(A) In a progressive wave, the phase diffe rence between two particles of a medium having a path difference `lambda` is `2pi`
(R): The phase difference is directly propor tional to path difference between particles in progressive wave

Options

A. Both 'A' and 'R' are true and 'R' is the correct, explanation of 'A'.
B. Both 'A' and 'R' are true and 'R' is not the correct explanation of 'A'
C. A' is true and 'R' is false
D. Both 'A' and 'R' are false

Answer

To solve the question, we need to analyze the statements (A) and (R) regarding the phase difference and path difference in a progressive wave.

### Step-by-Step Solution:

1. **Understanding Statement (A)**:
- Statement (A) states that in a progressive wave, the phase difference between two particles of a medium having a path difference of `λ` is `2π`.
- We know that the phase difference (Δφ) is related to the path difference (Δx) by the formula:
\[ \Delta \phi = \frac{2\pi}{\lambda} \Delta x \]
- If the path difference Δx = λ, substituting this into the equation gives:
\[ \Delta \phi = \frac{2\pi}{\lambda} \cdot \lambda = 2\pi \]
- Therefore, statement (A) is **true**.

2. **Understanding Statement (R)**:
- Statement (R) claims that the phase difference is directly proportional to the path difference between particles in a progressive wave.
- From the formula we derived earlier:
\[ \Delta \phi = \frac{2\pi}{\lambda} \Delta x \]
- Here, the constant \( \frac{2\pi}{\lambda} \) indicates that as the path difference (Δx) increases, the phase difference (Δφ) also increases. This shows a direct proportionality.
- Therefore, statement (R) is also **true**.

3. **Evaluating the Relationship Between (A) and (R)**:
- Since both statements (A) and (R) are true, we need to determine if (R) is the correct explanation of (A).
- Since (R) explains that the relationship between path difference and phase difference is directly proportional, and this is the underlying reason why (A) holds true, we conclude that (R) is indeed the correct explanation of (A).

### Conclusion:
Both statements (A) and (R) are true, and (R) is the correct explanation of (A).

### Final Answer:
Both A and R are true, and R is the correct explanation of A.

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