The relationship between phase difference `Deltaphi` and the path difference `Deltar` between two interfering waves is given by: (`lambda` = wavelength)

Options

A. `Deltax = (lambda/(2pi))Deltaphi`
B. `Deltar = (2pi)/lambda Deltaphi`
C. `Deltaphi = (lambda/(2pi))Deltax`
D. `Deltaphi = 2(2pi)Deltax`

Answer

To establish the relationship between the phase difference (Δφ) and the path difference (Δr) between two interfering waves, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Concept**:
- When two waves interfere, they can have a phase difference due to the difference in the distance they travel. This difference in distance is known as the path difference (Δr).

2. **Define Path Difference**:
- Let’s denote the path difference between the two waves as Δr, which can be expressed as:
\[ \Delta r = x_2 - x_1 \]
- Here, \(x_1\) and \(x_2\) are the distances traveled by the two waves to a point of interference.

3. **Phase Difference Definition**:
- The phase difference (Δφ) between two waves is related to how far one wave is ahead or behind the other in terms of their oscillations.

4. **Relating Phase Difference to Path Difference**:
- The phase difference can be expressed in terms of the wave number (k) and the path difference (Δr):
\[ \Delta \phi = k \cdot \Delta r \]
- Here, \(k\) is the wave number, defined as:
\[ k = \frac{2\pi}{\lambda} \]
- Substituting the expression for \(k\) into the equation for Δφ gives:
\[ \Delta \phi = \frac{2\pi}{\lambda} \cdot \Delta r \]

5. **Final Relationship**:
- Therefore, the relationship between the phase difference and the path difference is:
\[ \Delta \phi = \frac{2\pi}{\lambda} \cdot \Delta r \]

### Summary:
The relationship between the phase difference (Δφ) and the path difference (Δr) for two interfering waves is given by:
\[ \Delta \phi = \frac{2\pi}{\lambda} \cdot \Delta r \]