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Unformatted text preview: 11. (a) The intensity for a singleslit diﬀraction pattern is given by
I = Im sin2 α
α2 where α = (πa/λ) sin θ, a is the slit width and λ is the wavelength. The angle θ is measured from
the forward direction. We require I = Im /2, so
sin2 α = 12
α.
2 (b) We evaluate sin2 α and α2 /2 for α = 1.39 rad and compare the results. To be sure that 1.39 rad is
closer to the correct value for α than any other value with three signiﬁcant digits, we could also try
1.385 rad and 1.395 rad.
(c) Since α = (πa/λ) sin θ,
θ = sin−1 αλ
πa . Now α/π = 1.39/π = 0.442, so
θ = sin−1 0.442λ
a . The angular separation of the two points of half intensity, one on either side of the center of the
diﬀraction pattern, is
0.442λ
.
∆θ = 2θ = 2 sin−1
a
(d) For a/λ = 1.0,
for a/λ = 5.0,
and for a/λ = 10, ∆θ = 2 sin−1 (0.442/1.0) = 0.916 rad = 52.5◦ ,
∆θ = 2 sin−1 (0.442/5.0) = 0.177 rad = 10.1◦ ,
∆θ = 2 sin−1 (0.442/10) = 0.0884 rad = 5.06◦ . ...
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This note was uploaded on 11/12/2011 for the course PHYS 2001 taught by Professor Sprunger during the Fall '08 term at LSU.
 Fall '08
 SPRUNGER
 Physics

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