Which of the following best explains why destructive interference occurs at certain points in a double-slit experiment?

Question

BlackTom AI · Accepted Answer

Destructive interference occurs when two coherent waves arrive at a point exactly out of phase by π (180°), so their displacements cancel.

1) Relevant knowledge points and concepts
- Interference depends on phase difference Δφ between waves, which is related to path difference Δx by Δφ = (2π/λ)Δx.
- Constructive interference occurs when Δφ = 2mπ (m integer) → Δx = mλ.
- Destructive interference occurs when Δφ = (2m+1)π → Δx = (2m+1)λ/2.
- Coherence (same frequency and stable phase relationship) is required for stable interference patterns.

2) Complete reasoning step by step
- Start from the relation Δφ = (2π/λ)Δx.
- For cancellation, require Δφ = π, 3π, 5π, ... i.e. Δφ = (2m+1)π.
- Substitute: (2π/λ)Δx = (2m+1)π.
- Solve for Δx: Δx = ((2m+1)π) (λ/2π) = (2m+1)λ/2.

3) Calculations with intermediate steps
- Δφ = (2π/λ)Δx.
- Set Δφ = (2m+1)π.
- Δx = (2m+1)π × (λ/2π) = (2m+1)(λ/2).
- Example: m = 0 → Δx = λ/2 gives Δφ = π and total amplitude A_total = A − A = 0 (complete destructive cancellation for equal amplitudes).

4) Why the correct answer is correct
- Option d states destructive interference occurs when the path difference is an odd multiple of half a wavelength, i.e. Δx = (2m+1)λ/2, which exactly matches the derived condition for Δφ = (2m+1)π and thus cancellation.

5) Why the other options are incorrect
- a: Different frequencies do not produce a fixed destructive pattern; they produce beats or a time-varying interference, not steady cancellation at fixed positions.  
- b: Integer multiples of λ (Δx = mλ) give Δφ = 2mπ → constructive interference, not destructive.  
- c: Even multiples of λ are just special cases of integer multiples (2mλ = m'λ) and also give constructive interference.

6) Conclusion
This reasoning confirms option d ("Destructive interference occurs when the path difference between the two waves is an odd multiple of half the wavelength.") is correct.

Which of the following best explains why destructive interference occurs at certain points in a double-slit experiment?单项选择题

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