Let denote a the width of the slit.
In single slit experiment amplitude of electric field of
light in direction α is calculated as superposition
of amplitudes of dE of small elements dx of the slit.
dE = Eo exp(2πi r(x) / λ) dx =
= Eo exp(2πi [x sin α] / λ) dx
E(α) = integral [-a/2 .. +a/2] of Eo exp(2πi [x sin α] / λ) dx =
= int [-a/2 .. +a/2] of 1/(2πi) λ/sinα Eo exp(2πi [x sin α] / λ) d [2πisinα/λ x]] =
= int [-πiasinα/λ .. +πiasinα/λ] of 1/(2πi) λ/sinα Eo exp(z) dz =
-1/π λ/sinα sin(π a sinα / λ) Eo
The intensity A(α) of light is proportional to square of electic
field E(α):
A(α) = 1/π² (λ/sinα)² sin²(πa sinα / λ) Eo² = Ao (λ/sinα)² sin²(πa sinα / λ)
Minima of intensity occur when
πa sinα / λ = n π, n = 1,2,3...
Answer:
a = n λ/sinα =
3 x 633 nm / (0.5 x 17.9 mm / 80 cm) = 170 μm
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