At the first minimum adjacent to the central maximum of a single-slit diffraction pattern the phase difference between the Huygens wavelet from the edge of the slit and the wavelet from the mid-point of the slit is

Options

A. (a) `pi/8` radian
B. (b) `pi/4` radian
C. (c) `2pi` radian
D. (d) `pi` radian

Explanation

To solve the problem of finding the phase difference between the Huygens wavelet from the edge of the slit and the wavelet from the midpoint of the slit at the first minimum adjacent to the central maximum of a single-slit diffraction pattern, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup:**
- We have a single slit of width \( A \).
- The slit can be thought of as being made up of many point sources (Huygens' principle).
- We are interested in the phase difference between the wavelet coming from the midpoint of the slit and the wavelet coming from the edge of the slit.

2. **Identifying the Condition for First Minimum:**
- The condition for the first minimum in single-slit diffraction is given by:
\[ A \sin \theta = \lambda \]
- Here, \( \lambda \) is the wavelength of the light used, and \( \theta \) is the angle at which the first minimum occurs.

3. **Calculating the Path Difference:**
- The path difference \( \Delta x \) between the wavelet from the edge of the slit and the wavelet from the midpoint of the slit can be expressed as:
\[ \Delta x = \frac{A}{2} \sin \theta \]
- This is because the midpoint of the slit is \( A/2 \) away from the edge.

4. **Relating Path Difference to Phase Difference:**
- The phase difference \( \Delta \phi \) corresponding to a path difference \( \Delta x \) is given by:
\[ \Delta \phi = \frac{2\pi}{\lambda} \Delta x \]
- Substituting the expression for \( \Delta x \):
\[ \Delta \phi = \frac{2\pi}{\lambda} \left(\frac{A}{2} \sin \theta\right) = \frac{\pi A \sin \theta}{\lambda} \]

5. **Substituting for First Minimum Condition:**
- From the first minimum condition \( A \sin \theta = \lambda \), we can substitute \( \sin \theta \) in the phase difference equation:
\[ \sin \theta = \frac{\lambda}{A} \]
- Therefore, substituting this into the phase difference equation gives:
\[ \Delta \phi = \frac{\pi A \left(\frac{\lambda}{A}\right)}{\lambda} = \frac{\pi}{2} \]

6. **Conclusion:**
- The phase difference between the wavelet from the edge of the slit and the wavelet from the midpoint of the slit at the first minimum is:
\[ \Delta \phi = \pi \text{ radians} \]

### Final Answer:
The phase difference between the Huygens wavelet from the edge of the slit and the wavelet from the midpoint of the slit at the first minimum is \( \pi \) radians.