This document summarizes problems from Chapter 37 on interference, diffraction, and polarization. It includes problems on Young's experiment involving constructive and destructive interference using different path lengths. Other problems cover diffraction gratings, resolving power of instruments, and the limits of angular resolution. Key concepts covered are the conditions for interference, using gratings to disperse light, and factors determining an instrument's ability to distinguish small details.

1. Chapter 37. Interference, Diffraction, and Polarization. Physics, 6th Edition
Chapter 37. Interference, Diffraction, and Polarization
Young’s Experiment; Interference
37-1. Light from a laser has a wavelength of 632 nm. Two rays from this source follow paths
that differ in length. What is the minimum path difference require to cause (a)
constructive interference, (b) destructive interference?
(a) Constructive: ∆p = nλ = (1)(632 nm); ∆p = 632 nm.
(b) Destructive: ∆p = nλ/2 = ½(632 nm); ∆p = 316 nm
37-2. Find the difference in path length required in Problem 37-1 to provide the very next
instances of constructive and destructive interference.
(a) 2nd Constructive: ∆p = nλ = (2)(632 nm); ∆p = 1264 nm.
3λ 3(632 nm)
(b) 2nd Destructive: ∆p = nλ/2 = = ; ∆p = 948 nm
2 2
37-3. Monochromatic light illuminates two parallel slits 0.2 mm apart. On a screen 1.0 m from
the slits, the first bright fringe is separated from the central fringe by 2.50 mm. What is
the wavelength of the light?
yd yd (2.5 mm)(0.2 mm)
= nλ ; λ= = ; λ = 500 nm
x nx (1)(1000 mm)
37-4. Monochromatic light from a sodium flame illuminates two slits separated by 1.0 mm. A
viewing screen is 1.0 m from the slits, and the distance from the central bright fringe to
the bright fringe nearest it is 0.589 mm. What is the frequency of the light?
yd c cx (3 x 108 m/s)(1 m)
= (1)λ = ; f = = ; f = 5.09 x 1014 Hz
x f yd (5.89 x 10-4 m)(0.001 mm)
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2. Chapter 37. Interference, Diffraction, and Polarization. Physics, 6th Edition
37-5. Two slits 0.05 mm apart are illuminated by green (λ = 520 nm). A diffraction pattern is
formed on a viewing screen 2.0 m away. What is the distance from the center of the screen
to the first bright fringe? What is the distance to the third dark fringe?
yd
The first bright fringe occurs when n = 1: = nλ ( n = 1, 2, 3, . . .)
x
(1)λ x (5.2 x 10-4 mm)(2000 mm)
y= = ; y = 20.8 mm
d (0.05 mm)
yd λ
The third dark fringe occurs when n = 5: =n (n = 1, 3, 5, . . .)
x 2
5λ x 5(5.2 x 10-4 mm)(2000 mm)
y= = ; y = 52.0 mm
2d (2)(0.05 mm)
*37-6. For the situation described in Problem 37-5, what is the separation of the two first-order
bright fringes located on each side of the central band?
(1)λ x
y= = 20.8 mm; ∆y = 2(20.8 mm) ; ∆y = 41.6 mm
d
*37-7. Young’s experiment is performed using monochromatic light of wavelength 500 nm. The
slit separation is 1.20 mm and the screen is 5.00 m away. How far apart are the bright
fringes? The separation of each fringe is same as for n = 1.
(1)λ x (5.00 x 10-7 m)(5.00 m)
y= = ; y = 2.08 mm
d (1.2 x 10-3m)
*37-8. In Young’s experiment it is noted that the second dark fringe appears at a distance of 2.5
cm from the central bright fringe. Assume that the slit separation is 60 µm and that the
screen is 2.0 m away. What is the wavelength of the incident light? (Note that n = 3)
yd λ 2 yd 2(0.025 m)(60 x 10-6 m)
= (3) ; λ= = ; λ = 500 nm
x 2 3x (3)(2.0 m)
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3. Chapter 37. Interference, Diffraction, and Polarization. Physics, 6th Edition
The Diffraction Grating
37-9. A diffraction grating having 300 lines/mm is illuminated by light of wavelength 589 nm.
What are the angles at which the first and second-order bright fringes are formed?
1
d= = 3.33 x 10-3 mm or d = 3.33 x 10-6 m
300 lines/mm
nλ (1)(589 x 10-9 m)
d sin θ = nλ ; sin θ = = ; θ = 10.20
d 3.33 x 10-6 m
nλ (2)(589 x 10-9 m)
d sin θ = nλ ; sin θ = = ; θ = 20.70
d 3.33 x 10-6 m
37-10. A diffraction grating has 250,000 lines/m. What is the wavelength of incident light if the
second order bright fringe occurs at 12.60.
1 d sin θ
d= 5
= 4.00 x 10-6 m; d sin θ = nλ ; λ=
2.50 x 10 lines/m n
(4.00 x 10-6 m) sin12.60
λ= ; λ = 436 nm
2
37-11. A small sodium lamp emits light of wavelength 589 nm, which illuminates a grating
marked with 6000 lines/cm. Calculate the angular deviation of the first and second order
bright fringes?
1
d= = 1.67 x 10-4 cm or d = 1.67 x 10-6 m
6000 lines/cm
nλ (1)(589 x 10-9 m)
d sin θ = nλ ; sin θ = = ; θ = 20.70
d 1.67 x 10-6 m
nλ (2)(589 x 10-9 m)
d sin θ = nλ ; sin θ = = ; θ = 45.00
d 1.67 x 10-6 m
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4. Chapter 37. Interference, Diffraction, and Polarization. Physics, 6th Edition
37-12. A parallel beam of light illuminates a diffraction grating with 6000 lines/cm. The second-
order bright fringe is located 32.0 cm from the central image on a screen 50 cm from the
grating. Calculate the wavelength of the light?
1
d= = 1.67 x 10-6 m; d sin θ = nλ (n = 1, 2, 3, . . .)
6000 lines/cm
32 cm d sin θ (1.67 x 10-6 m)sin32.60
tan θ = ; θ = 32.60 ; λ= = ; λ = 449 nm
50 cm 2 2
*37-13. The visible light spectrum ranges in wavelength from 400 to 700 nm. find the angular
width of the first-order spectrum produced by passing white light through a grating
marked with 20,000 lines/in. ( Angular width = θ700 - θ400 )
1  1m 
d= = 5.00 x 10-5 in.; d = (5 x 10-5in.)   = 1.27 x 10 m
-6
20, 000 lines/in.  39.37 in. 
nλ4 (1)(400 x 10-9 m)
d sin θ = nλ ; sin θ 4 = = ; θ4 = 18.40
d 1.27 x 10-6 m
nλ7 (1)(700 x 10-9 m)
d sin θ = nλ ; sin θ 7 = = ; θ7 = 33.40
d 1.27 x 10-6 m
Angular width = 33.40 – 18.40; ∆θ = 15.00
*37-14. An infrared spectrophotometer uses gratings to disperse infrared light. One grating is
ruled with 240 lines/mm. What is the maximum wavelength that can be studied with this
grating? (The maximum dispersion occurs when θ = 900 and sin θ = 1 )
1
d= = 4.17 x 10-3 mm or d = 4.17 x 10-6 m
240 lines/mm
d sin θ (4.17 x 10-6 m)(1)
λ= = ; λmax = 4.17 µm
1 1
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5. Chapter 37. Interference, Diffraction, and Polarization. Physics, 6th Edition
Resolving Power of Instruments
37-15. Light of wavelength 600 nm falls on a circular opening of diameter 0.32 mm. A
diffraction pattern forms on a screen 80 cm away. What is the distance from the center of
the pattern to the first dark fringe?
R 1.22λ 1.22λ f 1.22(600 x 10-9 m)(0.80 m)
θ0 = = ; R= = ; R = 1.83 mm
f D D 0.32 x 10-3 m
37-16. The limiting angle of resolution for an objective lens in an optical instrument is 3 x 10-4
rad for a light 650 nm light source. What is the diameter of the circular opening?
1.22λ 1.22(650 x 10-9 m)
θ0 = ; D= ; D = 2.64 mm
D 3 x 10-4 rad
37-17. A certain radio telescope has a parabolic reflector that is 70 m in diameter. Radio waves
from outer space have a wavelength of 21 cm. Calculate the theoretical limit of resolution
for this telescope?
1.22λ 1.22(0.21 m)
θ0 = = ; θo = 3.66 x 10-3 rad
D 70 m
37-18. Using a telescope whose objective lens has a diameter of 60 m, how far apart can two
objects be resolved if they are located a distance in space equal to that from the earth to
the sun (93 million miles)?
Assume that light is at the central wavelength of 500 nm
1.22λ s0 1.22λ p 1.22(500 x 10-9 m)(9.3 x 107 mi)
θ0 = = ; s0 = = ;
D p D 60 m
so = 0.946 mi or 4,990 ft
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6. Chapter 37. Interference, Diffraction, and Polarization. Physics, 6th Edition
37-19. What is the angular limit of resolution of a person’s eye when the diameter of the opening
is 3 mm? Assume that the wavelength of the light is 500 nm.
1.22λ 1.22(500 x 10-9 m)
θ0 = = ; θo = 2.03 x 10-4 rad
D 3 x 10-3m
37-20. At what distance could the eyes of Problem 37-19 resolve wires in a door screen that are
separated by 2.5 mm?
s0 s0 2.5 x 10-3 m
θ0 = ; p= = ; p = 12.3 m
p θ 0 2.03 x 10-4 rad
Challenge Problems
37-21. In Young’s experiment 600-nm light illuminates a slit located 2.0 m from a screen. The
second bright fringe formed on the screen is 5 mm from the central maximum. What is
the slit width?
yd nλ x 2(600 x 10-9 m)(2.0 m)
= nλ ; d = = ; d = 480 µm
x y 5 x 10-3m
37-22. A transmission grating is ruled with 5000 lines/cm. For light of wavelength 550 nm,
what is the angular deviation of the third order bright fringe?
1
d= = 2.00 x 10-4 cm or d = 2.00 x 10-6 m
5000 lines/cm
nλ (3)(550 x 10-9 m)
d sin θ = nλ ; sin θ = = ; θ = 55.60
d 2.00 x 10-6 m
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7. Chapter 37. Interference, Diffraction, and Polarization. Physics, 6th Edition
37-23. Monochromatic light passes through two slits separated by 0.24 mm. In the pattern
formed on a screen 50 cm away, the distance between the first bright fringe of the left of
the central maximum and the first fringe on the right is 2.04 mm. What is the wavelength
of the light? (The deviation of the first bright fringe is half of 2.04 mm or 1.02 mm.)
yd yd (1.02 x 10-3 m)(0.24 x 10-3 m)
= nλ ; λ = = ; λ = 490 nm
x nx (1)(0.50 m)
37-24. A transmission grating ruled with 6000 lines/cm forms a second-order bright fringe at an
angle of 530 from the central fringe. What is the wavelength of the incident light?
1
d= = 1.67 x 10-4 cm or d = 1.67 x 10-6 m
6000 lines/cm
d sin θ (1.67 x 10-6 m) sin 530
d sin θ = nλ ; λ= = ; λ = 667 nm
n 2
*37-25. If the separation of the two slits in Young’s experiment is 0.10 mm and the distance to
the screen is 50 cm, find the distance between the first dark fringe and the third bright
fringe when the slits are illuminated with light of wavelength 600 nm.
yd yd λ
Bright : = nλ (n = 1, 2, 3, . . .) Dark : =n (n = 1, 3, 5, . . .)
x x 2
(3)λ x (1)λ x  λx 
∆y = yB 3 − yD1 = − = (2.5)  ;
d 2d  d 
2.5(600 x 10-9 m)(0.5 m)
∆y = ; ∆y = 7.50 mm
0.10 x 10-3 m
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8. Chapter 37. Interference, Diffraction, and Polarization. Physics, 6th Edition
*37-26. Light from a mercury-arc lamp is incident on a diffraction grating ruled with 7000 lines
per inch. The spectrum consists of a yellow line (579 nm) and a blue line (436 nm).
Compute the angular separation (in radians) of these lines in the third-order spectrum.
1  1m 
d= = 1.43 x 10-4 in.; d = (1.43 x 10-4in.)   = 3.63 x 10 m
-6
7000 lines/in.  39.37 in. 
nλ y (3)(579 x 10-9 m)
d sin θ = nλ ; sin θ y = = ; θ4 = 28.60
d 3.63 x 10-6 m
nλb (3)(436 x 10-9 m)
d sin θ = nλ ; sin θ b = = ; θ7 = 21.10
d 3.63 x 10-6 m
∆θ = 28.60 – 21.10; ∆θ = 7.480
*37-27. A telescope will be used to resolve two points on a mountain 160 km away. If the
separation of the points is 2.0 m, what is the minimum diameter for the objective lens?
Assume that the light has an average wavelength of 500 nm.
1.22λ s0 1.22λ p 1.22(500 x 10-9 m)(1.6 x 105 m)
θ0 = = ; D= = ; D = 4.86 cm
D p s0 2m
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9. Chapter 37. Interference, Diffraction, and Polarization. Physics, 6th Edition
Critical Thinking Problems
37-28. A Michelson interferometer, as shown in Fig. 37-18, can be used to measure small
distance. The beam splitter partially reflects and partially transmits monochromatic light
of wavelength λ from the source S. One mirror M1 is fixed and another M2 is movable.
The light rays reaching the eye from each mirror differ, causing constructive and
destructive interference patterns to move across the scope as the mirror M2 is moved a
distance x. Show that his distance is given by
λ
x=m
2
where m is the number of dark fringes that cross an indicator line on the scope as the
M2
mirror moves a distance x. x
The difference in path lengths for each ray of
light will be the distance x and back, or 2x, Source.S
M1
Thus, there is constructive interference when:
Fixed
2x = mλ m = 0, 1, 2, . . . mirror
λ Observer
Thus, x=m m = 0, 1, 2, . . .
2
37-29. A Michelson interferometer (see previous problem) is used to measure the advance of a
small screw. How far has the screw advanced if krypton-86 light (λ = 606 nm) is used
and 4000 fringes move across the field of view as the screw advances?
mλ (4000)(606 x 10-9 m)
x= = ; x = 1.21 mm
2 2
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10. Chapter 37. Interference, Diffraction, and Polarization. Physics, 6th Edition
*37-30. A diffraction grating has 500 lines/mm ruled on its glass surface. White light passes
through the grating and forms several spectra on a screen 1.0 m away. Is the deviation of
colors with a grating different than those experienced for prisms? On the screen, what is
the distance between the first order blue line (400 nm) and the first order red line (680
nm). How many complete spectra (400-700 nm) are possible for these conditions?
(a) When white light passes through a prism, the blue light bends more than the red
light since the angle of refraction is less for longer wavelengths. (See Fig. 35-8 in
text.) The reverse is true for gratings. The longer wavelengths are deviated more.
1
(b) d = = 2.00 x 10-3 mm; d = 2.00 x 10-6 m
500 lines/mm
θ
nλr (1)(680 x 10-9 m) 1m
d sin θ = nλ ; sin θ r = = ; θr = 19.90
d 2.00 x 10-6 m
nλb (1)(400 x 10-9 m) Red Blue
d sin θ = nλ ; sin θ b = = ; θ7 = 11.50
d 2.00 x 10-6 m
y
tan θ = ; y = x tan θ ; ∆y = yr − yb = x tan θ r − x tan θ b
x
∆y = (1 m) tan 19.90– (1 m) tan 11.50; ∆y = 15.9 cm
(c) The maximum number of complete spectra is found by setting θ = 900 for red line.
d sin 900 2.00 x 10-6 m
d sin θ = nλ ; n= = ; n = 2.86
λ 700 x 10-9 m
Therefore, only two complete spectra are possible for this grating.
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11. Chapter 37. Interference, Diffraction, and Polarization. Physics, 6th Edition
*37-31. The tail lights of an automobile are separated by 1.25 m. Assume that the pupil of a
person’s eye has a diameter of 5 mm and that the light has an average wavelength of 640
nm. At night, on a long straight highway, how far away can the two tail lights be
resolved? Suppose you squint your eyes forming a slit where the limiting angle changes
from θ0 = 1.22 λ/D to θ0 = λ/d. What is the new distance for resolution of the images?
1.22λ s0 s0 D (1.25 m)(0.005 m)
θ0 = = ; p= = ; p = 8000 m
D p 1.22λ 1.22(640 x 10-9 m)
λ s0 s0 D (1.25 m)(0.005 m)
θ0 = = ; p= = ; p = 9765 m
d p λ (640 x 10-9 m)
*37-32. The intensity of unpolarized light is reduced by one-half when it passes through a
polarizer. In the case of the plane-polarized light reaching the analyzer, the intensity I of
the transmitted beam is given by
I = I 0 cos2 θ
where I0 is the maximum intensity transmitted and θ is the angle through which the
analyzer has been rotated. Consider three Polaroid plates stacked so that the axis of each
is turned 300 with respect to the spreading plate. By what percentage will the incident
light be reduced in intensity when it passes through all three plates?
The first plate transmits ½ of the incident light intensity. Each succeeding plate
transmits a vector resolution at angle 300 and transmits a fraction, cos 300 = 0.866, of the
amplitude, or (cos 300)2 of the intensity. Since cos 300 = 0.866, (cos 300)2 = 0.75.
Thus the intensity of the stack is:
(0.5)(0.75)(0.75 = 0.281 or 28.1%
264