INTERFERENCE AND DIFFRACTION
32
E
XERCISES
Section 32.2 Double-Slit Interference
10.
The experimental arrangement and geometrical approximations valid for Equation 32.2a are satisfied for the
situation and data given, so
bright
/
(7.1 cm/2.2 m)(15
m/1)
484 nm.
y
d mL
λ
μ
=
=
=
(In particular,
and
d
λ
<<
2
1
3.23
10
1.85
θ
−
=
×
=
°
is small.)
11.
I
NTERPRET
This problem is about double-slit interference. We are interested in the spacing between adjacent
bright fringes.
D
EVELOP
We assume that the geometrical arrangement of the source, slits, and screen is that for which
Equations 32.2a and 32.2b apply. The location of bright fringes is given by
bright
L
y
m
d
λ
=
where
m
is the order number.
E
VALUATE
The spacing of bright fringes is
(550 nm)(75 cm)
(
1)
1.65 cm
0.025 mm
L
L
L
y
m
m
d
d
d
λ
λ
λ
Δ
=
+
−
=
=
=
A
SSESS
Since
,
d
λ
<<
the spacing between bright fringes is much smaller than
L
, as it should.
12.
The particular geometry of this type of double-slit experiment is described in the paragraphs preceding Equations
32.2a and 32.2b.
(a)
The spacing of bright fringes on the screen is
/
, so
(0.12 mm)(5 mm)/
y
L d
L
λ
Δ
=
=
(633 nm)
94.8 cm.
=
(b)
For two different wavelengths, the ratio of the spacings is
/
/
y
y
;
λ λ
′
′
Δ
Δ
=
therefore
(5 mm)(480/633)
3.79 mm.
y
′
Δ
=
=
13.
I
NTERPRET
This problem is about double-slit interference. We are interested in the wavelength of the
light source.
D
EVELOP
For small angles, the interference fringes are evenly spaced, with
/
d
θ
λ
Δ
=
(see Equation 31.1a).
E
VALUATE
Substituting the values given, we obtain
(0.37 mm)(0.065
)(
/180
)
420 mm.
d
λ
θ
π
=
Δ
=
°
° =
A
SSESS
The wavelength
λ
is much smaller than the slit spacing
d
, as expected.
14.
The interference minima fall at angles given by Equation 32.1b; therefore
1
2
(4
)
/sin
4.5(546 nm)/
d
λ
θ
=
+
=
sin 0.113
1.25 mm.
° =
(Note that
gives the first dark fringe.)
0
m
=
Section 32.3 Multiple-Slit Interference and Diffraction Gratings
15.
I
NTERPRET
The setup is a multi-slit interference experiment. We want to know the number of minima
(destructive interferences) between two adjacent maxima.
D
EVELOP
In an
N
-slit system with slit separation
d
(illuminated by normally incident plane waves), the main
maxima occur for angles (see Equation 32.1a)
sin
/ ,
m
d
θ
λ
=
and minima for angles (see Equation 32.4)
sin
θ
=
/
m
Nd
λ
′
(excluding
equal to zero or multiples of
N
).
m
′
E
VALUATE
Between two adjacent maxima, say
and (
1)
,
m
mN
m
N
′ =
+
there are
1
N
−
minima. The number of
integers between
is
and (
1)
mN
m
N
+
(
1)
1
m
N
mN
N
1
+
−
−
=
−
32.1

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