Why is (2n-1)π used in destructive interference of light wave ?

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No difference btween them (2n-1)  in this n = 1,2,3,4,5,6...and(2n+1) in this n= 0,1,2,3,4,5...

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I think its (2n+1)π and its bcz the intensity of light is minimum when the cosine angle is (2n+1)π

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But the ques was why -ve sign is used

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Introduction

In the phenomenon of destructive interference of light waves, (2n-1)π is used as a phase difference between the interfering waves. This phase difference plays a crucial role in determining the destructive interference pattern. Let's understand why (2n-1)π is used in detail.

Understanding Destructive Interference

When two or more waves superpose, they can either constructively interfere, resulting in an increased amplitude, or destructively interfere, leading to a decrease in amplitude. In the case of light waves, destructive interference occurs when the crests of one wave align with the troughs of another wave, resulting in cancellation of the wave amplitudes.

Phase Difference

The phase difference between two interfering waves determines whether they will constructively or destructively interfere. It represents the shift in the wave cycles between the two waves. In the case of destructive interference, the phase difference is crucial.

The Role of (2n-1)π

The value (2n-1)π is used to represent the phase difference in destructive interference. Here's why:

1. π Radians or 180 Degrees
- One complete cycle of a wave corresponds to 2π radians or 360 degrees.
- For destructive interference, we need the crests of one wave to align with the troughs of the other.
- This alignment occurs when the wave cycles are shifted by half a cycle, which is π radians or 180 degrees.

2. Integer Value of n
- The value of n represents the order of the destructive interference pattern.
- Each order of destructive interference corresponds to a different set of crests and troughs aligning.
- In order to cover all possible phase differences for destructive interference, we use different integer values of n.

3. (2n-1)π
- By using (2n-1)π, we ensure that the phase difference covers all possibilities for destructive interference.
- The term (2n) accounts for the complete cycles and the term (-1) represents the shift by half a cycle.
- This allows the wave cycles to align in a way that cancels out their amplitudes, resulting in destructive interference.

Conclusion

In summary, (2n-1)π is used in destructive interference of light waves because it represents the phase difference required for the crests and troughs of the interfering waves to align and cancel out each other's amplitudes. The term (2n) accounts for complete cycles, while the term (-1) represents the shift by half a cycle. By using different integer values of n, we cover all possibilities for destructive interference patterns.

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