The slit width \( a \) for the first minimum at \( 30^\circ \) is \( 1.3 \times 10^{-6} \, \text{m} \).
- (ii) The slit width \( a \) for the first maximum at \( 30^\circ \) is \( 1.95 \times 10^{-6} \, \text{m} \).