Assertion: Two identical waves due to two coherent sources interfere at a point with a phase difference of `(2pi)/3`, then the resultant intensity at this point is equal to the individual intensity of the sources.
Reason: A phase difference of `(2pi)/3` is equivalent to a path difference of `lambda/3`.

Options

A. (a)If both Assertion and Reason are true and the Reason is correct explanation of the Assertion.
B. (b)If both Assertion and Reason are true and the Reason is not the correct explanation of the Assertion.
C. (c) If Assertion is true, but the Reason is false.
D. (d) If Assertion is false, but the Reason is true.

Explanation

To solve the question, we need to analyze both the assertion and the reason provided.

### Step 1: Understanding the Assertion
The assertion states that when two identical waves from coherent sources interfere at a point with a phase difference of \( \frac{2\pi}{3} \), the resultant intensity at that point is equal to the individual intensity of the sources.

### Step 2: Formula for Resultant Intensity
The resultant intensity \( I \) of two coherent waves can be calculated using the formula:
\[ I = 4I_0 \cos^2\left(\frac{\phi}{2}\right) \]
where \( I_0 \) is the intensity of each individual wave and \( \phi \) is the phase difference.

### Step 3: Substitute the Phase Difference
Given \( \phi = \frac{2\pi}{3} \), we substitute this into the formula:
\[ I = 4I_0 \cos^2\left(\frac{2\pi}{3 \cdot 2}\right) = 4I_0 \cos^2\left(\frac{\pi}{3}\right) \]

### Step 4: Calculate \( \cos^2\left(\frac{\pi}{3}\right) \)
The value of \( \cos\left(\frac{\pi}{3}\right) \) is \( \frac{1}{2} \). Therefore,
\[ \cos^2\left(\frac{\pi}{3}\right) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 5: Substitute Back to Find Resultant Intensity
Now substituting back, we get:
\[ I = 4I_0 \cdot \frac{1}{4} = I_0 \]
This shows that the resultant intensity \( I \) is indeed equal to the individual intensity \( I_0 \).

### Step 6: Conclusion for Assertion
Thus, the assertion is **true**.

### Step 7: Understanding the Reason
The reason states that a phase difference of \( \frac{2\pi}{3} \) corresponds to a path difference of \( \frac{\lambda}{3} \).

### Step 8: Path Difference Calculation
The relationship between phase difference \( \phi \) and path difference \( \Delta x \) is given by:
\[ \Delta x = \frac{\lambda}{2\pi} \cdot \phi \]
Substituting \( \phi = \frac{2\pi}{3} \):
\[ \Delta x = \frac{\lambda}{2\pi} \cdot \frac{2\pi}{3} = \frac{\lambda}{3} \]
This confirms that the reason is also **true**.

### Step 9: Relationship Between Assertion and Reason
While both statements are true, the reason does not correctly explain the assertion. The assertion's correctness is derived from the intensity formula, not directly from the path difference.

### Final Answer
The assertion is true, the reason is true, but the reason is not the correct explanation for the assertion. Therefore, the answer is option B.

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