serway1 (1)

Image Formation 351

P36.24 See the solution.

P36.26 1.50 cm s

P36.28 20.0 cm

P36.30 (a) 6.40 cm (b) −0.250 (c) converging

P36.32 See the solution.

P36.34 (a) 39.0 mm (b) 39.5 mm

P36.36 1.16 mm s toward the lens

P36.38 (a) qa = 26.2 cm qd = 46.7 cm hbЈ = −8.75 cm hcЈ = −23.3 cm. See the solution. (b) The equa-
tion follows from hЈ/h = −q/p and 1/p + 1/q = 1/f. (c) The integral stated adds up the areas of

ribbons covering the whole image, each with vertical dimension | hЈ | and horizontal width dq.

328 cm2

P36.40 (a) at q = −34.7 cm virtual, upright, and diminished (b) at q = −36.1 cm virtual, upright, and
diminished

P36.42 f
1.41

P36.44 23.2 cm

P36.46 −575

P36.48 (a) See the solution. (b) h′ = − hf (c) −1.07 mm
P36.50 p

(a) 67.5 cm (b) The lenses can be displaced in two ways. The first lens can be displaced 1.28 cm
farther away from the object, and the second lens 17.7 cm toward the object. Alternatively, the first
lens can be displaced 0.927 cm toward the object and the second lens 4.44 cm toward the object.

P36.52 if M < 1, f = −Md , if M > 1, f = Md

(1 − M )2 ( M − 1)2

P36.54 160 cm to the left of the lens, inverted, M = −0.800

P36.56 25.3 cm to right of mirror, virtual, upright, enlarged 8.05 times

P36.58 (a) 263 cm (b) 79.0 cm

P36.60 (a) 1.40 kW m2 (b) 6.91 mW m2 (c) 0.164 cm (d) 58.1 W m2

P36.62 (a) 0.334 m or larger (b) Ra = 0.025 5 or larger (c) inverted
P36.64 R

(a) 1.99 (b) 10.0 cm to the left of the lens; Ϫ2.50

P36.66 See the solution; real, inverted, and actual size.

P36.68 See the solution.



37

Interference of Light Waves

CHAPTER OUTLINE ANSWERS TO QUESTIONS

37.1 Conditions for Interference Q37.1 (a) Two waves interfere constructively if their
37.2 Young’s Double-Slit Experiment
37.3 Light Waves in Interference path difference is zero, or an integral multiple
37.4 Intensity Distribution of the of the wavelength, according to δ = mλ, with
m = 0, 1, 2, 3, . . . .
Double-Slit Interference Pattern
37.5 Change of Phase Due to (b) Two waves interfere destructively if their path

Reflection
37.6 Interference in Thin Films
37.7 The Michelson Interferometer

difference is a half wavelength, or an odd

multiple of λ, described by δ = ⎛ m + 1⎞ λ, with
2 ⎝ 2⎠

m = 0, 1, 2, 3, . . . .

Q37.2 The light from the flashlights consists of many different
wavelengths (that’s why it’s white) with random time
differences between the light waves. There is no
coherence between the two sources. The light from
the two flashlights does not maintain a constant phase
relationship over time. These three equivalent statements
mean no possibility of an interference pattern.

*Q37.3 (i) The angles in the interference pattern are controlled by λ/d, which we estimate in each case:
(a) 0.45 µm/400 µm ≈ 1.1 × 10−3 (b) 0.7 µm/400 µm ≈ 1.6 × 10−3 (c) and (d) 0.7 µm/800 µm ≈
0.9 × 10−3. The ranking is b > a > c = d.

(ii) Now we consider Lλ/d: (a) 4 m (0.45 µm/400 µm) ≈ 4.4 mm (b) 4 m (0.7 µm/400 µm) ≈
7 mm (c) 4 m(0.7 µm/800 µm) ≈ 3 mm (d) 8 m(0.7 µm/800 µm) ≈ 7 mm. The ranking is
b = d > a > c.

*Q37.4 Yes. A single beam of laser light going into the slits divides up into several fuzzy-edged beams
diverging from the point halfway between the slits.

*Q37.5 Answer (c). Underwater, the wavelength of the light decreases according to λwater = λair . Since
nwater

the angles between positions of light and dark bands are proportional to λ, the underwater fringe

separations decrease.

Q37.6 Every color produces its own pattern, with a spacing between the maxima that is characteristic
of the wavelength. With white light, the central maximum is white. The first side maximum is
a full spectrum with violet on the inner edge and red on the outer edge on each side. Each side
maximum farther out is in principle a full spectrum, but they overlap one another and are hard to
distinguish. Using monochromatic light can eliminate this problem.

*Q37.7 With two fine slits separated by a distance d slightly less than λ, the equation d sin θ = 0 has
the usual solution θ = 0. But d sin θ = 1λ has no solution. There is no first side maximum.

d sin θ = (1/2)λ has a solution. Minima flank the central maximum on each side. Answer (b).

353

354 Chapter 37

Q37.8 As water evaporates from the “soap” bubble, the thickness of the bubble wall approaches
zero. Since light reflecting from the front of the water surface is phase-shifted 180° and light
reflecting from the back of the soap film is phase-shifted 0°, the reflected light meets the
conditions for a minimum. Thus the soap film appears black, as in the textbook illustration
accompanying this question.

*Q37.9 Answer (b). If the thickness of the oil film were smaller than half of the wavelengths of visible
light, no colors would appear. If the thickness of the oil film were much larger, the colors would
overlap to mix to white or gray.

*Q37.10 (i) Answer (b). If the oil film is brightest where it is thinnest, then nair < noil < nglass. With this
condition, light reflecting from both the top and the bottom surface of the oil film will undergo
phase reversal. Then these two beams will be in phase with each other where the film is very thin.
This is the condition for constructive interference as the thickness of the oil film decreases toward
zero. If the oil film is dark where it is thinnest, then nair < noil > nglass. In this case, reflecting light
undergoes phase reversal upon reflection from the front surface but no phase reversal upon
reflection from the back surface. The two reflected beams are 180° out of phase and interfere
destructively as the oil film thickness goes to zero.

(ii) Yes. It should be lower in index than both kinds of glass.

(iii) Yes. It should be higher in refractive index than both kinds of glass.

(iv) No.

Q37.11 If R is large, light reflecting from the lower surface of the lens can interfere with light reflect-
ing from the upper surface of the flat. The latter undergoes phase reversal on reflection while
the former does not. Where there is negligible distance between the surfaces, at the center of the
pattern you will see a dark spot because of the destructive interference associated with the 180°
phase shift. Colored rings surround the dark spot. If the lens is a perfect sphere the rings are
perfect circles. Distorted rings reveal bumps or hollows on the fine scale of the wavelength of
visible light.

Q37.12 A camera lens will have more than one element, to correct (at least) for chromatic aberration. It
will have several surfaces, each of which would reflect some fraction of the incident light. To
maximize light throughput the surfaces need antireflective coatings. The coating thickness is
chosen to produce destructive interference for reflected light of some wavelength.

*Q37.13 (i) Answer (c). The distance between nodes is one-half the wavelength.

(ii) Answer (d). Moving one mirror by 125 nm lengthens the path of light reflecting from it by
250 nm. Since this is half a wavelength, the action reverses constructive into destructive
interference.

(iii) Answer (e). The wavelength of the light in the film is 500 nm /2 = 250 nm. If the film is made
62.5 nm thicker, the light reflecting inside the film has a path length 125 nm greater. This is half a
wavelength, to reverse constructive into destructive interference.

*Q37.14 Answer (a). If the mirrors do not move the character of the interference stays the same. The light
does not get tired before entering the interferometer or undergo any change on the way from the
source to the half-silvered mirror.

Interference of Light Waves 355

SOLUTIONS TO PROBLEMS

Section 37.1 Conditions for Interference

Section 37.2 Young’s Double-Slit Experiment

Section 37.3 Light Waves in Interference

( )∆ybright
P37.1 = λL = 632.8 × 10−9 (5.00) m= 1.58 cm
P37.2 d
2.00 × 10−4

=( )( )ybrightλLm
d
λ = yd = 3.40 × 10−3 m 5.00 × 10−4 m = 515 nm
For m = 1, L 3.30 m

P37.3

400 m

300 m 1 000 m

FIG. P37.3

Note, with the conditions given, the small angle approximation does not work well. That is, sinθ ,
tanθ , and θ are significantly different. We treat the interference as a Fraunhofer pattern.

(a) At the m = 2 maximum, tanθ = 400 m = 0.400
1 000 m

θ = 21.8°

So λ = d sinθ = (300 m)sin 21.8° = 55.7 m

m2

(b) The next minimum encountered is the m = 2 minimum, and at that point,

d sin θ = ⎛ m + 1⎞ λ
⎝ 2⎠

which becomes d sinθ = 5 λ
2

or sin θ = 5 λ = 5 ⎛ 55.7 m ⎞ = 0.464
2 d 2 ⎝ 300 m ⎠

and θ = 27.7°

so y = (1 000 m) tan 27.7° = 524 m

Therefore, the car must travel an additional 124 m .
If we considered Fresnel interference, we would more precisely find

( )(a) λ = 1 5502 + 1 0002 − 2502 + 1 0002 = 55.2 m and (b) 123 m
2

356 Chapter 37

P37.4 λ = v = 354 m s = 0.177 m
f 2 000 s−1

(a) d sinθ = mλ so (0.300 m)sinθ = 1(0.177 m) and θ = 36.2°

(b) d sinθ = mλ so d sin 36.2° = 1(0.030 0 m) and d = 5.08 cm

)((c) 1.00 × 10−6 m sin 36.2° = (1) λ so λ = 590 nm

f = c = 3.00 × 108 m s = 508 THz
λ 5.90 × 10−7 m

P37.5 In the equation d sin θ = ⎛ m + 1 ⎞ λ y
⎝ 2 ⎠ L
d
The first minimum is described by m = 0

and the tenth by m = 9: sin θ = λ ⎛ 9 + 1⎞
d ⎝ 2⎠

Also, tanθ = y Source
L

but for small θ, sinθ ≈ tanθ

Thus, d = 9.5λ = 9.5λ L
sinθ y
FIG. P37.5
( )d
= 9.5 5 890 × 10−10 m (2.00 m) = 1.54 × 10−3 m = 1.54 mm

7.26 × 10−3 m

*P37.6 Problem statement: A single oscillator makes the two speakers of a boom box, 35.0 cm apart,
vibrate in phase at 1.62 kHz. At what angles, measured from the perpendicular bisector of the line
joining the speakers, will a distant observer hear maximum sound intensity? Minimum sound
intensity? The ambient temperature is 20°C.

We solve the first equation for λ, substitute into the others, and solve for each angle to find this
answer: The wavelength of the sound is 21.2 cm. Interference maxima occur at angles of 0° and
37.2° to the left and right. Minima occur at angles of 17.6° and 65.1°. No second-order or higher-
order maximum exists. No angle exists, smaller or larger than 90°, for which sinθ2 loud = 1.21. No
location exists in the Universe that is two wavelengths farther from one speaker than from the other.

P37.7 (a) For the bright fringe, y
L = 1.20 m
ybright = mλ L where m = 1 Source d = 0.250 m
d

( )y =
546.1 × 10−9 m (1.20 m) = 2.62 × 10−3 m = 2.62 mm

0.250 × 10−3 m

(b) For the dark bands, ydark = λL ⎛ m + 1⎞ ; m = 0, 1, 2, 3, .. . bright bright bright
d ⎝ 2⎠ dark dark dark

y2 − y1 = λL ⎣⎢⎡⎛⎝1 + 1⎞ − ⎛ 0 + 1⎞ ⎤ = λL (1) y1
d 2⎠ ⎝ 2⎠ ⎦⎥ d y2

(546.1 × 10−9 m)(1.20 m) FIG. P37.7

= 0.250 × 10−3 m

∆y = 2.62 mm

Interference of Light Waves 357

P37.8 Location of A = central maximum,

Location of B = first minimum.

[ ]So, = λL ⎛ + 1⎞ − = 1 λL =
∆y = ymin − ymax d ⎝ 0 2⎠ 0 2 d 20.0 m

Thus, d = λL = (3.00 m)(150 m) = 11.3 m

2 ( 20.0 m) 40.0 m

P37.9 Taking m = 0 and y = 0.200 mm in Equations 37.3 and 37.4 gives

( )( )L ≈
2dy = 2 0.400 × 10−3 m 0.200 × 10−3 m = 0.362 m Bright
λ 442 × 10−9 m Dark
0.2 mm Bright
L ≈ 36.2 cm 0.2 mm Dark
Bright
Geometric optics or a particle theory of light would incorrectly L
predict bright regions opposite the slits and darkness in between.
But, as this example shows, interference can produce just the opposite.

FIG. P37.9

P37.10 At 30.0°, d sinθ = mλ

) )( (3.20 × 10−4 m sin 30.0° = m 500 × 10−9 m so m = 320

There are 320 maxima to the right, 320 to the left, and one for m = 0 straight ahead.

There are 641 maxima .

P37.11 Observe that the pilot must not only home in on the airport, but must be headed in the right
direction when she arrives at the end of the runway.

(a) λ = c = 3 × 108 m s = 10.0 m
f 30 × 106 s−1

(b) The first side maximum is at an angle given by d sinθ = (1) λ.

(40 m)sinθ = 10 m θ = 14.5° tanθ = y
L

y = L tanθ = (2 000 m) tan14.5° = 516 m

(c) The signal of 10-m wavelength in parts (a) and (b) would show maxima at 0°, 14.5°, 30.0°,

48.6°, and 90°. A signal of wavelength 11.23 m would show maxima at 0°, 16.3°, 34.2°,

and 57.3°. The only value in common is 0°. If λ1 and λ2 were related by a ratio of small

integers (a just musical consonance!) in λ1 = n1 , then the equations d sinθ = n2λ1 and
λ2 n2

d sinθ = n1λ2 would both be satisfied for the same nonzero angle. The pilot could come flying

in with that inappropriate bearing, and run off the runway immediately after touchdown.

P37.12 In d sinθ = mλ d y = mλ y = mλL
L d

)(dy
)(dt
= mλ dL = 1 633 × 10−9 m 3 m s= 6.33 mm s
d dt 0.3 × 10−3 m

358 Chapter 37

P37.13 φ = 2π d sin θ = 2π d ⎛ y⎞
λ λ ⎝ L⎠

2π
5.00 × 10−7 m
) )( ((a)
φ= 1.20 × 10−4 m sin (0.500°) = 13.2 rad

2π ⎛ 5.00 × 10−3 m⎞
5.00 × 10−7 m ⎜⎝ 1.20 m ⎠⎟
( ) ( )(b)
φ= 1.20 × 10−4 m = 6.28 rad

( ( ) )θ
(c) If φ = 0.333 rad = 2π d sinθ = sin−1 ⎛ λφ ⎞ ⎡ 5.00 × 10−7 m (0.333 rad) ⎤
λ ⎜⎝ 2π d ⎟⎠ = sin−1 ⎢
2π 1.20 × 10−4 m ⎥
⎣⎢ ⎥⎦

θ = 1.27 × 10−2 deg

(d) If d sinθ = λ sin−1 ⎛ λ ⎞ sin−1 ⎡ 5 × 10−7 m ⎤
4 ⎜⎝ 4d ⎟⎠ ⎢ 1.20 × 10−4 m ⎥
( )θ= = ⎣⎢ 4 ⎥⎦

θ = 5.97 × 10−2 deg

P37.14 The path difference between rays 1 and 2 is: δ = d sinθ1 − d sinθ2
For constructive interference, this path difference must be equal to an integral number of
wavelengths: d sinθ1 − d sinθ2 = mλ, or

d (sinθ1 − sinθ2 ) = mλ

*P37.15 (a) The path difference δ = d sinθ and when L >> y
(b)
( )( )δ = yd = 1.80 × 10−2 m 1.50 × 10−4 m = 1.93 × 10−6 m = 1.93 µm
L 1.40 m

δ = 1.93 × 10−6 m = 3.00, or δ = 3.00λ
λ 6.43 × 10−7 m

(c) Point P will be a maximum because the path difference is an integer multiple of the
wavelength.

Section 37.4 Intensity Distribution of the Double-Slit Interference Pattern

P37.16 (a) I = cos2 ⎛ φ ⎞ (Equation 37.12)
I max ⎝ 2 ⎠ φ = 2 cos−1 I = 2 cos−1 0.640 = 1.29 rad

Therefore, I max

(b) δ = λφ = (486.1 nm)(1.29 rad) = 99.6 nm

2π 2π

Interference of Light Waves 359

P37.17 Iav = I max cos2 ⎛ π d sinθ ⎞
⎝ λ⎠

For small θ, sinθ = y
L

and Iav = 0.750Imax

y = λ L cos−1 Iav
π d Imax

( ( ) )y =
6.00 × 10−7 (1.20 m ) cos−1 0.750Imax = 48.0 µm
I max
π 2.50 × 10−3 m

P37.18 I = I max cos2 ⎛ π yd ⎞
⎝ λL ⎠

( )( )I
( )Imax
= cos2 ⎡π 6.00 × 10−3 m 1.80 × 10−4 m ⎤
⎢ ⎥= 0.968
⎣⎢ 656.3 × 10−9 m (0.800 m) ⎥⎦

*P37.19 We use trigonometric identities to write

6 sin(100pt) + 8 sin(100pt + p/2) = 6 sin(100pt) + 8 sin(100pt) cos(p/2) + 8 cos(100pt)sin(p/2)

E1 + E2 = 6 sin(100p t) + 8 cos(100p t)
and ER sin(100p t + φ ) = ER sin(100p t) cos φ + ER cos (100p t) sin φ

The equation E1 + E2 = ER sin(100p t + φ ) is satisfied if we require just

6 = E cos φ and 8 = E sin φ
R R

or 62 + 82 = ER2(cos2 φ + sin2 φ) ER = 10

and tan φ = sin φ/cos φ = 8/6 = 1.33 φ = 53.1°

*P37.20 In Iav = I max cos2 ⎛ π d sinθ ⎞ for angles between −0.3° IրImax
⎝ λ⎠

and +0.3° we take sin θ = θ to find

I = I max cos2 ⎛ π 250 µm θ ⎞ I/Imax = cos2 (1438 θ)
⎜⎝ 0.5461 µm ⎟⎠
−0.2 0 0.2
This equation is correct assuming θ is in radians; but q , degrees

we can then equally well substitute in values for θ in FIG. P37.20
degrees and interpret the argument of the cosine function

as a number of degrees. We get the same answers for θ negative and for θ positive. We evaluate

θ, degrees −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

I/I 0.101 1.00 0.092 0.659 0.652 0.096 1 0.096 0.652 0.659 0.092 1.00 0.101
max

The cosine-squared function has maximum values of 1 at θ = 0, at 1438 θ = 180° with θ = 0.125°,
and at 1438 θ = 360° with θ = 0.250°. It has minimum values of zero halfway between the maximum

values. The graph then has the appearance shown.

360 Chapter 37

P37.21 (a) From Equation 37.9,

φ = 2π d sinθ = 2π d ⋅ y
λ λ y2 + D2

)) )( ( (φ
≈ 2π yd = 2π 0.850 × 10−3 m 2.50 × 10−3 m = 7.95 rad
λD
600 × 10−9 m (2.80 m)

(b) I = cos2 ⎣⎡(π d λ )sinθ ⎤⎦ = cos2 (φ 2)
I max cos2 ⎣⎡(π d λ )sinθmax ⎤⎦
cos2 mπ

I = cos2 φ = cos2 ⎛ 7.95 rad ⎞ = 0.453
I max 2 ⎝ 2 ⎠

*P37.22 (a) The resultant amplitude is

Er = E0 sin ω t + E0 sin (ω t + φ)+ E0 sin (ω t + 2φ ), where φ = 2π d sin θ
λ

Er = E0 (sinω t + sinω t cos φ + cosω t sin φ + sinω t cos 2φ + cosω t sin 2φ )

)Er = E0 (sinω t )(1 + cosφ + 2 cos2 φ − 1 + E0 (cosω t )(sin φ + 2 sin φ cosφ)

Er = E0 (1 + 2 cos φ)(sinω t cosφ + cosω t sin φ) = E0 (1 + 2 cosφ)sin (ω t + φ)

Then the intensity is I ∝ Er2 = E02 (1 + 2 cos φ )2 ⎛ 1⎞
⎝ 2⎠

where the time average of sin2 (ω t + φ ) is 1.

2

From one slit alone we would get intensity I max ∝ E02 ⎛ 1⎞ so
⎝ 2⎠

I = I max ⎢⎣⎡1 + 2 cos ⎛ 2π d sinθ ⎞ ⎤2
⎝ λ ⎠ ⎥⎦

(b) Look at the N = 3 graph in the textbook Figure 37.8. Minimum intensity is zero, attained
where cos φ = − 1 . One relative maximum occurs at cos φ = −1.00, where
2
I = Imax [1 − 2]2 = Imax
The larger local maximum happens where cos φ = +1.00, giving I = Imax[1 + 2]2 = 9.00Imax.

The ratio of intensities at primary versus secondary maxima is 9.00 .

Interference of Light Waves 361

Section 37.5 Change of Phase Due to Reflection

Section 37.6 Interference in Thin Films

P37.23 (a) The light reflected from the top of the oil film undergoes phase
reversal. Since 1.45 > 1.33, the light reflected from the bottom
undergoes no reversal. For constructive interference of reflected
light, we then have

2nt = ⎛ m + 1⎞ λ
⎝ 2⎠

or λm = 2nt 2) = 2(1.45)(280 nm) FIG. P37.23

m + (1 m + (1 2)

Substituting for m gives: m = 0, λ0 = 1 620 nm (infrared)
m = 1, λ1 = 541 nm (green)
m = 2, λ2 = 325 nm (ultraviolet)

Both infrared and ultraviolet light are invisible to the human eye, so the dominant color in
reflected light is green .

(b) The dominant wavelengths in the transmitted light are those that produce destructive
interference in the reflected light. The condition for destructive interference upon
reflection is

2nt = mλ

or λm = 2nt = 812 nm
m m

Substituting for m gives: m = 1, λ1 = 812 nm (near infrared)
m = 2, λ2 = 406 nm (violet)

m = 3, λ3 = 271 nm (ultraviolet)

Of these, the only wavelength visible to the human eye (and hence the dominant
wavelength observed in the transmitted light) is 406 nm. Thus, the dominant color

in the transmitted light is violet .

P37.24 Light reflecting from the first surface suffers phase reversal. Light reflecting from the second
surface does not, but passes twice through the thickness t of the film. So, for constructive
interference, we require

λn + 2t = λn
2

where λn = λ is the wavelength in the material.
n

Then 2t = λn = λ
2 2n

λ = 4nt = 4 (1.33)(115 nm) = 612 nm

362 Chapter 37

P37.25 Since 1 < 1.25 < 1.33, light reflected both from the top and from the bottom surface of the oil

suffers phase reversal.

For constructive interference we require 2t = mλcons
n

and for destructive interference, 2t = ⎡⎣m + (1 2)⎤⎦ λdes

n

Then λcons = 1 + 1 = 640 nm = 1.25 and m = 2
λdest 2m 512 nm

Therefore, t = 2(640 nm) = 512 nm
2 (1.25)

P37.26 Treating the anti-reflectance coating like a camera-lens coating,

2t = ⎛ m + 1⎞ λ
⎝ 2⎠ n

Let m = 0: t = λ = 3.00 cm = 0.500 cm
4n
4 (1.50)

This anti-reflectance coating could be easily countered by changing the wavelength of the radar
to 1.50 cm. Then the coating would exhibit maximum reflection!

P37.27 2nt = ⎛ m + 1 ⎞ λ so t = ⎛ m + 1⎞ λ
⎝ 2 ⎠ ⎝ 2⎠ 2n

Minimum t = ⎛ 1 ⎞ (500 nm) = 96.2 nm
⎝ 2 ⎠ 2 (1.30 )

P37.28 Since the light undergoes a 180° phase change at each surface of the film, the condition for
constructive interference is 2nt = mλ, or λ = 2nt . The film thickness is

m
t = 1.00 × 10−5 cm = 1.00 × 10−7 m = 100 nm. Therefore, the wavelengths intensified in the
reflected light are

λ = 2(1.38)(100 nm) = 276 nm where m = 1, 2, 3, . . .

mm
or λ1 = 276 nm, λ2 = 138 nm, . . . . All reflection maxima are in the ultraviolet and beyond.

No visible wavelengths are intensified.

P37.29 (a) For maximum transmission, we want destructive interference in the light reflected from the

front and back surfaces of the film.

If the surrounding glass has refractive index greater than 1.378, light reflected from the

front surface suffers no phase reversal and light reflected from the back does undergo phase

reversal. This effect by itself would produce destructive interference, so we want the

distance down and back to be one whole wavelength in the film: 2t = λ .
n
λ 656.3 nm
t = 2n = = 238 nm
2 (1.378)

(b) The filter will undergo thermal expansion. As t increases in 2nt = λ, so does λ increase .

(c) Destructive interference for reflected light happens also for λ in 2nt = 2λ ,

or λ = 1.378(238 nm) = 328 nm (near ultraviolet)

Interference of Light Waves 363

P37.30 If the path length difference ∆ = λ, the transmitted light will be bright. Since ∆ = 2d = λ,

dmin = λ = 580 nm = 290 nm
2 2

P37.31 For destructive interference in the air,

2t = mλ

For 30 dark fringes, including the one where the plates
meet,

t = 29(600 nm) = 8.70 × 10−6 m

2
Therefore, the radius of the wire is
r = t = 8.70 µm = 4.35 µm

22

FIG. P37.31

P37.32 The condition for bright fringes is

2t + λ = m λ m = 1, 2, 3, . . . θ
2n n
R

From the sketch, observe that

t = R (1 − cosθ ) ≈ R ⎛ − 1 + θ2 ⎞ = R⎛ r ⎞2 = r2 t
⎜⎝1 2 ⎟⎠ 2⎝ R⎠ 2R

The condition for a bright fringe becomes r2 = ⎝⎛⎜ m − 1 ⎟⎞⎠ λ r
R 2 n FIG. P37.32

Thus, for fixed m and λ, nr2 = constant

Therefore, n r2 = nair ri2 and nliquid = (1.00 ) (1.50 cm )2 = 1.31
liquid f (1.31 cm )2

P37.33 For total darkness, we want destructive interference for reflected light for both 400 nm and 600 nm.
With phase reversal at just one reflecting surface, the condition for destructive interference is

2nairt = mλ m = 0, 1, 2, . . .

The least common multiple of these two wavelengths is 1 200 nm, so we get no reflected light at

2(1.00)t = 3(400 nm) = 2(600 nm) = 1 200 nm, so t = 600 nm at this second dark fringe.

By similar triangles, 600 nm = 0.050 0 mm
or the distance from the contact point is x 10.0 cm

( )x = ⎛ 0.100 m ⎞
600 × 10−9 m ⎝ 5.00 × 10−5 m ⎠ = 1.20 mm

364 Chapter 37

Section 37.7 The Michelson Interferometer

( )P37.34 Distance = 2 3.82 × 10−4 m = 1 700λ λ = 4.49 × 10−7 m = 449 nm

The light is blue .

P37.35 When the mirror on one arm is displaced by ∆ , the path difference changes by 2∆ . A shift
resulting in the reversal between dark and bright fringes requires a path length change of one-half
wavelength. Therefore, 2∆ = mλ , where in this case, m = 250.

2

)(∆ = m λ = (250) 6.328 × 10−7 m = 39.6 µm
44

P37.36 Counting light going both directions, the number of wavelengths originally in the cylinder is

m1 = 2L . It changes to m2 = 2L = 2nL as the cylinder is filled with gas. If N is the number of
λ λn λ

bright fringes passing, N = m2 − m1 = 2L (n − 1), or the index of refraction of the gas is
λ
n = 1+ Nλ
2L

Additional Problems

*P37.37 The same light source will radiate into the syrup light with wavelength 560 nm/1.38 = 406 nm.
The first side bright fringe is separated from the central bright fringe by distance y described by

d sinθ = 1λ dy/L ≈ λ y = λ L / d = 406 × 10−9 m(1.20 m)/(30 × 10−6m) = 1.62 cm

P37.38 (a) Where fringes of the two colors coincide we have d sinθ = mλ = m′λ ′, requiring λ = m′.
λ′ m

(b) λ = 430 nm, λ ′ = 510 nm

∴ m′ = 430 nm = 43, which cannot be reduced any further. Then m = 51 , m′ = 43.
m 510 nm 51

( )θm
= sin−1 ⎛ mλ ⎞ = sin−1 ⎡ (51) 430 × 10−9 m ⎤
⎜⎝ d ⎟⎠ ⎢ ⎥ = 61.3°
⎢⎣ 0.025 × 10−3 m ⎦⎥

ym = L tanθm = (1.5 m) tan 61.3° = 2.74 m

P37.39 The wavelength is λ = c = 3.00 × 108 ms = 5.00 m.
f 60.0 × 106 s−1

Along the line AB the two traveling waves going in opposite directions add to give a standing

wave. The two transmitters are exactly 2.00 wavelengths apart and the signal from B, when it

arrives at A, will always be in phase with transmitter B. Since B is 180° out of phase with A, the

two signals always interfere destructively at the position of A.

The first antinode (point of constructive interference) is located at distance

λ = 5.00 m = 1.25 m from the node at A.
44

Interference of Light Waves 365

P37.40 Along the line of length d joining the source, two identical waves s NAN s

moving in opposite directions add to give a standing wave. λ

An antinode is halfway between the sources. If d > λ , there is 4
22
FIG. P37.40
space for two more antinodes for a total of three. If d > λ , there
2

will be at least five antinodes, and so on. To repeat, if d > 0, the
λ

number of antinodes is 1 or more. If d > 1, the number of
λ

antinodes is 3 or more. If d > 2, the number of antinodes is 5 or more. In general,
λ

the number of antinodes is 1 plus 2 times the greatest integer less than or equal to d .
λ

If d < λ , there will be no nodes. If d > λ , there will be space for at least two nodes, as shown
24 24

in the picture. If d > 3λ , there will be at least four nodes. If d > 5λ , six or more nodes will fit
24 24

in, and so on. To repeat, if 2d < λ, the number of nodes is 0. If 2d > λ, the number of nodes is 2

or more. If 2d > 3λ, the number of nodes is 4 or more. If 2d > 5λ, the number of nodes is 6 or

more. Again, if ⎛ d + 1⎞ > 1, the number of nodes is at least 2. If ⎛ d + 1⎞ > 2, the number of
⎝ λ 2⎠ ⎝ λ 2⎠

nodes is at least 4. If ⎛ d + 1⎞ > 3, the number of nodes is at least 6. In general,
⎝ λ 2⎠

the number of nodes is 2 times the greatest nonzero integer less than ⎛ d + 1⎞ .
⎝ λ 2⎠

Next, we enumerate the zones of constructive interference. They are described by d sinθ = mλ,
m = 0, 1, 2, . . . with θ counted as positive both left and right of the maximum at θ = 0 in the
center. The number of side maxima on each side is the greatest integer satisfying sinθ ≤ 1,

d1 ≥ mλ, m ≤ d . So the total
λ

number of bright fringes is one plus 2 times the greatest integer less than or equal to d .
λ

It is equal to the number of antinodes on the line joining the sources.

The interference minima are to the left and right at angles described by d sin θ = ⎛ m + 1⎞ λ,
⎝ 2⎠

m = 0, 1, 2, . . . . With sin θ < 1, d1 > ⎛ mmax + 1⎞ λ, mmax < d − 1 or mmax +1< d + 1. Let
⎝ 2⎠ λ 2 λ 2

n = 1, 2, 3, . . . . Then the number of side minima is the greatest integer n less than d + 1 .
λ2

Counting both left and right,

the number of dark fringes is two times the greatest positive integer less than ⎛ d + 1⎞ .
⎝ λ 2⎠

It is equal to the number of nodes in the standing wave between the sources.

366 Chapter 37

P37.41 My middle finger has width d = 2 cm.

(a) Two adjacent directions of constructive interference for 600-nm light are described by

d sinθ = mλ

θ0 = 0

( ) ( )2 × 10−2 m sinθ1 = 1 6 × 10−7 m

Thus, θ1 = 2 × 10−3 degree

and θ1 − θ0 ~10−3 degree

(b) Choose θ1 = 20°

(2 × 10−2 m)sin 20° = (1) λ

λ = 7 mm

Millimeter waves are microwaves .

f = c: f = 3 × 108 m s ~1011 Hz
λ 7 × 10−3 m

*P37.42 Constructive interference occurs where m = 0, 1, 2, 3, . . . , −1, −2, −3, . . . , for

⎛ 2π x1 − 2π ft + π⎞ − ⎛ 2π x2 − 2π ft + π⎞ = 2π m 2π (x1 − x2 ) + ⎛ π − π ⎞ = 2π m
⎝ λ 6⎠ ⎝ λ 8⎠ ⎝ 6 8 ⎠
λ

(x1 − x2 ) + 1 − 1 = m x1 − x2 = ⎛ m − 1⎞ 650 nm with m = 0, 1, 2, 3, ... , − 1, − 2, − 3, ...
⎝ 48 ⎠
650 nm 12 16

*P37.43 A bright line for the green light requires dy/L = mλ 1. A blue interference maximum requires
1
dy/L = m2λ2 for integers m1 and m2. Then m1 540 nm = m2 450 nm. The smallest integers
satisfying the equation are m1 = 5 and m2 = 6. Then for both

dy/L = 2 700 nm y = (1.4 m) 2.7 µm/ 150µm = 2.52 cm

P37.44 If the center point on the screen is to be a dark spot rather than bright, passage through the plastic
must delay the light by one-half wavelength. Calling the thickness of the plastic t,

t + 1 = t = nt or t= λ where n is the index of refraction for the plastic.
λ 2 λn λ
2(n − 1)

Interference of Light Waves 367

P37.45 No phase shift upon reflection from the upper surface (glass to air) of the film, but there will be a
shift of λ due to the reflection at the lower surface of the film (air to metal). The total phase

2
difference in the two reflected beams is

then δ = 2nt + λ
2

For constructive interference, δ = mλ

or 2(1.00)t + λ = mλ

2

Thus, the film thickness for the mth order bright fringe is

tm = ⎛ m − 1⎞ λ = m ⎛ λ⎞ − λ
⎝ 2⎠ 2 ⎝⎜ 2 ⎠⎟ 4

and the thickness for the m − 1 bright fringe is:

t m −1 = (m − 1) ⎛ λ⎞ − λ
⎜⎝ 2 ⎟⎠ 4

Therefore, the change in thickness required to go from one bright fringe to the next is

∆t = tm − t m −1 = λ
2

To go through 200 bright fringes, the change in thickness of the air film must be:

200 ⎛ λ⎞ = 100λ
⎝⎜ 2 ⎠⎟

Thus, the increase in the length of the rod is

( )∆L = 100λ = 100 5.00 × 10−7 m = 5.00 × 10−5 m

From ∆L = Liα ∆T
we have:
α = ∆L = 5.00 × 10−5 m C) = 20.0 × 10−6 °C−1
Li ∆T
(0.100 m)(25.0 °

P37.46 Since 1 < 1.25 < 1.34, light reflected from top and bottom surfaces of the oil undergoes phase
reversal. The path difference is then 2t, which must be equal to

mλn = mλ
n

for maximum reflection, with m = 1 for the given first-order condition and n = 1.25. So

t = mλ = 1(500 nm) = 200 nm
2n 2 (1.25)

The volume we assume to be constant: 1.00 m3 = (200 nm) A

= 1.00 m3 = 5.00 × 106 m2 = 5.00 km2
200 10−9 m
( )A

368 Chapter 37

P37.47 One radio wave reaches the receiver R directly from the
distant source at an angle θ above the horizontal. The other

wave undergoes phase reversal as it reflects from the water

at P.

Constructive interference first occurs for a path difference of

d=λ (1)
2

It is equally far from P to R as from P to R′, the mirror

image of the telescope.

The angles θ in the figure are equal because they each form FIG. P37.47
part of a right triangle with a shared angle at R′.

So the path difference is d = 2(20.0 m)sinθ = (40.0 m)sinθ

The wavelength is λ = c = 3.00 × 108 ms = 5.00 m
f 60.0 × 106 Hz

Substituting for d and λ in Equation (1), (40.0 m)sinθ = 5.00 m

2
Solving for the angle θ, sinθ = 5.00 m and θ = 3.58° .

80.0 m

P37.48 For destructive interference, the path length must differ by mλ. We may treat this problem as
a double slit experiment if we remember the light undergoes a π -phase shift at the mirror. The

2
second slit is the mirror image of the source, 1.00 cm below the mirror plane. Modifying

Equation 37.7,

( ( ) )ydark
= mλ L = 1 5.00 × 10−7 m (100 m) = 2.50 mm
d
2.00 × 10−2 m

P37.49 2 (15.0 km)2 + h2 = 30.175 km
(15.0 km)2 + h2 = 227.63
FIG. P37.49
h = 1.62 km

P37.50 For dark fringes, 2nt = mλ

and at the edge of the wedge, t = 84 (500 nm)

2

When submerged in water, 2nt = mλ

m = 2(1.33)(42)(500 nm)

500 nm

so m + 1 = 113 dark fringes

FIG. P37.50

Interference of Light Waves 369

P37.51 From Equation 37.14, I = cos2 ⎛ π yd ⎞
I max ⎝⎜ λL ⎟⎠

Let λ2 equal the wavelength for which I → I2 = 0.640
I max I max

π yd L
cos−1 I2 Imax
Then ( )λ2 = 12

But π yd = λ1 cos−1 ⎛ I1 ⎞1 2 = (600 nm) cos−1 (0.900) = 271 nm
L ⎜⎝ I max ⎠⎟

271 nm
cos−1 0.6401 2
Substituting this value into the expression for λ2, ( )λ2 = = 421 nm

Note that in this problem, cos−1 ⎛ I ⎞1 2 must be expressed in radians.
⎝⎜ I max ⎠⎟

P37.52 At entrance, 1.00 sin 30.0° = 1.38 sinθ2 θ2 = 21.2°
Call t the unknown thickness. Then

cos 21.2° = t a= t
a cos 21.2°

tan 21.2° = c c = t tan 21.2°
t b = 2t tan 21.2° sin 30.0°

sin θ1 = b FIG. P37.52
2c

The net shift for the second ray, including the phase reversal on reflection of the first, is

2an − b − λ
2

where the factor n accounts for the shorter wavelength in the film. For constructive interference,
we require

The minimum thickness will be given by 2an − b − λ = mλ
λ = 2an − b = 2 nt − 2t (tan 21.2°) sin 30.0° 2
2 cos 21.2°
2an − b − λ = 0
2

590 nm = ⎛ 2 × 1.38 − 2 tan 21.2° sin 30.0° ⎞ t = 2.57t t = 115 nm
2 ⎝ cos 21.2° ⎠

370 Chapter 37

P37.53 The shift between the two reflected waves is δ = 2na − b − λ
2

where a and b are as shown in the ray diagram, n is the

index of refraction, and the term λ is due to phase reversal
2

at the top surface. For constructive interference, δ = mλ
where m has integer values. This condition becomes

2na − b = ⎛ m + 1⎞ λ (1) FIG. P37.53
⎝ 2⎠

From the figure’s geometry, a= t
cosθ2

c = a sinθ2 = t sinθ2
cosθ2

b = 2c sin φ1 = 2t sinθ2 sin φ1
cosθ2

Also, from Snell’s law, sin φ1 = n sinθ2

Thus, b = 2nt sin2 θ2
cosθ2

With these results, the condition for constructive interference given in Equation (1) becomes:

2nt sin2 θ2
cosθ2
( )⎛ t ⎞ 2nt ⎛ 1⎞
cosθ2 ⎝ 2⎠
2n ⎜⎝ cosθ2 ⎟⎠
− = 1 − sin2 θ2 = m + λ

or 2nt cosθ2 = ⎛ m + 1⎞ λ
⎝ 2⎠

*P37.54 (a) For a linear function taking the value 1.90 at y = 0 and 1.33 at y = 20 cm we write
n( y) = 1.90 + (1.33 − 1.90)y/20 cm or n(y) = 1.90 − 0.0285 y/cm

0.0285y2 20 cm

20 cm 20 cm

n(y)dy = [1.90
∫ ∫(b) − 0.0285y / cm]dy = 1.90y − 20
00

= 38.0 cm − 5.7 cm

= 32.3 cm

(c) The beam will continuously curve downward.

Interference of Light Waves 371

P37.55 (a) Minimum: 2nt = mλ2 for m = 0, 1, 2, . . .
for m′ = 0, 1, 2, . . .
Maximum: 2nt = ⎛ m′ + 1⎞ λ1
⎝ 2⎠

for λ1 > λ2, ⎛ m′ + 1⎞ < m
⎝ 2⎠

so m′ = m − 1

Then 2nt = mλ2 = ⎛ m − 1⎞ λ1
so ⎝ 2⎠

2mλ2 = 2mλ1 − λ1

m = λ1 λ2 )

2(λ1 −

(b) m = 500 = 1.92 → 2 (wavelengths measured to ±5 nm)

2(500 − 370)

Minimum: 2nt = mλ2

2(1.40)t = 2(370 nm) t = 264 nm

Maximum: 2nt = ⎛ m − 1+ 1⎞ λ = 1.5λ
⎝ 2⎠

2(1.40)t = 1.5(500 nm) t = 268 nm

Film thickness = 266 nm

P37.56 From the sketch, observe that d
x
⎛ d ⎞ 2 4h2 + d2
⎝ 2 ⎠ 2
x= h2 + = x

Including the phase reversal due to reflec- h
tion from the ground, the total shift between
the two waves is δ = 2x − d − λ . d/2

2 FIG. P37.56

(a) For constructive interference, the total shift must be an integral number of wavelengths, or

δ = mλ where m = 0, 1, 2, 3, . . .

Thus, 2x − d = ⎛ m + 1⎞ λ or λ = 4x − 2d
⎝ 2⎠ 2m + 1

For the longest wavelength, m = 0, giving λ = 4x − 2d = 2 4h2 + d2 − 2d

(b) For destructive interference, δ = ⎛ m − 1⎞ λ where m = 0, 1, 2, 3, . . . .
⎝ 2⎠

Thus, 2x − d = mλ or λ = 2x − d
m

For the longest wavelength, m = 1 giving λ = 2x − d = 4h2 + d2 − d

372 Chapter 37

P37.57 Call t the thickness of the film. The central maximum L
corresponds to zero phase difference. Thus, the added
distance ∆r traveled by the light from the lower slit Thin
film

must introduce a phase difference equal to that intro- y' Zero
duced by the plastic film. The phase difference φ is
θ order
⎛ ⎞ dα m=0
⎜⎝ t ⎟⎠ ∆r
φ = 2π λa (n − 1) Screen

The corresponding difference in path length ∆r is

⎛ λa ⎞ ⎛ t ⎞ ⎛ λa ⎞ FIG. P37.57
⎜⎝ 2π ⎠⎟ ⎜⎝ λa ⎟⎠ ⎜⎝ 2π ⎟⎠
∆r = φ = 2π (n − 1) = t (n − 1)

Note that the wavelength of the light does not appear in this equation. In the figure, the two rays

from the slits are essentially parallel. tanθ = ∆r = y′
Thus the angle θ may be expressed as dL

Eliminating ∆r by substitution, y′ = t (n − 1) gives y′ = t (n − 1) L

Ld d

P37.58 Bright fringes occur when 2t = λ ⎛ m + 1⎞
and dark fringes occur when n ⎝ 2⎠
The thickness of the film at x is
2t = ⎛ λ ⎞ m
⎝⎜ n ⎠⎟

t = ⎛ h⎞ x
⎝ ⎠

Therefore, xbright = λ ⎛ m + 1⎞ and xdark = λm .
2hn ⎝ 2⎠ 2hn
FIG. P37.58

P37.59 (a) Constructive interference in the reflected light requires 2t = ⎛ m + 1⎞ λ. The first bright ring
⎝ 2⎠

has m = 0 and the 55th has m = 54, so at the edge of the lens

)(54.5 650 × 10−9 m

t = = 17.7 µm
2

Now from the geometry in textbook Figure 37.12, we can find the distance t from the
curved surface down to the flat plate by considering distances down from the center
of curvature:

R2 − r2 = R − t or R2 − r2 = R2 − 2Rt + t 2

( ) ( )R = r2 + t2 =
( )2t
5.00 × 10−2 m 2 + 1.77 × 10−5 m 2 = 70.6 m

2 1.77 × 10−5 m

(b) 1 = ( n − 1) ⎛ 1 − 1⎞ = 0.520 ⎛ 1− 1 ⎞ so f = 136 m
f ⎜⎝ R2 R2 ⎟⎠ ⎝ ∞ −70.6 m⎠

Interference of Light Waves 373

P37.60 The shift between the waves reflecting from the top and

bottom surfaces of the film at the point where the film has

thickness t is δ = 2tnfilm + λ , with the factor of λ being due θ
2 2
R

to a phase reversal at one of the surfaces.

For the dark rings (destructive interference), the total shift

should be δ = ⎛ m + 1 ⎞ λ with m = 0, 1, 2, 3, .... nfilm t
⎝ 2 ⎠ r

This requires that t = mλ . R2 = r2 + ( R − t )2 FIG. P37.60
2nfilm
so r2 = 2Rt + t 2
To find t in terms of r and R,

Since t is much smaller than R, t 2 << 2Rt and r2 ≈ 2Rt = ⎛ mλ ⎞
Thus, where m is an integer, r ≈ mλR 2R ⎜⎝ 2nfilm ⎠⎟
nfilm

P37.61 Light reflecting from the upper interface of the air layer suffers
no phase change, while light reflecting from the lower interface
is reversed 180°. Then there is indeed a dark fringe at the outer t0
circumference of the lens, and a dark fringe wherever the air r
thickness t satisfies 2t = mλ, m = 0, 1, 2, . . . . R=8m

(a) At the central dark spot m = 50 and FIG. P37.61

( )t0 50λ
= 2 = 25 589 × 10−9 m = 1.47 × 10−5 m

(b) In the right triangle,

( )(8 m)2 = r2 + 8 m − 1.47 × 10−5 m 2 = r2 + (8 m)2

( )− 2(8 m) 1.47 × 10−5 m + 2 × 10−10 m2

( )The last term is negligible. r = 2(8 m) 1.47 × 10−5 m = 1.53 × 10−2 m

(c) 1 = (n − 1) ⎛ 1 − 1 ⎞ = (1.50 − 1) ⎛ 1− 1 ⎞
f ⎝⎜ R1 R2 ⎠⎟ ⎝ ∞ 8.00 m⎠

f = −16.0 m

374 Chapter 37

P37.62 For bright rings the gap t between surfaces is given by

2t = ⎛ m + 1⎞ λ . The first bright ring has m = 0 and the
⎝ 2⎠

hundredth has m = 99.

)(So, t = 1 (99.5) 500 × 10−9 m = 24.9 µm.
2

Call rb the ring radius. From the geometry of the figure
at the right,

( )t = r − r2 − rb2 − R − R2 − rb2

Since rb << r, we can expand in series: FIG. P37.62

t = r − r ⎛⎝⎜1 − 1 rb2 ⎞ − R + R ⎝⎛⎜1 − 1 rb2 ⎞ = 1 rb2 − 1 rb2
2 r2 ⎠⎟ 2 R2 ⎠⎟ 2 r 2 R

⎡ 2 24.9 × 10−6 m ⎤1 2
( )rb= ⎡ 2t ⎤1 2 = ⎢⎢⎣1 ⎥ = 1.73 cm
⎢⎣1 r −1 R ⎦⎥ 4.00 m − 1 12.0 m ⎦⎥

P37.63 (a) Bright bands are observed when 2nt = ⎛ m + 1⎞ λ.
⎝ 2⎠

Hence, the first bright band (m = 0) corresponds to nt = λ .
4

Since x1 = t1
x2 t2

we have x2 = x1 ⎛ t2 ⎞ = x1 ⎛ λ2 ⎞ = (3.00 cm) ⎛ 680 nm ⎞ = 4.86 cm
⎜⎝ t1 ⎟⎠ ⎜⎝ λ1 ⎠⎟ ⎝ 420 nm ⎠

(b) t1 = λ1 = 420 nm = 78.9 nm
4n
4 (1.33)

t2 = λ2 = 680 nm = 128 nm
4n
4 (1.33)

(c) θ ≈ tanθ = t1 = 78.9 nm = 2.63 × 10−6 rad
x1 3.00 cm

P37.64 Depth = one-quarter of the wavelength in plastic.

t = λ = 780 nm = 130 nm
4n
4 (1.50)

P37.65 2h sin θ = ⎛ m + 1 ⎞ λ bright
⎝ 2 ⎠ so

) )( (h
2h ⎛ ∆y ⎞ = 1 λ = Lλ = (2.00 m) 606 × 10−9 m = 0.505 mm
⎝ 2L ⎠ 2 2∆y
2 1.2 × 10−3 m

Interference of Light Waves 375

P37.66 Superposing the two vectors, ER = E1 + E2

⎛ E0 ⎞ 2 ⎛ E0 ⎞ 2 2 E02 E02
⎝ 3 ⎠ ⎝ 3 ⎠ 3 9 9
ER = E1 + E 2 = E0 + cos φ + sin φ = E02 + E02 cos φ + cos2 φ + sin2 φ

ER = 10 E02 + 2 E02 cos φ
9 3

Since intensity is proportional to the square of the amplitude,

I = 10 I max + 2 I max cos φ
9 3
Using the trigonometric identity cos φ = 2 cos2 φ − 1, this becomes
2

I = 10 I max + 2 I max ⎛ 2 cos2 φ − 1⎠⎞ = 4 I max + 4 I max cos2 φ
9 3 ⎝ 2 9 3 2

or I = 4 I max ⎛⎝1 + 3 cos2 φ⎞
9 2⎠

P37.67 Represent the light radiated from each slit to point P as a phasor. The two 3E
have equal amplitudes E. Since intensity is proportional to amplitude
squared, they add to amplitude 3E. θ E
E φ
Then cosθ = 3 E 2 , θ = 30°. Next, the obtuse angle
E FIG. P37.67

between the two phasors is 180 − 30 − 30 = 120°, and

φ = 180 − 120°= 60°. The phase difference between the

two phasors is caused by the path difference δ = SS2 − SS1

according to δ = φ , δ = λ 60° = λ . Then
λ 360° 360° 6

L2 + d2 − L = λ
6

L2 + d2 = L2 + 2Lλ + λ 2
6 36

The last term is negligible, so

⎛ 2Lλ ⎞1 2 2(1.2 m)620 × 10−9 m = 0.498 mm
⎜⎝ 6 ⎟⎠
d = = 6

376 Chapter 37

ANSWERS TO EVEN PROBLEMS

P37.2 515 nm

P37.4 (a) 36.2° (b) 5.08 cm (c) 508 THz

P37.6 Question: A single oscillator makes the two speakers of a boom box, 35.0 cm apart, vibrate in
phase at 1.62 kHz. At what angles, measured from the perpendicular bisector of the line joining
the speakers, will a distant observer hear maximum sound intensity? Minimum sound intensity?
The ambient temperature is 20°C. Answer: The wavelength of the sound is 21.2 cm. Interference
maxima occur at angles of 0° and 37.2° to the left and right. Minima occur at angles of 17.6° and
65.1°. No second-order or higher-order maximum exists. No angle exists, smaller or larger than
90°, for which sinq2 loud = 1.21. No location exists in the Universe that is two wavelengths farther
from one speaker than from the other.

P37.8 11.3 m

P37.10 641

P37.12 6.33 mm/s

P37.14 See the solution.

P37.16 (a) 1.29 rad (b) 99.6 nm

P37.18 0.968

P37.20 See the solution.

P37.22 (a) See the solution. (b) The cosine function takes on the extreme value −1 to describe the
secondary maxima. The cosine function takes on the extreme value +1 to describe the primary
maxima. The ratio is 9.00.

P37.24 612 nm

P37.26 0.500 cm

P37.28 No reflection maxima in the visible spectrum

P37.30 290 nm

P37.32 1.31

P37.34 449 nm; blue

P37.36 1 + Nl/2L

P37.38 (a) See the solution. (b) 2.74 m

Interference of Light Waves 377

P37.40 number of antinodes = number of constructive interference zones

=1 plus 2 times the greatest positive integer ≤ d
λ

number of nodes = number of destructive interference zones

= 2 times the greatest positive integer < ⎛ d + 1⎞
⎝ λ 2⎠

P37.42 x1 − x2 = ⎛ m − 1⎞ 650 nm with m = 0, 1, 2, 3, . . . , − 1, − 2, − 3, . . .
⎝ 48 ⎠

P37.44 λ

2(n − 1)

P37.46 5.00 km2

P37.48 2.50 mm

P37.50 113

P37.52 115 nm

P37.54 (a) n(y) = 1.90 − 0.0285 y/cm (b) 32.3 cm (c) The beam will continuously curve downward.

P37.56 (a) 2(4h2 + d )2 1/2 − 2d (b) (4h2 + d )2 1/2 − d

P37.58 See the solution.

P37.60 See the solution.

P37.62 1.73 cm

P37.64 130 nm

P37.66 See the solution.



38

Diffraction Patterns and Polarization

CHAPTER OUTLINE ANSWERS TO QUESTIONS

38.1 Introduction to Diffraction Q38.1 Audible sound has wavelengths on the order of meters
Patterns or centimeters, while visible light has a wavelength
on the order of half a micrometer. In this world of
38.2 Diffraction Patterns from Narrow breadbox-sized objects, λ is large for sound, and sound
Slits
a
38.3 Resolution of Single-Slit and diffracts around behind walls with doorways. But λ
Circular Apertures
a
38.4 The Diffraction Grating is a tiny fraction for visible light passing ordinary-size
38.5 Diffraction of X-Rays by Crystals objects or apertures, so light changes its direction by
38.6 Polarization of Light Waves only very small angles when it diffracts.

Another way of phrasing the answer: We can see by a
small angle around a small obstacle or around the edge
of a small opening. The side fringes in Figure 38.1 and
the Arago spot in the center of Figure 38.3 show this
diffraction. We cannot always hear around corners.
Out-of-doors, away from reflecting surfaces, have some-
one a few meters distant face away from you and whisper.
The high-frequency, short-wavelength, information-
carrying components of the sound do not diffract around
his head enough for you to understand his words.

Suppose an opera singer loses the tempo and cannot
immediately get it from the orchestra conductor. Then
the prompter may make rhythmic kissing noises with her
lips and teeth. Try it—you will sound like a birdwatcher
trying to lure out a curious bird. This sound is clear on
the stage but does not diffract around the prompter’s box
enough for the audience to hear it.

Q38.2 The wavelength of visible light is extremely small in comparison to the dimensions of your hand,
so the diffraction of light around an obstacle the size of your hand is totally negligible. However,
sound waves have wavelengths that are comparable to the dimensions of the hand or even larger.
Therefore, significant diffraction of sound waves occurs around hand-sized obstacles.

*Q38.3 Answer (d). The power of the light coming through the slit decreases, as you would expect. The

central maximum increases in width as the width of the slit decreases. In the condition sinθ = λ
a

for destructive interference on each side of the central maximum, θ increases as a decreases.

*Q38.4 We consider the fraction λL/a. In case (a) and in case (f) it is λ0L/a. In case (b) it is 2λ0L/3a.
In case (c) it is 3λ0L/2a. In case (d) it is l0L/2a. In case (e) it is l02L/a. The ranking is e > c > a =
f > b > d.

*Q38.5 Answer (b). The wavelength will be much smaller than with visible light.

379

380 Chapter 38

*Q38.6 Answer (c). The ability to resolve light sources depends on diffraction, not on intensity.

Q38.7 Consider incident light nearly parallel to the horizontal

ruler. Suppose it scatters from bumps at distance d apart

to produce a diffraction pattern on a vertical wall a y

distance L away. At a point of height y, where θ = y q
L dL

gives the scattering angle θ, the character of the FIG. Q38.7
interference is determined by the shift δ between beams

scattered by adjacent bumps, where δ = d − d.
cosθ

Bright spots appear for δ = mλ, where m = 0, 1, 2, 3, . . . . For small θ, these equations combine

and reduce to mλ = ym2 d . Measurement of the heights ym of bright spots allows calculation of the
2L2

wavelength of the light.

*Q38.8 Answer (b). No diffraction effects are observed because the separation distance between adjacent
ribs is so much greater than the wavelength of x-rays. Diffraction does not limit the resolution of
an x-ray image. Diffraction might sometimes limit the resolution of an ultrasonogram.

*Q38.9 Answer (a). Glare, as usually encountered when driving or boating, is horizontally polarized.
Reflected light is polarized in the same plane as the reflecting surface. As unpolarized light hits
a shiny horizontal surface, the atoms on the surface absorb and then reemit the light energy as
a reflection. We can model the surface as containing conduction electrons free to vibrate eas-
ily along the surface, but not to move easily out of surface. The light emitted from a vibrating
electron is partially or completely polarized along the plane of vibration, thus horizontally.

Q38.10 Light from the sky is partially polarized. Light from the blue sky that is polarized at 90° to the
polarization axis of the glasses will be blocked, making the sky look darker as compared to the
clouds.

*Q38.11 Answer (a). The grooves in a diffraction grating are not electrically conducting. Sending light
through a diffraction grating is not like sending a vibration on a rope through a picket fence. The
electric field in light does not have an amplitude in real space. Its amplitude is in newtons per
coulomb, not in millimeters.

Q38.12 First think about the glass without a coin and about one particular point P on the screen. We can
divide up the area of the glass into ring-shaped zones centered on the line joining P and the light
source, with successive zones contributing alternately in-phase and out-of-phase with the light
that takes the straight-line path to P. These Fresnel zones have nearly equal areas. An outer zone
contributes only slightly less to the total wave disturbance at P than does the central circular
zone. Now insert the coin. If P is in line with its center, the coin will block off the light from
some particular number of zones. The first unblocked zone around its circumference will send
light to P with significant amplitude. Zones farther out will predominantly interfere destructively
with each other, and the Arago spot is bright. Slightly off the axis there is nearly complete
destructive interference, so most of the geometrical shadow is dark. A bug on the screen crawling
out past the edge of the geometrical shadow would in effect see the central few zones coming out
of eclipse. As the light from them interferes alternately constructively and destructively, the bug
moves through bright and dark fringes on the screen. The diffraction pattern is shown in
Figure 38.3 in the text.

Diffraction Patterns and Polarization 381

Q38.13 Since the obsidian is opaque, a standard method of measuring incidence and refraction angles
and using Snell’s Law is ineffective. Reflect unpolarized light from the horizontal surface
of the obsidian through a vertically polarized filter. Change the angle of incidence until you
observe that none of the reflected light is transmitted through the filter. This means that the
reflected light is completely horizontally polarized, and that the incidence and reflection angles
are the polarization angle. The tangent of the polarization angle is the index of refraction of the
obsidian.

Q38.14 The fine hair blocks off light that would otherwise go through a fine slit and produce a diffraction
pattern on a distant screen. The width of the central maximum in the pattern is inversely
proportional to the distance across the slit. When the hair is in place, it subtracts the same
diffraction pattern from the projected disk of laser light. The hair produces a diffraction minimum
that crosses the bright circle on the screen. The width of the minimum is inversely proportional
to the diameter of the hair. The central minimum is flanked by narrower maxima and minima.
Measure the width 2y of the central minimum between the maxima bracketing it, and use
Equation 38.1 in the form y = λ to find the width a of the hair.

La

Q38.15 The condition for constructive interference is that the three radio signals arrive at the city in

phase. We know the speed of the waves (it is the speed of light c), the angular bearing θ of the

city east of north from the broadcast site, and the distance d between adjacent towers. The wave

from the westernmost tower must travel an extra distance 2d sinθ to reach the city, compared to

the signal from the eastern tower. For each cycle of the carrier wave, the western antenna would

transmit first, the center antenna after a time delay d sinθ , and the eastern antenna after an
c
additional equal time delay.

Q38.16 It is shown in the correct orientation. If the horizontal width of the opening is equal to or less than

the wavelength of the sound, then the equation a sinθ = (1) λ has the solution θ = 90°, or has no

solution. The central diffraction maximum covers the whole seaward side. If the vertical height

of the opening is large compared to the wavelength, then the angle in a sinθ = (1) λ will be small,

and the central diffraction maximum will form a thin horizontal sheet.

Featured in the motion picture M*A*S*H (20th Century Fox, Aspen Productions, 1970)

is a loudspeaker mounted on an exterior wall of an Army barracks. It has an approximately

rectangular aperture, and it is installed incorrectly. The longer side is horizontal, to maximize

sound spreading in a vertical plane and to minimize sound radiated in different horizontal

directions.

SOLUTIONS TO PROBLEMS

Section 38.1 Introduction to Diffraction Patterns

Section 38.2 Diffraction Patterns from Narrow Slits

P38.1 sin θ = λ = 6.328 × 10−7 = 2.11 × 10−3
a 3.00 × 10−4

y = tanθ ≈ sinθ ≈ θ (for small θ)
1.00 m

2y = 4.22 mm

382 Chapter 38

P38.2 The positions of the first-order minima are y ≈ sinθ = ± λ . Thus, the spacing between these two
P38.3 La

minima is ∆y = 2 ⎛ λ ⎞ L and the wavelength is
⎝⎜ a ⎠⎟

λ = ⎛ ∆y ⎞ ⎛ a⎞ = ⎛ 4.10 × 10−3 m⎞ ⎛ 0.550 × 10−3 m⎞ = 547 nm
⎝ 2⎠ ⎝ L⎠ ⎝⎜ 2 ⎠⎟ ⎜⎝ 2.06 m ⎠⎟

y ≈ sinθ = mλ ∆y = 3.00 × 10−3 nm
La
∆m = 3 − 1 = 2 and a = ∆mλ L
∆y

( )2 690 × 10−9 m (0.500 m) 2.30 × 10−4 m
( )a = 3.00 × 10−3 m =

P38.4 For destructive interference,
P38.5 sinθ = m λ = λ = 5.00 cm = 0.139
a a 36.0 cm

and θ = 7.98°
y = tanθ
L

gives y = L tanθ = (6.50 m) tan 7.98° = 0.912 m

y = 91.2 cm

If the speed of sound is 340 m/s, λ = v = 340 m s = 0.523 m
f 650 s−1

Diffraction minima occur at angles described by a sinθ = mλ

(1.10 m)sinθ1 = 1(0.523 m) θ1 = 28.4°

(1.10 m)sinθ2 = 2(0.523 m) θ2 = 72.0°

(1.10 m)sinθ3 = 3(0.523 m) θ3 nonexistent

Maxima appear straight ahead at 0° and left and right at an angle given approximately by

(1.10 m)sinθx = 1.5(0.523 m) θx ≈ 46°

There is no solution to a sinθ = 2.5λ, so our answer is already complete, with three sound
maxima.

Diffraction Patterns and Polarization 383

*P38.6 (a) The rectangular patch on the wall is wider
than it is tall. The aperture will be taller
than it is wide. For horizontal spreading we
have

tan θ width = ywidth = 0.110 m 2 = 0.012 2
L 4.5 m

awidth sinθwidth = 1λ

awidth = 632.8 × 10−9 m = 5.18 × 10−5 m
0.012 2

For vertical spreading, similarly

tan θheight = 0.006 m 2 = 0.000 667
4.5 m

aheight = 1λ = 632.8 × 10−9 m FIG. P38.6
sin θ h 0.000 667

= 9.49 × 10−4 m

(b) The central bright patch is horizontal. The aperture is vertical.

A smaller distance between aperture edges causes a wider diffraction angle. The longer dimension
of each rectangle is 18.3 times larger than the smaller dimension.

P38.7 sinθ ≈ y = 4.10 × 10−3 m
P38.8 L 1.20 m

( )We define φ = π a sinθ
λ
π 4.00 × 10−4 m ⎛ 4.10 × 10−3 m⎞ = 7.86 rad
= 546.1 × 10−9 m ⎜⎝ 1.20 m ⎠⎟

I = ⎡ sin (φ) ⎤2 = ⎡ sin (7.86) ⎤2 = 1.62 × 10−2
I max
⎢⎣ φ ⎥⎦ ⎣⎢ 7.86 ⎦⎥

Equation 38.1 states that sinθ = mλ , where m = ±1, ± 2, ± 3, . . . .
a

The requirement for m = 1 is from an analysis of the extra path distance

traveled by ray 1 compared to ray 3 in the textbook

Figure 38.5. This extra distance must be equal to λ for destructive
2

interference. When the source rays approach the slit at an angle b, there is

a distance added to the path difference (of ray 1 compared to ray 3) of

a sin β. Then, for destructive interference, FIG. P38.8
2
sinθ = 2λ − sin β
a sin β + a sinθ = λ so sinθ = λ 0 − sin β. a
222 a

Dividing the slit into 4 parts leads to the 2nd order minimum:

Dividing the slit into 6 parts gives the third order minimum: sinθ = 3λ − sin β
a

Generalizing, we obtain the condition for the mth order minimum: sinθ = mλ − sin β
a

384 Chapter 38

*P38.9 The diffraction envelope shows a broad central maximum I
flanked by zeros at a sinθ = 1l and a sinθ = 2l. That is,
the zeros are at (π a sinθ)/λ = π, −π, 2π, −2π, . . . . Not- −p 0 p p asinq
ing that the distance between slits is d = 9µm = 3a, we say FIG. P38.9 l

that within the diffraction envelope the interference pattern
shows closely spaced maxima at d sinθ = mλ, giving
(π 3a sinθ)/λ = mπ or (πa sinθ)/λ = 0, π/3, −π/3, 2π/3,
−2π/3. The third-order interference maxima are missing
because they fall at the same directions as diffraction

minima, but the fourth order can be visible at
(πa sinθ)/λ = 4π/3 and –4π/3 as diagrammed.

P38.10 (a) Double-slit interference maxima are at angles given
by d sinθ = mλ.

For m = 0, θ0 = 0°

For m = 1, (2.80 µm)sinθ = 1(0.501 5 µm): θ1 = sin−1 (0.179) = 10.3°

Similarly, for m = 2, 3, 4, 5 and 6, θ2 = 21.0° , θ3 = 32.5° , θ4 = 45.8°

θ5 = 63.6° , and θ6 = sin−1 (1.07) = nonexistent

Thus, there are 5 + 5 + 1 = 11 directions for interference maxima.

(b) We check for missing orders by looking for single-slit diffraction minima, at a sinθ = mλ.

For m = 1, (0.700 µm)sinθ = 1(0.501 5 µm) and θ1 = 45.8°

Thus, there is no bright fringe at this angle. There are only nine bright fringes , at

θ = 0°, ± 10.3°, ± 21.0°, ± 32.5°, and ± 63.6° .

(c) I = I max ⎡ sin (π a sinθ λ ) ⎤2

⎢⎣ π sinθ λ ⎦⎥

At θ = 0°, sinθ → 1 and I → 1.00
θ Imax

At θ = 10.3°, π a sinθ = π (0.700 µm)sin10.3° = 0.785 rad = 45.0°
λ
0.501 5 µm

I = ⎡ sin 45.0° ⎤2 = 0.811
I max ⎣⎢ 0.785 ⎦⎥

Similarly, at θ = 21.0°, π a sinθ = 1.57 rad = 90.0° and I = 0.405
At θ = 32.5°, λ Imax
At θ = 63.6°,
π a sinθ = 2.36 rad = 135° and I = 0.090 1
λ Imax

π a sinθ = 3.93 rad = 225° and I = 0.032 4
λ Imax

Diffraction Patterns and Polarization 385

Section 38.3 Resolution of Single-Slit and Circular Apertures

P38.11 We assume Rayleigh’s criterion applies to the predator’s eye with pupil narrowed. (It is a few

times too optimistic for a normal human eye with pupil dilated.)

sin θ = λ = 5.00 × 10−7 m = 1.00 × 10−3 rad
a 5.00 × 10−4

*P38.12 (a) 1.22λ /D = 1.22(589 nm)/(9 000 000 nm) = 79.8 µrad

(b) For a smaller angle of diffraction we choose the smallest visible wavelength, violet at
400 nm, to obtain 1.22λ/D = 1.22(400 nm)/(9 000 000 nm) = 54.2 µrad .

(c) The wavelength in water is shortened to its vacuum value divided by the index of refraction.
The resolving power is improved, with the minimum resolvable angle becoming

1.22λ/D = 1.22(589 nm /1.33)/(9 000 000 nm) = 60.0 µrad
Better than water for many purposes is oil immersion.

P38.13 Undergoing diffraction from a circular opening, the beam spreads into a cone of half-angle

θmin = 1.22 λ = 1.22 ⎛ 632.8 × 10 −9 m ⎞ = 1.54 × 10−4 rad
D ⎝⎜ 0.005 00 m ⎠⎟

The radius of the beam ten kilometers away is, from the definition of radian measure,

( )rbeam = θmin 1.00 × 104 m = 1.544 m

and its diameter is dbeam = 2rbeam = 3.09 m .

*P38.14 When the pupil is open wide, it appears that the resolving power of human vision is limited by

the coarseness of light sensors on the retina. But we use Rayleigh’s criterion as a handy indicator

of how good our vision might be. We take θmin = d = 1.22 λ, where θmin is the smallest angular
L D

separation of two objects for which they are resolved by an aperture of diameter D, d is the

separation of the two objects, and L is the maximum distance of the aperture from the two objects

at which they can be resolved.

Two objects can be resolved if their angular separation is greater than θmin. Thus, θmin should be
as small as possible. Therefore, light with the smaller of the two given wavelengths is easier to

resolve, i.e., blue .

( )( )L =
Dd 5.20 × 10−3 m 2.80 × 10−2 m = 1.193 × 10−4 m2
=
1.22λ 1.22λ λ

Thus L = 186 m for λ = 640 nm, and L = 271 m for λ = 440 nm. The viewer with the assumed

diffraction-limited vision could resolve adjacent tubes of blue in the range 186 m to 271 m , but
cannot resolve adjacent tubes of red in this range.

P38.15 When the pupil is open wide, it appears that the resolving power of human vision is limited by
the coarseness of light sensors on the retina. But we use Rayleigh’s criterion as a handy indicator
of how good our vision might be. According to this criterion, two dots separated center-to-center
by 2.00 mm would overlap

when θmin = d = 1.22 λ
L D

( ( )( ) )L
Thus, = dD = 2.00 × 10−3 m 4.00 × 10−3 m = 13.1 m
1.22λ 1.22 500 × 10−9 m

386 Chapter 38

P38.16 When the pupil is open wide, it appears that the resolving power of human D = 250 × 103 m
λ = 5.00 × 10−7 m
vision is limited by the coarseness of light sensors on the retina. But we use d = 5.00 × 10−3 m

Rayleigh’s criterion as a handy indicator of how good our vision might be.

λ ⎛ 5.00 × 10−7 m⎞
d ⎜⎝ 5.00 × 10−3 m ⎟⎠
( )x
= 1.22 D = 1.22 250 × 103 m = 30.5 m

P38.17 The concave mirror of the spy satellite is probably about 2 m in diameter, and is surely not more
than 5 m in diameter. That is the size of the largest piece of glass successfully cast to a precise
shape, for the mirror of the Hale telescope on Mount Palomar. If the spy satellite had a larger mir-
ror, its manufacture could not be kept secret, and it would be visible from the ground. Outer space
is probably closer than your state capitol, but the satellite is surely above 200-km altitude, for
reasonably low air friction. We find the distance between barely resolvable objects at a distance
of 200 km, seen in yellow light through a 5-m aperture:

y = 1.22 λ
LD

y = (200 000 m ) (1.22 ) ⎛ 6 × 10−7 m⎞ = 3 cm
⎜⎝ 5m ⎠⎟

(Considering atmospheric seeing caused by variations in air density and temperature, the distance

between barely resolvable objects is more like (200 000 m)(1 s) ⎛ 3 1° ⎞ ⎜⎛⎝ π rad ⎟⎠⎞ = 97 cm.)
⎜⎝ 600 s ⎠⎟ 180°

Thus the snooping spy satellite cannot see the difference between III and II or IV on a license
plate. It cannot count coins spilled on a sidewalk, much less read the dates on them.

P38.18 1.22 λ = d λ = c = 0.020 0 m
DL f

D = 2.10 m L = 9 000 m

d = 1.22 (0.020 0 m)(9 000 m) = 105 m

2.10 m

Section 38.4 The Diffraction Grating

P38.19 d = 1.00 cm = 1.00 × 10−2 m = 5.00 µm
2 000 2 000

( )sinθ
= mλ = 1 640 × 10−9 m = 0.128 θ= 7.35°
d 5.00 × 10−6 m

Diffraction Patterns and Polarization 387

P38.20 The principal maxima are defined by

d sinθ = mλ m = 0, 1, 2, . . . q 0.488 m
For m = 1, λ = d sinθ

where q is the angle between the central (m = 0) and the first 1.72 m
order (m = 1) maxima. The value of q can be determined from
the information given about the distance between maxima and the

grating-to-screen distance. From the figure,

tanθ = 0.488 m = 0.284 FIG. P38.20
1.72 m

so θ = 15.8°
and sinθ = 0.273

The distance between grating “slits” equals the reciprocal of the number of grating lines per
centimeter

d = 1 = 1.88 × 10−4 cm = 1.88 × 103 nm
5 310 cm−1

( )The wavelength is λ = d sinθ = 1.88 × 103 nm (0.273) = 514 nm

P38.21 The grating spacing is d = 1.00 × 10−2 m = 2.22 × 10−6 m
4 500

In the 1st-order spectrum, diffraction angles are given by

sinθ = λ : sin θ1 = 656 × 10−9 m = 0.295
d 2.22 × 10−6 m

so that for red θ1 = 17.17° FIG. P38.21
and for blue
sin θ 2 = 434 × 10−9 m = 0.195
2.22 × 10−6 m

so that θ2 = 11.26°

The angular separation is in first-order, ∆θ = 17.17° − 11.26° = 5.91°

In the second-order spectrum, ∆θ = sin−1 ⎛ 2λ 1 ⎞ − sin−1 ⎛ 2λ 2 ⎞ = 13.2°
⎜⎝ d ⎠⎟ ⎜⎝ d ⎠⎟

Again, in the third order, ∆θ = sin−1 ⎛ 3λ1 ⎞ − sin−1 ⎛ 3λ2 ⎞ = 26.5°
⎝⎜ d ⎠⎟ ⎜⎝ d ⎟⎠

Since the red does not appear in the fourth-order spectrum, the answer is complete.

388 Chapter 38

P38.22 sinθ = 0.350: d = λ = 632.8 nm = 1.81 × 103 nm
sinθ 0.350

Line spacing = 1.81 µm

P38.23 (a) d = 1 = 2.732 × 10−4 cm = 2.732 × 10−6 m = 2 732 nm
3 660 lines cm

λ = d sinθ : At θ = 10.09°, λ = 478.7 nm
m

At θ = 13.71°, λ = 647.6 nm

At θ = 14.77°, λ = 696.6 nm

(b) d = λ and 2λ = d sinθ2 so sin θ 2 = 2λ = λ 2λ = 2 sinθ1
sin θ1 d sin θ1

Therefore, if θ1 = 10.09° then sinθ2 = 2 sin(10.09°) gives θ2 = 20.51° .

Similarly, for θ1 = 13.71°, θ2 = 28.30° and for θ1 = 14.77°, θ2 = 30.66° .

P38.24 sinθ = mλ
d

Therefore, taking the ends of the visible spectrum to be λv = 400 nm and λr = 750 nm, the ends
of the different order spectra are defined by:

End of second order: sin θ 2 r = 2λr = 1 500 nm
Start of third order: d d

sin θ3 v = 3λv = 1 200 nm
d d

Thus, it is seen that θ2r > θ3v and these orders must overlap regardless of the value of the

grating spacing d.

*P38.25 The sound has wavelength λ = v = 343 m s s = 9.22 × 10−3 m. Each diffracted beam is described
f 37.2 × 103

by d sinθ = mλ, m = 0, 1, 2, . . .

The zero-order beam is at m = 0, θ = 0. The beams in the first order of interference are to the

left and right at θ = sin−1 1λ = sin−1 9.22 × 10−3 m = sin−1 0.709 = 45.2°. For a second-order beam
d 1.3 × 10−2 m

we would need θ = sin−1 2λ = sin−1 2(0.709) = sin−1 1.42. No angle, smaller or larger than 90°,

d

has a sine greater than 1. Then a diffracted beam does not exist for the second order or any higher

order. The whole answer is then, three beams, at 0° and at 45.2° to the right and to the left .

Diffraction Patterns and Polarization 389

P38.26 For a side maximum, tan θ = y = 0.4 µm
L 6.9 µm

θ = 3.32°

d sinθ = mλ (1)(780 × 10−9 m)

d = = 13.5 µm
sin 3.32°

The number of grooves per millimeter = 1 × 10−3 m
13.5 × 10−6 m

= 74.2 FIG. P38.26

P38.27 d = 1.00 × 10−3 m mm = 4.00 × 10−6 m = 4 000 nm d sinθ = mλ ⇒ m = d sinθ
250 lines mm λ

(a) The number of times a complete order is seen is the same as the number of orders in which
the long wavelength limit is visible.

mmax = d sinθmax = (4 000 nm)sin 90.0° = 5.71
λ
700 nm

or 5 orders is the maximum

(b) The highest order in which the violet end of the spectrum can be seen is:

mmax = d sinθmax = (4 000 nm)sin 90.0° = 10.0
λ
400 nm

or 10 orders in the short-wavelength region

P38.28 (a) The several narrow parallel slits make a diffraction

grating. The zeroth- and first-order maxima are separated

according to

d sinθ = (1) λ sin θ = λ = 632.8 × 10−9 m
d 1.2 × 10−3m

θ = sin−1 (0.000 527) = 0.000 527 rad

y = L tanθ = (1.40 m)(0.000 527) = 0.738 mm

(b) Many equally spaced transparent lines appear on the FIG. P38.28

film. It is itself a diffraction grating. When the same

light is sent through the film, it produces interference

maxima separated according to

d sinθ = (1) λ sin θ = λ = 632.8 × 10−9 m = 0.000 857
d 0.738 × 10−3 m

y = L tanθ = (1.40 m)(0.000 857) = 1.20 mm

An image of the original set of slits appears on the screen. If the screen is removed, light
diverges from the real images with the same wave fronts reconstructed as the original
slits produced. Reasoning from the mathematics of Fourier transforms, Gabor showed
that light diverging from any object, not just a set of slits, could be used. In the picture,
the slits or maxima on the left are separated by 1.20 mm. The slits or maxima on the right
are separated by 0.738 mm. The length difference between any pair of lines is an integer
number of wavelengths. Light can be sent through equally well toward the right or toward
the left. Soccer players shift smoothly between offensive and defensive tactics.

390 Chapter 38

P38.29 d = 1 = 2.38 × 10−6 m = 2 380 nm
4 200 cm

d sinθ = mλ or θ = sin−1 ⎛ mλ ⎞ and y = L tan θ = L tan ⎣⎡⎢sin−1 ⎛ mλ ⎞ ⎤
⎜⎝ d ⎠⎟ ⎝⎜ d ⎟⎠ ⎦⎥

Thus, ∆y = L ⎧ ⎣⎢⎡sin−1 ⎛ mλ2 ⎞ ⎤ − tan ⎡⎣⎢sin−1 ⎛ mλ1 ⎞ ⎤⎫
For m = 1, ⎨tan ⎜⎝ d ⎠⎟ ⎦⎥ ⎜⎝ d ⎠⎟ ⎥⎦ ⎬
For m = 2, ⎩ ⎭

∆y = (2.00 m) ⎪⎨⎧tan ⎡ −1 ⎛ 589.6 ⎞ ⎤ − tan ⎡⎢sin−1 ⎛ 589 ⎞ ⎤⎪⎫ = 0.554 mm
⎪⎩ ⎢sin ⎝⎜ 2 380⎟⎠ ⎥ ⎣ ⎝⎜ 2 380 ⎠⎟ ⎥⎬
⎣ ⎦ ⎦⎭⎪

∆y = (2.00 m) ⎪⎧⎨tan ⎢⎡sin−1 ⎛ 2 (589.6) ⎞ ⎤ − tan ⎡ −1 ⎛ 2 (589) ⎞ ⎤⎪⎫ = 1.54 mm
⎩⎪ ⎣ ⎝⎜ ⎥ ⎢sin ⎜⎝ ⎥⎬
2 380 ⎟⎠ ⎦ ⎣ 2 380 ⎟⎠ ⎦⎭⎪

For m = 3, ∆y = (2.00 m) ⎧⎪⎨tan ⎡ −1 ⎛ 3 ( 589.6 ) ⎞ ⎤ − tan ⎡ −1 ⎛ 3 ( 589) ⎞ ⎤⎫⎪ = 5.04 mm
⎪⎩ ⎢sin ⎝⎜ ⎥ ⎢sin ⎜⎝ ⎥⎬
⎣ 2 380 ⎟⎠ ⎦ ⎣ 2 380 ⎠⎟ ⎦⎪⎭

Thus, the observed order must be m = 2 .

Section 38.5 Diffraction of X-Rays by Crystals

P38.30 2d sinθ = mλ: ( )λ = 2d sinθ = 2 0.353 × 10−9 m sin 7.60° = 9.34 × 10−11 m = 0.093 4 nm
m1

(( ))sinθ
P38.31 2d sinθ = mλ: = mλ = 1 0.140 × 10−9 m = 0.249
and 2d 2 0.281 × 10−9 m

θ = 14.4°

P38.32 Figure 38.23 of the text shows the situation.

2d sinθ = mλ or λ = 2d sinθ
m

m = 1: λ1 = 2 ( 2.80 m)sin 80.0° = 5.51 m

1

m = 2: λ2 = 2 ( 2.80 m)sin 80.0° = 2.76 m

2

m = 3: λ3 = 2 ( 2.80 m)sin 80.0° = 1.84 m

3

*P38.33 The crystal cannot produce diffracted beams of visible light. The wavelengths of visible light are
some hundreds of nanometers. There is no angle whose sine is greater than 1. Bragg’s law
2d sin q = ml cannot be satisfied for a wavelength much larger than the distance between atomic
planes in the crystal.

Diffraction Patterns and Polarization 391

Section 38.6 Polarization of Light Waves

P38.34 The average value of the cosine-squared function is one-half, so the first polarizer transmits

1 the light. The second transmits cos2 30.0° = 3.
24

If = 1 × 3 = 3
2 4 Ii 8 Ii

P38.35 I = Imax cos2 θ ⇒ θ = cos−1 I
I max

(a) I = 1 ⇒ θ = cos−1 1 = 54.7°
Imax 3.00 3.00

(b) I = 1 ⇒ θ = cos−1 1 = 63.4°
Imax 5.00 5.00

(c) I = 1 ⇒ θ = cos−1 1 = 71.6°
Imax 10.0 10.0

P38.36 By Brewster’s law, n = tanθ p = tan (48.0°) = 1.11

P38.37 sin θ c = 1 or n = 1 = 1 = 1.77
n Thus, sinθc sin 34.4°

Also, tanθ p = n. θ p = tan−1 (n) = tan−1 (1.77) = 60.5°

P38.38 sin θ c = 1 and tanθ p = n
n

Thus, sinθc = 1 or cotθ p = sinθc .
tanθ p

P38.39 (a) At incidence, n1 sinθ1 = n2 sinθ2 and θ1′ = θ1. For complete
polarization of the reflected light,

(90 − θ1′) + (90 − θ2 ) = 90° q1 q '
1

θ1′ + θ2 = 90 = θ1 + θ2 n1
n2 q 3 q 3'
Then n1 sinθ1 = n2 sin (90 − θ1 ) = n2 cosθ1

sin θ1 = n2 = tan θ1 q2
cosθ1 n1
n1
At the bottom surface, θ3 = θ2 because the normals to the q4
surfaces of entry and exit are parallel.
FIG. P38.39(a)
Then n2 sinθ3 = n1 sinθ4 and θ3′ = θ3

n2 sinθ2 = n1 sinθ4 and θ4 = θ1

The condition for complete polarization of the reflected light is

90 − θ3′ + 90 − θ4 = 90° θ2 + θ1 = 90°

This is the same as the condition for θ1 to be Brewster’s angle at the top surface.
continued on next page

392 Chapter 38

(b) We consider light moving in a plane perpendicular to n1 q1 q1
the line where the surfaces of the prism meet at the Φ n2
unknown angle Φ. We require

n1 sinθ1 = n2 sinθ2 n3 q 2 q 3 q 3
θ1 + θ2 = 90°

So n1 sin (90 − θ2 ) = n2 sinθ2 n1 = tan θ 2 q4
n2

And n2 sinθ3 = n3 sinθ4 θ3 + θ4 = 90° FIG. P38.39(b)
n2 sinθ3 = n3 cosθ3
tan θ3 = n3
n2

In the triangle made by the faces of the prism and the ray in the prism,

Φ + 90 + θ2 + (90 − θ3 ) = 180

So one particular apex angle is required, and it is

Φ = θ3 − θ2 = tan−1 ⎛ n3 ⎞ − tan−1 ⎛ n1 ⎞
⎜⎝ n2 ⎠⎟ ⎜⎝ n2 ⎟⎠

Here a negative result is to be interpreted as meaning the same as a positive result.

P38.40 For incident unpolarized light of intensity Imax:
The average value of the cosine-squared function is one
half, so the intensity after transmission by the first disk is

I = 1 I max
2

After transmitting 2nd disk: I = 1 I max cos2 θ
2
FIG. P38.40

After transmitting 3rd disk: I = 1 I max cos2 θ cos2 (90° −θ)
2

where the angle between the first and second disk is θ = ω t .

Using trigonometric identities cos2 θ = 1 (1 + cos 2θ )

2

and cos2 (90° − θ ) = sin2 θ = 1 (1 − cos 2θ )

2

we have I = 1 I max ⎡(1 + cos 2θ ) ⎤ ⎡(1 − cos 2θ ) ⎤
2
⎣⎢ 2 ⎦⎥ ⎣⎢ 2 ⎥⎦

( )I 1 1 ⎛ 1⎞ )
= 8 I max 1 − cos2 2θ = 8 I max ⎝ 2⎠ (1 − cos 4θ

Since θ = ω t, the intensity of the emerging beam is given by I = 1 I max (1 − 4ω t ) .
16

Diffraction Patterns and Polarization 393

*P38.41 (a) Let I0 represent the intensity of unpolarized light incident on the first polarizer. In Malus’s
law the average value of the cosine-squared function is 1/2, so the first filter lets through 1/2
of the incident intensity. Of the light reaching them, the second filter passes cos245° = 1/2 and
the third filter also cos245° = 1/2. The transmitted intensity is then I0(1/2)(1/2)(1/2) = 0.125 I0.
The reduction in intensity is by a factor of 0.875 of the incident intensity.

(b) By the same logic as in part (a) we have transmitted I0(1/2)(cos230°)(cos230°)(cos230°) =
I0(1/2)(cos230°)3 = 0.211 I0. Then the fraction absorbed is 0.789 .

(c) Yet again we compute transmission I0(1/2)(cos215°)6 = 0.330 I0. And the fraction absorbed
is 0.670 .

(d) We can get more and more of the incident light through the stack of ideal filters, approach-
ing 50%, by reducing the angle between the transmission axes of each one and the next.

Additional Problems

*P38.42 The central bright fringe is wider than the side bright fringes, so the light must have been

diffracted by a single slit . For precision, we measure from the second minimum on one side

of the center to the second minimum on the other side:

2y = (11.7 − 6.3) cm = 5.4 cm y = 2.7 cm

tanθ = y = 0.027 m = 0.010 4
L 2.6 m

θ = 0.595° = 0.010 4 rad

a sinθ = mλ

( ) ( )a =
mλ 2 632.8 × 10−9 m 2 632.8 × 10−9 m = 1.22 × 10−4 m
= =
sinθ sin 0.595° 0.010 4

P38.43 Let the first sheet have its axis at angle q to the original plane of polarization, and let each further
sheet have its axis turned by the same angle.

The first sheet passes intensity Imax cos2 θ
The second sheet passes
and the nth sheet lets through Imax cos4 θ
Try different integers to find
(a) So n = 6 Imax cos2n θ ≥ 0.90Imax where θ = 45°
n

cos2×5 ⎝⎜⎛ 45° ⎞⎠⎟ = 0.885 cos2×6 ⎝⎜⎛ 45° ⎞⎟⎠ = 0.902
5 6

(b) θ = 7.50°

394 Chapter 38

P38.44 Consider vocal sound moving at 340 m/s and of frequency 3 000 Hz. Its wavelength is

λ = v = 340 m s = 0.113 m
f 3 000 Hz

If your mouth, for horizontal dispersion, behaves similarly to a slit 6.00 cm wide, then
a sinθ = mλ predicts no diffraction minima. You are a nearly isotropic source of this sound. It
spreads out from you nearly equally in all directions. On the other hand, if you use a megaphone
with width 60.0 cm at its wide end, then a sinθ = mλ predicts the first diffraction minimum at

θ = sin−1 ⎛ mλ ⎞ = sin−1 ⎜⎛⎝ 0.113 m ⎟⎞⎠ = 10.9°
⎝⎜ a ⎟⎠ 0.600 m

This suggests that the sound is radiated mostly toward the front into a diverging beam of angular
diameter only about 20°. With less sound energy wasted in other directions, more is available for
your intended auditors. We could check that a distant observer to the side or behind you receives
less sound when a megaphone is used.

*P38.45 (a) We assume the first side maximum is at a sinq = 1.5 l. (Its location is determined more
(b) precisely in problem 66.) Then the required fractional intensity is

I = ⎡ sin(π a sinθ/λ) ⎤2 = ⎡ sin(1.5π ) ⎤2 = 1 = 0.0450
I max ⎢⎣ π a sinθ/λ ⎦⎥ ⎢⎣ 1.5π ⎥⎦ 2.25π 2

Proceeding as in part (a) we have a sinq/l = 2.5 and

I = ⎡ sin(π a sinθ/λ) ⎤2 = ⎡ sin(2.5π ) ⎤2 = 1 = 0.0162
I max ⎢⎣ π a sinθ/λ ⎦⎥ ⎢⎣ 2.5π ⎥⎦ 6.25π 2

*P38.46 The energy in the central maximum we can estimate in Figure 38.6 as proportional to
(height)(width) = Imax(2p)

As in problem 45, the maximum height of the first side maximum is approximately

Imax[sin (3p /2)/(3p /2)]2 = 4Imax/9p 2

Then the energy in one side maximum is proportional to p (4Imax/9p 2),
and that in both of the first side maxima together is proportional to 2p (4Imax/9p 2).

Similarly and more precisely, and always with the same proportionality constant, the energy in
both of the second side maxima is proportional to 2p (4Imax/25p 2).

The energy in all of the side maxima together is proportional to

2p (4Imax/p 2)[1/32 + 1/52 + 1/72 + …] = 2p (4Imax/p 2)[p 2/8 – 1] = (8Imax/p )[p 2/8 – 1]

= Imax(p – 8/p) = 0.595Imax

The ratio of the energy in the central maximum to the total energy is then

Imax(2p)/{Imax(2p) + 0.595Imax} = 2p /6.88 = 0.913 = 91.3%
Our calculation is only a rough estimate, because the shape of the central maximum in particular
is not just a vertically-stretched cycle of a cosine curve. It is slimmer than that.

Diffraction Patterns and Polarization 395

P38.47 The first minimum is at a sinθ = (1) λ
This has no solution if
or if λ >1
a
a < λ = 632.8 nm

*P38.48 With light in effect moving through vacuum, Rayleigh’s criterion limits the resolution according to

d/L = 1.22 l/D D = 1.22 lL/d = 1.22 (885 × 10–9 m) 12 000 m/2.3 m = 5.63 mm

The assumption is absurd. Over a horizontal path of 12 km in air, density variations associated
with convection (“heat waves” or what an astronomer calls “seeing”) would make the motor-
cycles completely unresolvable with any optical device.

P38.49 d = 400 1 = 2.50 × 10−6 m
mm −1
2 × 541 × 10−9 m ⎞
(a) d sinθ = mλ θa = sin −1 ⎛ 2.50 × 10−6 m ⎟⎠ = 25.6°
⎜⎝

(b) λ = 541 × 10−9 m = 4.07 × 10−7 m θb = sin−1 ⎛ 2 × 4.07 × 10−7 m ⎞ = 19.0°
1.33 ⎜⎝ 2.50 × 10−6 m ⎟⎠

(c) d sinθa = 2λ and d sinθb = 2λ
n

combine by substitution to give n sinθb = (1) sinθa

P38.50 (a) λ = v : λ = 3.00 × 108 ms = 0.214 m
f 1.40 × 109 s−1

θmin = 1.22 λ: θmin = 1.22 ⎛ 0.214 m ⎞ = 7.26 µrad
D ⎝ 3.60 × 104 m ⎠

θmin = 7.26 µrad ⎛ 180 × 60 × 60 s⎞ = 1.50 arc seconds
⎝ π ⎠

(b) θmin = d: ( )d = θmin L = 7.26 × 10−6 rad (26 000 ly) = 0.189 ly
L

(c) It is not true for humans, but we assume the hawk’s visual acuity is limited only by
Rayleigh’s criterion.

θmin = 1.22 λ θmin = 1.22 ⎛ 500 × 10−9 m⎞ = 50.8 µrad (10.5 seconds of arc)
D ⎝⎜ 12.0 × 10−3 m ⎠⎟

( )(d) d = θmin L = 50.8 × 10−6 rad (30.0 m) = 1.52 × 10−3 m = 1.52 mm

396 Chapter 38

P38.51 With a grazing angle of 36.0°, the angle of incidence is 54.0°

tanθp = n = tan 54.0° = 1.38

In the liquid, λn = λ = 750 nm = 545 nm .
n 1.38

P38.52 (a) Bragg’s law applies to the space lattice of melanin rods. Consider the planes d = 0.25 µm
apart. For light at near-normal incidence, strong reflection happens for the wavelength
given by 2d sinθ = mλ. The longest wavelength reflected strongly corresponds to m = 1:

)(2 0.25 × 10−6 m sin 90° = 1λ λ = 500 nm. This is the blue-green color.

(b) For light incident at grazing angle 60°, 2d sinθ = mλ

)(gives 1λ = 2 0.25 × 10−6 m sin 60° = 433 nm. This is violet.

(c) Your two eyes receive light reflected from the feather at different angles, so they receive
light incident at different angles and containing different colors reinforced by constructive
interference.

(d) The longest wavelength that can be reflected with extra strength by these melanin rods is
the one we computed first, 500 nm blue-green.

(e) If the melanin rods were farther apart (say 0.32 µm) they could reflect red with
constructive interference.

P38.53 (a) d sinθ = mλ

)(or d =
mλ 3 500 × 10−9 m = 2.83 µm
=
sinθ sin 32.0°

Therefore, lines per unit length = 1 = 1 m
d 2.83 × 10−6

or lines per unit length = 3.53 × 105 m−1 = 3.53 × 103 cm−1 .

( )(b)
sin θ = mλ = m 500 × 10−9 m = m (0.177)
d 2.83 × 10−6 m

For sinθ ≤ 1.00, we must have m (0.177) ≤ 1.00

or m ≤ 5.66

Therefore, the highest order observed is m=5

Total number of primary maxima observed is 2m + 1 = 11

Diffraction Patterns and Polarization 397

*P38.54 (a) For the air-to-water interface, qp

tanθ p = nwater = 1.33 θp = 53.1° Air
nair 1.00 q q 2 q3 Water

and (1.00)sinθp = (1.33)sinθ2

θ2 = sin−1 ⎛ sin 53.1° ⎞ = 36.9°
⎝ 1.33 ⎠

For the water-to-slab interface, tanθ p = tanθ3 = nslab =n FIG. P38.54
nwater 1.33

so θ3 = tan−1(n/1.33).

The angle between surfaces is θ = θ3 − θ2 = tan−1(n/1.33) − 36.9°.

(b) If we imagine n → ∞, then θ → 53.1°. The material of the slab in this case is higher in opti-

cal density than any gem. The light in the water skims along the upper surface of the slab.

(c) If we imagine n = 1, then θ = 0. The slab is so low in optical density that it is like air. The
light strikes parallel surfaces as it enters and exits the water, both at the polarizing angle.

P38.55 A central maximum and side maxima in seven orders of interference appear.
If the seventh order is just at 90°,

d sinθ = mλ )(d (1) = 7 654 × 10−9 m d = 4.58 µm

If the seventh order is at less than 90°, the eighth order might be nearly
ready to appear according to

d (1) = 8(654 × 10−9 m) d = 5.23 µm

FIG. P38.55

Thus 4.58 µm < d < 5.23 µm .

P38.56 (a) We require θmin = 1.22 λ = radius of diffraction disk = D
Then D L 2L

D2 = 2.44λ L

)((b) D = 2.44 500 × 10−9 m (0.150 m) = 428 µm

P38.57 For the limiting angle of resolution between lines we

(( ))assume
θmin = 1.22 λ = 1.22 550 × 10−9 m = 1.34 × 10−4 rad.
D 5.00 × 10−3 m

Assuming a picture screen with vertical dimension , the minimum viewing distance for no

visible lines is found from θmin = 485. The desired ratio is then
L

1 1
485θmin 1.34 × 10−4
=( )L = = 15.4
485 rad

When the pupil of a human eye is wide open, its actual resolving power is significantly poorer
than Rayleigh’s criterion suggests.

398 Chapter 38

P38.58 (a) Applying Snell’s law gives n2 sin φ = n1 sinθ. From the sketch,
we also see that:

θ + φ + β = π , or φ = π − (θ + β )

Using the given identity: FIG. P38.58(a)

sin φ = sin π cos(θ + β ) − cosπ sin (θ + β )

which reduces to: sin φ = sin (θ + β )

Applying the identity again: sin φ = sinθ cos β + cosθ sin β

Snell’s law then becomes: n2 (sinθ cos β + cosθ sin β ) = n1 sinθ

or (after dividing by cosq ): n2 (tanθ cos β + sin β ) = n1 tanθ

Solving for tanθ gives: tanθ = n2 sin β
n1 − n2 cos β

(b) If β = 90.0°, n1 = 1.00, and n2 = n, the above result becomes:

tanθ = n (1.00) , or n = tanθ, which is Brewster’s law.

1.00 − 0

P38.59 (a) From Equation 38.1, θ = sin−1 ⎛ mλ ⎞
In this case m = 1 and ⎝⎜ a ⎟⎠
Thus,
λ = c = 3.00 × 108 ms = 4.00 × 10−2 m
(b) From Equation 38.2, f 7.50 × 109 Hz
When θ = 15.0°,
and θ = sin −1 ⎛ 4.00 × 10−2 m⎞ = 41.8°
⎜⎝ 6.00 × 10−2 m ⎟⎠

I = ⎡ sin (φ) ⎤2 where φ = π a sinθ
I max λ
⎢⎣ φ ⎥⎦

φ = π (0.060 0 m)sin15.0° = 1.22 rad

0.040 0 m

I = ⎡ sin (1.22 rad) ⎤2 = 0.593
I max
⎢⎣ 1.22 rad ⎥⎦

(c) sinθ = λ so θ = 41.8°:
a

This is the minimum angle subtended by the two sources at the slit. Let
a be the half angle between the sources, each a distance = 0.100 m
from the center line and a distance L from the slit plane. Then,

L= cot α = (0.100 m ) cot ⎛ 41.8° ⎞ = 0.262 m FIG. P38.59(c)
⎝ 2 ⎠

Diffraction Patterns and Polarization 399

*P38.60 (a) The first sheet transmits one-half the intensity of the originally unpolarized light, because
(b) the average value of the cosine-squared function in Malus’s law is one-half. Then

(I )= 1 cos2 45.0° (cos2 45.0°) = 1
Imax 2 8

No recipes remain. The two experiments follow precisely analogous steps, but the results
are different. The middle filter in part (a) changes the polarization state of the light that
passes through it, but the recipe selections do not change individual recipes. The result for
light gives us a glimpse of how quantum-mechanical measurements differ from classical
measurements.

P38.61 (a) Constructive interference of light of wavelength l on the screen is described by

( )d sinθ = mλ where tanθ = y so sinθ = y . Then (d ) y L2 + y2 −1 2 = mλ.
L L2 + y2

Differentiating with respect to y gives

−1 ⎛ 1⎞ −3 2 (0 + 2y) = m dλ
( ) ( )d1 2 (d) ⎝ 2⎠
L2 + y2 + y − L2 + y2 dy

d − (d) y2 = m dλ = (d ) L2 + (d) y2 − (d) y2
+ y2 dy
L2 + y2 L2 + y2 32
( ) ( ) ( )L212 32

(d ) L2

L2 + y2
( )dλ = m 32

dy

10−2 m ⎛ 0.55 × 10−6 m⎞
8 000 ⎝⎜ 1.25 × 10−6 m ⎠⎟
( )(b)
Here d sinθ = mλ gives sin θ =1 550 × 10−9 m ,θ = sin −1 = 26.1°

y = L tanθ = 2.40 m tan 26.1° = 1.18 m

dL2 1.25 × 10−6 m (2.40 m)2
L2 + y2 1 (2.4 m)2 + (1.18 m)2 3
( ) ( )Now dλ= m 32 = 2 = 3.77 × 10−7 = 3.77 nm cm .
dy

P38.62 (a) The angles of bright beams diffracted from the grating are given by (d)sinθ = mλ. The

angular dispersion is defined as the derivative dθ : (d) cosθ dθ = m dθ = m
dλ dλ d cosθ
dλ

(b) For the average wavelength 578 nm,

d sinθ = mλ )(0.02 m sinθ = 2 578 × 10−9 m

8 000

θ = sin−1 2 × 578 × 10−9 m = 27.5°
2.5 × 10−6 m

The separation angle between the lines is

∆θ = dθ ∆λ = d m ∆λ = 2.5 × 10−6 2 2.11 × 10−9 m
dλ cosθ m cos 27.5°

= 0.001 90 = 0.001 90 rad = 0.001 90 rad ⎛ 180° ⎞ = 0.109°
⎝ π rad⎠

400 Chapter 38

P38.63 (a) The E and O rays, in phase at the surface of the plate, will have a phase difference

θ = ⎛ 2π ⎞ δ
⎝ λ ⎠

after traveling distance d through the plate. Here d is the difference in the optical path
lengths of these rays. The optical path length between two points is the product of the
actual path length d and the index of refraction. Therefore,

δ = dnO − dnE

The absolute value is used since nO may be more or less than unity. Therefore,
nE

θ = ⎛ 2π ⎞ dnO − dnE = ⎛ 2π ⎞ d nO − nE
⎝ λ ⎠ ⎝ λ ⎠

( )(b) d = λθ = 550 × 10−9 m (π 2) = 1.53 × 10−5 m = 15.3 µm
2π nO − nE 2π 1.544 − 1.553

P38.64 (a) From Equation 38.2, I = ⎡ sin (φ) ⎤2
I max
⎣⎢ φ ⎥⎦

where we define φ ≡ π a sinθ
Therefore, when λ

I =1 we must have sin φ = 1 , or sin φ = φ
Imax 2 φ2 2

(b) Let y1 = sin φ and y2 = φ.
2

A plot of y1 and y2 in the range 1.00 ≤ φ ≤ π is shown to
2

the right.

The solution to the transcendental equation is found to

be φ = 1.39 rad .

FIG. P38.64(b)

(c) π a sinθ = φ gives sin θ = ⎛φ⎞ λ = 0.443 λ.
λ ⎝π⎠ a a

If λ is small, then θ ≈ 0.443 λ .
aa

This gives the half-width, measured away from the maximum at θ = 0. The pattern is
symmetric, so the full width is given by

∆θ = 0.443 λ − ⎛ −0.443 λ ⎞ = 0.886λ
a ⎜⎝ a ⎠⎟ a