Assertion: Three waves of equal amplitudes interfere at a point. Phase difference between two successive waves is `pi/2`. Then, resultant intensity is same as the intensity due to individual wave.
Reason: Two different light sources are never coherent.

Options

A. If both Assertion and Reason are true and the Reason is correct explanation of the Assertion.
B. If both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
C. If Assertion is true, but the Reason is false.
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: Analyze the Assertion**
The assertion states that three waves of equal amplitudes interfere at a point, with a phase difference of \( \frac{\pi}{2} \) between two successive waves. We need to determine if the resultant intensity is the same as the intensity due to an individual wave.

- Let the amplitude of each wave be \( A \).
- The intensity \( I \) of a wave is proportional to the square of its amplitude, given by \( I \propto A^2 \).
- If we have three waves with equal amplitudes and a phase difference of \( \frac{\pi}{2} \), we can represent the waves mathematically as:
- Wave 1: \( A \sin(\omega t) \)
- Wave 2: \( A \sin(\omega t + \frac{\pi}{2}) \)
- Wave 3: \( A \sin(\omega t + \pi) \)

**Step 2: Calculate Resultant Amplitude**
To find the resultant amplitude, we can use the principle of superposition. The resultant amplitude \( R \) can be calculated using the vector addition of the individual wave amplitudes.

- The first two waves can be represented as vectors in the complex plane:
- Wave 1: \( A \) along the x-axis
- Wave 2: \( A \) along the y-axis (90 degrees to Wave 1)

The resultant of these two waves is:
\[ R_{12} = \sqrt{A^2 + A^2} = \sqrt{2A^2} = A\sqrt{2} \]

Now, adding the third wave (which is \( -A \) along the x-axis):
\[ R = R_{12} + (-A) = A\sqrt{2} - A \]

**Step 3: Calculate Resultant Intensity**
The resultant amplitude \( R \) can be simplified:
\[ R = A(\sqrt{2} - 1) \]
The intensity due to the resultant wave is:
\[ I_R \propto R^2 = (A(\sqrt{2} - 1))^2 = A^2(2 - 2\sqrt{2} + 1) = A^2(3 - 2\sqrt{2}) \]

Since the intensity of an individual wave is:
\[ I = A^2 \]

We can see that \( I_R \) is not equal to \( I \) unless \( \sqrt{2} - 1 = 1 \), which is not true. Therefore, the assertion is **false**.

**Step 4: Analyze the Reason**
The reason states that two different light sources are never coherent. Coherence requires that the sources have a constant phase relationship and the same frequency.

This statement is **true**; however, it does not explain the assertion about the resultant intensity of the waves.

**Conclusion:**
- The assertion is **false**.
- The reason is **true**, but it does not provide a correct explanation for the assertion.

**Final Answer:**
Assertion: False; Reason: True; Reason is not a correct explanation for the assertion.

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