p37_011.pdf

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11. (a) The intensity for a single-slit diffraction pattern is given by
I
=
I
m
sin
2
α
α
2
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
=
I
m
/2,
so
sin
2
α
=
1
2
α .
2
(b) We evaluate sin
2
α
and
α
2
/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
Now
α/π
= 1.39/π = 0.442, so
θ
= sin
−1
0.442λ
a
.
αλ
πa
.
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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