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 derive the relationship between phase difference (ΔΦ) and path difference (Δr) for two interfering waves, we can follow these steps:

### Step 1: Understand the Definitions
- **Path Difference (Δr)**: This is the difference in distance traveled by two waves arriving at a point.
- **Phase Difference (ΔΦ)**: This is the difference in phase of the two waves when they arrive at that point.

### Step 2: Use the Relationship between Path Difference and Wavelength
The path difference (Δr) is related to the wavelength (λ) of the waves. The relationship is given by:
\[ \Delta r = n \lambda \]
where \( n \) is an integer representing the number of wavelengths.

### Step 3: Relate Phase Difference to Path Difference
The phase difference (ΔΦ) can be expressed in terms of the path difference. The relationship is given by:
\[ \Delta \Phi = \frac{2\pi}{\lambda} \Delta r \]
This equation states that the phase difference is proportional to the path difference, scaled by the factor \( \frac{2\pi}{\lambda} \).

### Step 4: Rearranging the Equation
To express ΔΦ in terms of Δr, we can rearrange the equation:
\[ \Delta \Phi = \frac{2\pi}{\lambda} \Delta r \]

### Conclusion
Thus, the relationship between phase difference (ΔΦ) and path difference (Δr) is given by:
\[ \Delta \Phi = \frac{2\pi}{\lambda} \Delta r \]