Asked by kelvin calin
Which is the destructive interference formula for diffraction grating problems?
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For an N‑slit diffraction grating (slit spacing d) the intensity is
I ∝ (sin^2(Nβ))/(sin^2 β) with β = π d sinθ / λ.
Principal (constructive) maxima occur at d sinθ = m λ (m integer).
Destructive minima occur when the numerator vanishes but the denominator does not:
Nβ = nπ => d sinθ = (n/N) λ, with n an integer that is not a multiple of N.
(If n is a multiple of N the denominator also vanishes and you get the principal maximum condition above.)
Special cases:
- For a double slit (N = 2) this gives d sinθ = (n/2) λ, so odd n gives the familiar d sinθ = (m + 1/2) λ minima.
- For a single slit (slit width a) the single‑slit minima follow a sinθ = m λ (m ≠ 0), which is a different diffraction effect (envelope) superposed on the grating pattern.
I ∝ (sin^2(Nβ))/(sin^2 β) with β = π d sinθ / λ.
Principal (constructive) maxima occur at d sinθ = m λ (m integer).
Destructive minima occur when the numerator vanishes but the denominator does not:
Nβ = nπ => d sinθ = (n/N) λ, with n an integer that is not a multiple of N.
(If n is a multiple of N the denominator also vanishes and you get the principal maximum condition above.)
Special cases:
- For a double slit (N = 2) this gives d sinθ = (n/2) λ, so odd n gives the familiar d sinθ = (m + 1/2) λ minima.
- For a single slit (slit width a) the single‑slit minima follow a sinθ = m λ (m ≠ 0), which is a different diffraction effect (envelope) superposed on the grating pattern.
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