Download chapter 37 interference and diffraction

CHAPTER 37 INTERFERENCE AND DIFFRACTION
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Section 37-2:
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Section 37-3: Multiple-Slit Interference and
Diffraction Gratings
Double-Slit Interference
Problem
3. A double-slit experiment has slit spacing 0.12 mm.
(a) What should be the slit-tc-screen distance L if
the bright fringes are to be 5.0 mm apart when the
slits are illuminated with 633-nm laser light?
(b) What will be the fringe spacing with 480-nm
light?
Solution
The particular geometry of this type of double-slit
experiment is described in the paragraphs preceding
Equations 37-2a and b. (a) The spacing of bright
fringes on the screen is Ay = XL/d, so L =
(0.12 mm)(5
__-mm)/(633 nm] = 94.8 cm. (b) For two
different wavelengths, the ratio of the spacings is
a y f / A y = Af/A; therefore Ay' = (5 mm)(480/633) =
3.79 mm.
Problem
7. Light shines on a pair of slits whose spacing is three
times the wavelength. Find the locations of the
first- and second-order bright fringes on a screen
50 cm from the slits. Hint: Do Equations 37-2
apply7
Solution
Since d = 3X, the angles are not small, and Equations 37-2 do not apply. The interference maxima
occur a t angles given by Equation 37-la, 6' =
sin-](mX/d) = sill-'(m/3), so only two orders are
present, for values of m = 1 and 2 (6' < 90'). If we
assume that the slit/screen geometry is as shown in
Fig. 37-6, then y = L tan 6' = L tan(sin-' (77213)) =
~ m /
(Consider a right triangle with
hypotenuse of 3 and opposite side m, or use tan 6' =
sin 6'1 d m . )For m = 1 and 2, and L = 50 cm,
this gives yl = (50 cm)(l/&) = 17.7 cm, and
yz = (50 c m ) ( 2 / 4 ) = 44.7 cm.
Jw.
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Problem
9. For a double-slit experiment with slit spacing
0.25 mm and wavelength 600 nm, at what angular
position is the path difference equal to one-fourth
of the wavelength?
Solution
If we set the path difference equal to a quarter
wavelength, we obtain d sin 6' = X/4, or 6' w sin 6' =
600 nm/4(0.25 mm) = 6 x 1 0 ~rad~ 0.0344".
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Problem
13. In a 5-slit system, how many minima lie betwee*
the zeroth-order and first-order maxima?
Solution
In an N-slit system with slit separation d (illuminated
by normally incident plane waves), the main maxima
occur for angles sin 0 = mX/d, and minima for
sin 8 = mJX/Nd (excluding m' equal to zero or
multiples of N ). Between two adjacent maxima, say
m' = m N and (m 1)N, there are N - 1 minima.
(The number of integers between mN and (m + l ) N is
(m l ) N - m N - 1 = N - 1, because the limits are
not included.) For N = 5, the number of minima is 4.
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Problem
17. Green light at 520 nm is diffracted by a grating
with 3000 lines per cm. Through what angle is the
light diffracted in (a) first and (b) fifth order?
Solution
For light normally incident on a diffraction grating,
maxima occur a t angles 6' = sin-'(mX/d), where d is
the grating spacing (equal to the reciprocal of the
number of lines per meter), and m is the order. (a) In
first order, 6'' = sin-'(520 nm x 3000/cm) = 8.97",
and (b) in fifth order, O5 = sinq1(5 sin 8.97') = 51.3'.
Problem
23. Estimate the number of lines per cm in the
grating used to produce Fig. 37-15.
Solution
The number of lines per cm (lid in cm-') is easily
estimated from the angular position of the central
550-nm line in a particular order, as shown in the
figure; that is, l/d = sin 6'lmA. For example, in fifth
order, this line is at 6' = 61' (average of right and left
values), so l/d = sin61°/5(550 nm) = 3.18x103/cm or
about 3200 lines/cm.
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FIGURE 37-15 Problem 23 Solution.
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