Capítulo 35 (5th edition) con soluciones la naturaleza de la luz y las leyes de la optica geometrica

Chapter 35 Solutons

35.1 The Moon's radius is 1.74 × 10 6 m and the Earth's radius is 6.37 × 10 6 m. The total distance
traveled by the light is:

d = 2(3.84 × 10 8 m – 1.74 × 10 6 m – 6.37 × 10 6 m) = 7.52 × 10 8 m

7.52 × 10 8 m
This takes 2.51 s, so v = = 2.995 × 10 8 m/s = 299.5 Mm/s
2.51 s

∆x 2(1.50 × 10 8 km)(1000 m/km)
35.2 ∆x = ct ; c= t = (22.0 min)(60.0 s/min)
= 2.27 × 10 8 m/s = 227 Mm/s

35.3 The experiment is most convincing if the wheel turns fast enough to pass outgoing light
through one notch and returning light through the next: t = 2l c

cθ (2.998 × 10 8 )[2 π / (720)]
θ = ω t = ω 
2l
so ω= = = 114 rad/s
 c 2l 2(11.45 × 10 3 )
The returning light would be blocked by a tooth at one-half the angular speed, giving another
data point.

35.4 (a) For the light beam to make it through both slots, the time for the light to travel the distance d
must equal the time for the disk to rotate through the angle θ, if c is the speed of light,

d θ dω
= , so c=
c ω θ

(b) We are given that

1.00°  π rad  rev  2 π rad 
d = 2.50 m , θ= = 2 .91 × 10 − 4 rad, ω = 5555 = 3.49 × 10 4 rad s
60.0  180°  s  1.00 rev 

c= =
(
dω ( 2.50 m ) 3.49 × 10 rad s
4
)
= 3.00 × 108 m s = 300 Mm/s
−4
θ 2.91 × 10 rad

n1
35.5 Using Snell's law, sin θ 2 = sin θ1
n2
λ1
θ 2 = 25.5° λ2 = = 442 nm
n1

© 2000 by Harcourt, Inc. All rights reserved.

Chapter 35 Solutions 323

c 3.00 × 10 8 m/s
35.6 (a) f= = = 4.74 × 10 14 Hz
λ 6.328 × 10 − 7 m

λ air 632.8 nm
(b) λ glass = = = 422 nm
n 1.50

cair 3.00 × 10 8 m/s
(c) vglass = = = 2.00 × 10 8 m/s = 200 Mm/s
n 1.50

35.7 n1 sin θ1 = n2 sin θ 2

sin θ1 = 1.333 sin 45.0°

sin θ1 = (1.33)(0.707) = 0.943
Figure for Goal Solution
θ1 = 70.5° → 19.5° above the horizon

Goal Solution
An underwater scuba diver sees the Sun at an apparent angle of 45.0° from the vertical. What is the
actual direction of the Sun?

G: The sunlight refracts as it enters the water from the air. Because the water has a higher index of
refraction, the light slows down and bends toward the vertical line that is normal to the interface.
Therefore, the elevation angle of the Sun above the water will be less than 45° as shown in the
diagram to the right, even though it appears to the diver that the sun is 45° above the horizon.

O: We can use Snell’s law of refraction to find the precise angle of incidence.

A: Snell’s law is: n1 sin θ1 = n2 sin θ 2

which gives sin θ1 = 1.333 sin 45.0°

sin θ1 = (1.333)(0.707) = 0.943

The sunlight is at θ1 = 70.5° to the vertical, so the Sun is 19.5° above the horizon.

L: The calculated result agrees with our prediction. When applying Snell’s law, it is easy to mix up the
index values and to confuse angles-with-the-normal and angles-with-the-surface. Making a sketch
and a prediction as we did here helps avoid careless mistakes.


324 Chapter 35 Solutions

*35.8 (a) n1 sin θ1 = n2 sin θ 2

1.00 sin 30.0° = n sin 19.24°

n = 1.52

c 3.00 × 10 8 m/s
(c) f= = = 4.74 × 10 14 Hz in air and in syrup.
λ 6.328 × 10 − 7 m

c 3.00 × 10 8 m/s
(d) v= = = 1.98 × 10 8 m/s = 198 Mm/s
n 1.52

v 1.98 × 10 8 m/s
(b) λ= = = 417 nm
f 4.74 × 10 14 / s

c 3.00 × 108 m s
35.9 (a) Flint Glass: v = = = 1.81 × 108 m s = 181 Mm/s
n 1.66

c 3.00 × 108 m s
(b) Water: v= = = 2 .25 × 108 m s = 225 Mm/s
n 1.333

c 3.00 × 108 m s
(c) Cubic Zirconia: v= = = 1.36 × 108 m s = 136 Mm/s
n 2.20

35.10 n1 sin θ1 = n2 sin θ 2 ; 1.333 sin 37.0° = n2 sin 25.0°

c c
n2 = 1.90 = v ; v = 1.90 = 1.58 × 10 8 m/s = 158 Mm/s

 n 1 sin θ 1 
35.11 n1 sin θ1 = n2 sin θ 2 ; θ 2 = sin–1  n 
 2 

 (1.00)(sin 30°) 
θ 2 = sin −1   = 19.5°
 1.50 

θ 2 and θ 3 are alternate interior angles formed by the ray cutting
parallel normals. So, θ 3 = θ 2 = 19.5° .

1.50 sin θ 3 = (1.00) sin θ 4

θ4 = 30.0°


λ0 436 nm
35.12 (a) Water λ = = = 327 nm
n 1.333

λ0 436 nm
(b) Glass λ = = = 287 nm
n 1.52

*35.13 sin θ1 = nw sin θ 2

1 1
sin θ 2 = sin θ1 = sin(90.0° − 28.0°) = 0.662
1.333 1.333

θ 2 = sin −1 0.662 = 41.5°

d 3.00 m
h= = = 3.39 m
tan θ 2 tan 41.5°

35.14 (a) From geometry, 1.25 m = d sin 40.0°

so d = 1.94 m

(b) 50.0° above horizontal , or parallel to the
incident ray

*35.15 The incident light reaches the left-hand mirror at
distance

(1.00 m) tan 5.00° = 0.0875 m

above its bottom edge. The reflected light first reaches
the right-hand mirror at height

2(0.0875 m) = 0.175 m

It bounces between the mirrors with this distance
between points of contact with either.

1.00 m
Since 0.175 m = 5.72, the light reflects

five times from the right-hand mirror and six times from the left.



*35.16 At entry, n1 sin θ1 = n2 sin θ 2 or 1.00 sin 30.0° = 1.50 sin θ 2

θ 2 = 19.5°

The distance h the light travels in the medium is given by

(2.00 cm) (2.00 cm)
cos θ 2 = or h= = 2.12 cm
h cos 19.5°

The angle of deviation upon entry is α = θ1 − θ 2 = 30.0° − 19.5° = 10.5°

d
The offset distance comes from sin α = : d = (2.21 cm) sin 10.5° = 0.388 cm
h

2.00 cm
*35.17 The distance, h, traveled by the light is h = = 2.12 cm
cos 19.5°

c 3.00 × 108 m/s
The speed of light in the material is v= = = 2.00 × 108 m/s
n 1.50

h 2.12 × 10 − 2 m
Therefore, t= = = 1.06 × 10 − 10 s = 106 ps
v 2.00 × 108 m/s

*35.18 Applying Snell's law at the air-oil interface,

nair sin θ = noil sin 20.0° yields θ = 30.4°

Applying Snell's law at the oil-water interface

nw sin θ ′ = noil sin 20.0° yields θ ′ = 22.3°

*35.19 time difference = (time for light to travel 6.20 m in ice) – (time to travel 6.20 m in air)

6.20 m 6.20 m c
∆t = − but v=
vice c n

 1.309 1 (6.20 m)
∆t = (6.20 m) − = (0.309) = 6.39 × 10 −9 s = 6.39 ns
 c c c


*35.20 Consider glass with an index of refraction of 1.5, which is 3 mm thick The speed of light i n
the glass is
3 × 108 m/s
= 2 × 108 m/s
1.5

3 × 10 −3 m 3 × 10 −3 m
The extra travel time is − ~ 10–11 s
2 × 108 m / s 3 × 108 m / s

600 nm
For light of wavelength 600 nm in vacuum and wavelength = 400 nm in glass,
1.5

3 × 10 − 3 m 3 × 10 − 3 m
the extra optical path, in wavelengths, is − ~ 103 wavelengths
4 × 10 −7 m 6 × 10 −7 m

*35.21 Refraction proceeds according to (1.00) sin θ1 = (1.66) sin θ 2 (1)

(a) For the normal component of velocity to be constant, v1 cos θ1 = v2 cos θ 2

or (c ) cos θ1 = (c 1.66) cos θ 2 (2)

We multiply Equations (1) and (2), obtaining: sin θ1 cos θ1 = sin θ 2 cos θ 2

or sin 2θ1 = sin 2θ 2

The solution θ1 = θ 2 = 0 does not satisfy Equation (2) and must be rejected. The physical
solution is 2θ1 = 180° − 2θ 2 or θ 2 = 90.0° − θ1 . Then Equation (1) becomes:

sin θ1 = 1.66 cos θ1 , or tan θ1 = 1.66

which yields θ1 = 58.9°

(b) Light entering the glass slows down and makes a smaller angle with the normal. Both effects
reduce the velocity component parallel to the surface of the glass, so that component cannot
remain constant, or will remain constant only in the trivial case θ1 = θ 2 = 0

35.22 See the sketch showing the path of the light
ray. α and γ are angles of incidence at
mirrors 1 and 2.
For triangle abca, 2α + 2 γ + β = 180°
or β = 180° − 2(α + γ ) (1)
Now for triangle bcdb,

(90.0° − α ) + (90.0° − γ ) + θ = 180°
or θ =α+γ (2)



Substituting Equation (2) into Equation (1) gives β = 180° − 2θ

Note: From Equation (2), γ = θ − α . Thus, the ray will follow a path like that shown only if
α < θ . For α > θ , γ is negative and multiple reflections from each mirror will occur before
the incident and reflected rays intersect.

35.23 Let n(x) be the index of refraction at distance x below the top of the atmosphere and
n( x = h) = n be its value at the planet surface. Then,

 n − 1.000 
n( x ) = 1.000 + x
 h 

(a) The total time required to traverse the atmosphere is

h dx h n( x ) 1 h  n − 1.000   h (n − 1.000)  h 2  h  n + 1.000 
t=∫ =∫ dx = ∫ 1.000 + x  dx = +  2  = c
0 v 0 c c 0   h   c ch   2 

20.0 × 10 3 m  1.005 + 1.000 
t= = 66.8 µs
3.00 × 108 m s  2 

(b) The travel time in the absence of an atmosphere would be h / c . Thus, the time in the
presence of an atmosphere is

 n + 1.000 
= 1.0025 times larger or 0.250% longer .
 2 

35.24 Let n(x) be the index of refraction at distance x below the top of the atmosphere and
n( x = h) = n be its value at the planet surface. Then,

 n − 1.000 
n( x ) = 1.000 + x
 h 

(a) The total time required to traverse the atmosphere is

t=∫
h dx
=∫
hn ( x) dx = 1 h  n − 1.000   h (n − 1.000)  h 2  h  n + 1.000 
c ∫0 
1.000 + x  dx = +  2  = c
0 v 0 c   h   c ch   2 

(b) The travel time in the absence of an atmosphere would be h / c . Thus, the time in the
presence of an atmosphere is

 n + 1.000 
times larger
 2 


35.25 From Fig. 35.20 nv = 1.470 at 400 nm and nr = 1.458 at 700 nm
Then (1.00)sin θ = 1.470 sin θ v and (1.00)sin θ = 1.458 sin θ r

 sin θ   sin θ 
δ r − δ v = θ r − θ v = sin −1 − sin −1
 1.458   1.470 

 sin 30.0°   sin 30.0° 
∆δ = sin −1 − sin −1 = 0.171°
 1.458   1.470 

 n sin θ1 
35.26 n1 sin θ1 = n2 sin θ 2 so θ 2 = sin −1  1 
 n2 

 (1.00)(sin 30.0°) 
θ 2 = sin −1   = 19.5°
 1.50 

θ 3 = ([(90.0° − 19.5°) + 60.0°] − 180°) + 90.0° = 40.5°

 n sin θ 3  −1  (1.50)(sin 40.5°) 
n3 sin θ 3 = n4 sin θ 4 so θ 4 = sin −1  3  = sin   = 77.1°
 n4   1.00 

35.27 Taking Φ to be the apex angle and δ min to be the angle of minimum deviation, from
Equation 35.9, the index of refraction of the prism material is

 Φ + δ min 
sin
 2 
n=
sin(Φ 2)

Φ
Solving for δ min , δ min = 2 sin −1  n sin  − Φ = 2 sin −1[( 2.20) sin ( 25.0°)] − 50.0° = 86.8°
 2

35.28 n(700 nm) = 1.458

(a) (1.00) sin 75.0° = 1.458 sin θ 2; θ 2 = 41.5°

(b) Let θ 3 + β = 90.0°, θ 2 + α = 90.0°; then α + β + 60.0° = 180°

So 60.0° − θ 2 − θ 3 = 0 ⇒ 60.0° − 41.5° = θ 3 = 18.5°

(c) 1.458 sin 18.5° = 1.00 sin θ4 θ4 = 27.6°

(d) γ = (θ1 − θ 2 ) + [β − (90.0° − θ 4 )]

γ = 75.0° − 41.5° + (90.0° − 18.5°) − (90.0° − 27.6°) = 42.6°



sin θ1
35.29 For the incoming ray, sin θ 2 =
n
 sin 50.0° 
Using the figure to the right, (θ 2 )violet = sin −1 = 27.48°
 1.66 

 sin 50.0° 
(θ 2 )red = sin −1 = 28.22°
 1.62 

For the outgoing ray, θ 3 = 60.0° – θ 2
′ and sin θ 4 = nsin θ 3

(θ 4 )violet = sin −1[1.66 sin 32.52°] = 63.17°

(θ 4 )red = sin −1[1.62 sin 31.78°] = 58.56°

The dispersion is the difference ∆ θ 4 = (θ 4 )violet − (θ 4 )red = 63.17° – 58.56° = 4.61°

 Φ + δ min 
sin
 2 
35.30 n=
sin(Φ 2)

Φ + δ min
For small Φ, δ min ≈ Φ so is also a small angle. Then, using the small angle
2
approximation ( sin θ ≈ θ when θ << 1 rad), we have:

n≈
(Φ + δ min ) 2
=
Φ + δ min
or δ min ≈ (n − 1)Φ where Φ is in radians.
Φ2 Φ

35.31 At the first refraction, (1.00) sin θ1 = n sin θ 2
The critical angle at the second surface is given by

 1.00 
n sin θ 3 = 1.00 , or θ 3 = sin −1 = 41.8°.
 1.50 

But, θ 2 = 60.0° −θ 3 . Thus, to avoid total internal reflection at the
second surface (i.e., have θ 3 < 41.8°), it is necessary that θ 2 > 18.2°.
Since sin θ1 = n sin θ 2 , this requirement becomes

sin θ1 > (1.50) sin (18.2°) = 0.468 , or θ 1 > 27.9°


35.32 At the first refraction, (1.00) sin θ1 = n sin θ 2 . The critical angle at
the second surface is given by

n sin θ 3 = 1.00 , or θ 3 = sin −1 (1.00 n)

But (90.0° − θ 2 ) + (90.0° − θ 3 ) + Φ = 180°, which gives θ 2 = Φ − θ 3 .

Thus, to have θ 3 < sin −1 (1.00 n) and avoid total internal reflection at the second surface, it is
necessary that θ 2 > Φ − sin −1 (1.00 n) . Since sin θ1 = n sin θ 2 , this requirement becomes

  1.00     1.00   
sin θ1 > n sin Φ − sin −1 θ1 > sin −1  n sin Φ − sin −1 
 n  
or
  n     

Through the application of trigonometric identities, θ1 > sin −1  n2 − 1 sin Φ − cos Φ
 

sin(δ + φ )
35.33 n=
sin(φ / 2)
so 1.544 sin ( φ ) = sin(5° + φ ) = cos( φ ) sin 5° + sin( φ ) cos 5°
1
2
1
2
1
2
1
2

tan ( φ ) = 1.544 − 5° 5°
1
2
sin
cos
and φ = 18.1°

*35.34 Note for use in every part: Φ + (90.0° − θ 2 ) + (90.0° − θ 3 ) = 180°
so θ3 = Φ − θ2
At the first surface is α = θ1 − θ 2
At exit, the deviation is β = θ4 − θ3

The total deviation is therefore δ = α + β = θ1 + θ 4 − θ 2 − θ 3 = θ1 + θ 4 − Φ
 sin 48.6° 
(a) At entry: n1 sin θ1 = n2 sin θ 2 or θ 2 = sin −1 = 30.0°
 1.50 
Thus, θ 3 = 60.0° − 30.0° = 30.0°
At exit: 1.50 sin 30.0° = 1.00 sin θ 4 or θ 4 = sin −1[1.50 sin ( 30.0°)] = 48.6°

so the path through the prism is symmetric when θ1 = 48.6° .

(b) δ = 48.6° + 48.6° − 60.0° = 37.2°

sin 45.6°
(c) At entry: sin θ 2 = ⇒ θ 2 = 28.4° θ 3 = 60.0° − 28.4° = 31.6°
1.50
At exit: sin θ 4 = 1.50 sin ( 31.6°) ⇒ θ 4 = 51.7° δ = 45.6° + 51.7° − 60.0° = 37.3°

sin 51.6°
(d) At entry: sin θ 2 = ⇒ θ 2 = 31.5° θ 3 = 60.0° − 31.5° = 28.5°
1.50
At exit: sin θ 4 = 1.50 sin (28.5°) ⇒ θ 4 = 45.7° δ = 51.6° + 45.7° − 60.0° = 37.3°



35.35 n sin θ = 1. From Table 35.1,

θ = sin −1 
1 
(a) = 24.4°
 2.419 

θ = sin −1 
1 
(b) = 37.0°
 1.66 

θ = sin −1 
1 
(c) = 49.8°
 1.309 

n2 n 
35.36 sin θ c = ; θ c = sin −1  2 
n1  n1 

θ c = sin −1 
1.333 
(a) Diamond: = 33.4°
 2.419 

θ c = sin −1 
1.333 
(b) Flint glass: = 53.4°
 1.66 

(c) Ice: Since n 2 > n 1, there is no critical angle .

n2
35.37 sin θ c = (Equation 35.10)
n1

n2 = n1 sin 88.8° = (1.0003)(0.9998) = 1.000 08

nair 1.00
*35.38 sin θ c = = = 0.735 θ c = 47.3°
npipe 1.36

Geometry shows that the angle of refraction at the end is

θ r = 90.0° – θ c = 90.0° – 47.3° = 42.7°

Then, Snell's law at the end, 1.00 sin θ = 1.36 sin 42.7°

gives θ = 67.2°

35.39 For total internal reflection, n1 sin θ1 = n2 sin 90.0°

(1.50) sin θ1 = (1.33)(1.00) or θ1 = 62.4°


35.40 To avoid internal reflection and come out through the
vertical face, light inside the cube must have

θ 3 < sin −1 (1/ n)

So θ 2 > 90.0° − sin −1 (1/ n)

But θ1 < 90.0° and n sin θ 2 < 1

In the critical case, sin −1 (1/ n) = 90.0° − sin −1 (1/ n)

1/n = sin 45.0° n = 1.41

35.41 From Snell's law, n1 sin θ1 = n2 sin θ 2

At the extreme angle of viewing, θ 2 = 90.0°

(1.59)(sin θ1) = (1.00) · sin 90.0°

So θ1 = 39.0°

Therefore, the depth of the air bubble is

rd rp
<d<
tan θ1 tan θ1

or 1.08 cm < d < 1.17 cm

sin θ 2 v2
*35.42 (a) = and θ 2 = 90.0° at the critical angle
sin θ1 v1

sin 90.0° 1850 m s
= so θ c = sin −1 0.185 = 10.7°
sin θ c 343 m s

(b) Sound can be totally reflected if it is traveling in the medium where it travels slower: air

(c) Sound in air falling on the wall from most directions is 100% reflected , so the wall is a good
mirror.



*35.43 For plastic with index of refraction n ≥ 1.42 surrounded by air, the critical angle for total
internal reflection is given by
θ c = sin −1   ≤ sin −1 
1 1 
= 44.8°
 n  1.42 

In the gasoline gauge, skylight from above travels down the plastic. The rays close to the
vertical are totally reflected from both the sides of the slab and from facets at the lower end of
the plastic, where it is not immersed in gasoline. This light returns up inside the plastic and
makes it look bright. Where the plastic is immersed in gasoline, with index of refraction
about 1.50, total internal reflection should not happen. The light passes out of the lower end
of the plastic with little reflected, making this part of the gauge look dark. To frustrate total
internal reflection in the gasoline, the index of refraction of the plastic should be n < 2.12 ,
since
θ c = sin −1 ( ) = 45.0° .
1.50
2.12

*35.44 Assume the lifeguard’s path makes angle θ1 with the north-
south normal to the shoreline, and angle θ 2 with this normal
in the water. By Fermat’s principle, his path should follow
the law of refraction:

sin θ1 v1 7.00 m s  sin θ1 
= = = 5.00 or θ 2 = sin −1
sin θ 2 v2 1.40 m s  5 

The lifeguard on land travels eastward a distance x = (16.0 m ) tan θ1. Then in the water, h e
travels 26.0 m − x = (20.0 m ) tan θ 2 further east. Thus, 26.0 m = (16.0 m ) tan θ1 + (20.0 m ) tan θ 2

  sin θ1  
or 26.0 m = (16.0 m ) tan θ1 + ( 20.0 m ) tan sin −1
  5  
We home in on the solution as follows:
θ1 (deg) 50.0 60.0 54.0 54.8 54.81
right-hand side 22.2 m 31.2 m 25.3 m 25.99 m 26.003 m

The lifeguard should start running at 54.8° east of north .

*35.45 Let the air and glass be medium 1 and 2, respectively. By Snell's law, n2 sin θ 2 = n1 sin θ1
or 1.56 sin θ 2 = sin θ1
But the conditions of the problem are such that θ1 = 2θ 2 . 1.56 sin θ 2 = sin 2θ 2
We now use the double-angle trig identity suggested. 1.56 sin θ 2 = 2 sin θ 2 cos θ 2

1.56
or cos θ 2 = = 0.780
2

Thus, θ 2 = 38.7° and θ1 = 2θ 2 = 77.5°


*35.46 (a) θ1 = θ1 = 30.0°
′

n1 sin θ1 = n2 sin θ 2

(1.00) sin 30.0° = 1.55 sin θ 2

θ 2 = 18.8°

(b) θ1 = θ1 = 30.0°
′

 n sin θ1  −1  1.55 sin 30.0° 
θ 2 = sin −1  1  = sin  = 50.8°
 n2  1 

(c) and (d) The other entries are computed similarly, and are shown in the table below.

(c) air into glass, angles in degrees (d) glass into air, angles in degrees

incidence reflection refraction incidence reflection refraction
0 0 0 0 0 0
10.0 10.0 6.43 10.0 10.0 15.6
20.0 20.0 12.7 20.0 20.0 32.0
30.0 30.0 18.8 30.0 30.0 50.8
40.0 40.0 24.5 40.0 40.0 85.1
50.0 50.0 29.6 50.0 50.0 none*
60.0 60.0 34.0 60.0 60.0 none*
70.0 70.0 37.3 70.0 70.0 none*
80.0 80.0 39.4 80.0 80.0 none*
90.0 90.0 40.2 90.0 90.0 none*

*total internal reflection

1 3
35.47 For water, sin θ c = =
4/3 4

Thus θ c = sin −1 (0.750) = 48.6°

and d = 2 [(1.00 m)tan θ c ]
d = (2.00 m)tan 48.6° = 2.27 m



Goal Solution
A small underwater pool light is 1.00 m below the surface. The light emerging from the water forms a
circle on the water's surface. What is the diameter of this circle?

G: Only the light that is directed upwards and hits the water’s surface at less than the critical angle will
be transmitted to the air so that someone outside can see it. The light that hits the surface farther
from the center at an angle greater than θ c will be totally reflected within the water, unable to be
seen from the outside. From the diagram above, the diameter of this circle of light appears to be
about 2 m.

O: We can apply Snell’s law to find the critical angle, and the diameter can then be found from the
geometry.

A: The critical angle is found when the refracted ray just grazes the surface (θ2 = 90°). The index of
refraction of water is n2 = 1.33, and n1 = 1.00 for air, so

θ c = sin −1 
1 
n1 sin θ c = n2 sin 90° gives = sin −1 (0.750) = 48.6°
 1.333 

r
The radius then satisfies tan θ c =
(1.00 m)

So the diameter is d = 2r = 2(1.00 m ) tan 48.6° = 2.27 m

L: Only the light rays within a 97.2° cone above the lamp escape the water and can be seen by an outside
observer (Note: this angle does not depend on the depth of the light source). The path of a light ray
is always reversible, so if a person were located beneath the water, they could see the whole
hemisphere above the water surface within this cone; this is a good experiment to try the next time
you go swimming!

*35.48 Call θ1 the angle of incidence and of reflection
on the left face and θ 2 those angles on the right
face. Let α represent the complement of θ1 and β
be the complement of θ 2 . Now α = γ and β = δ
because they are pairs of alternate interior
angles. We have

A = γ +δ = α +β

and B = α + A + β = α + β + A = 2A


*35.49 (a) We see the Sun swinging around a circle in the extended plane of our parallel of latitude. Its
angular speed is

∆θ 2 π rad
ω= = = 7.27 × 10 −5 rad s
∆t 86 400 s

The direction of sunlight crossing the cell from the window changes at this rate, moving o n
the opposite wall at speed

( )
v = rω = ( 2.37 m ) 7.27 × 10 − 5 rad s = 1.72 × 10 − 4 m s = 0.172 mm s

(b) The mirror folds into the cell the motion that would occur in a room twice as wide:
v = rω = 2 (0.174 mm s) = 0.345 mm s

(c) and (d)

As the Sun moves southward and upward at 50.0°, we may regard the corner of the window
as fixed, and both patches of light move northward and downward at 50.0° .

*35.50 By Snell's law, n1 sin θ1 = n2 sin θ 2

c c c sin θ1 sin θ 2
With v = n , sin θ1 = sin θ 2 or =
v1 v2 v1 v2

sin 12.0° sin θ 2
This is also true for sound. Here, =
340 m/s 1510 m/s

θ2 = arcsin (4.44 sin 12.0°) = 67.4°

c 2.998 × 108 m s
*35.51 (a) n= = = 1.76 × 107
v  km   1.00 h   1.00 × 10 3 m 
hr   3600 s   1.00 km 
61.15
   

(b) n1 sin θ1 = n2 sin θ 2 so (1.76 × 10 ) sin θ
7
1 = (1.00) sin 90.0°

θ1 = 3.25 × 10 − 6 degree

This problem is misleading. The speed of energy transport is slow, but the speed of the
wavefront advance is normally fast. The condensate's index of refraction is not far from
unity.



*35.52 Violet light:

(1.00) sin 25.0° = 1.689 sin θ 2 ⇒ θ 2 = 14.490°

yv = ( 5.00 cm) tan θ 2 = ( 5.00 cm) tan 14.490° = 1.2622 cm

Red Light:

(1.00) sin 25.0° = 1.642 sin θ 2 ⇒ θ 2 = 14.915°

yR = ( 5.00 cm) tan 14.915° = 1.3318 cm

The emergent beams are both at 25.0° from the normal. Thus,

w = ∆y cos 25.0° where ∆y = 1.3318 cm − 1.2622 cm = 0.0396 cm

w = (0.396 mm ) cos 25.0° = 0.359 mm

35.53 Horizontal light rays from the setting
Sun pass above the hiker. The light rays
are twice refracted and once reflected, as
in Figure (b) below, by just the certain
special raindrops at 40.0° to 42.0° from
the hiker's shadow, and reach the hiker
as the rainbow.

The hiker sees a greater percentage of the
violet inner edge, so we consider the red Figure (a)
outer edge. The radius R of the circle of
droplets is

R = (8.00 km)(sin 42.0°) = 5.35 km

Then the angle φ , between the vertical and the radius where
the bow touches the ground, is given by

2.00 km 2.00 km
cos φ = = = 0.374 or φ = 68.1°
R 5.35 km

The angle filled by the visible bow is 360° – (2 × 68.1°) = 224°,
so the visible bow is
Figure (b)
224°
= 62.2% of a circle
360°


4
35.54 From Snell’s law, (1.00) sin θ1 = sin θ 2
3 eye
x θ1
x = R sin θ 2 = r sin θ1
z r
r sin θ 2 3 d θ2 Fish at depth d
so = = R Image at depth z
R sin θ1 4
Fish

apparent depth z r cos θ1 3 cos θ1
= = =
actual depth d R cos θ 2 4 1 − sin 2 θ
2

( )
2
3  9
But sin 2 θ 2 = sin θ1 = 1 − cos 2 θ1
4  16

z 3 cos θ1 3 cos θ1 3d cos θ1
So = = or z=
d 4
1−
9
+
9
cos 2 θ1
4 7 + 9 cos θ1 2
7 + 9 cos 2 θ1
16 16 16

35.55 As the beam enters the slab, (1.00) sin 50.0° = (1.48) sin θ 2 giving θ 2 = 31.2° . The beam then
strikes the top of the slab at x1 = 1.55 mm tan( 31.2°) from the left end. Thereafter, the beam
strikes a face each time it has traveled a distance of 2 x1 along the length of the slab. Since the
slab is 420 mm long, the beam has an additional 420 mm − x1 to travel after the first reflection.
The number of additional reflections is

420 mm − x1 420 mm − 1.55 mm tan( 31.2°)
= = 81.5
2x1 3.10 mm tan( 31.2°)
or 81 reflections since the answer must be an
integer. The total number of reflections made i n
the slab is then 82 .

2
S1  n2 − n1 
2
′  1.52 − 1.00 
= =  = 0.0426
S1  n2 + n1 
*35.56 (a)
   1.52 + 1.00 

2
S1  n2 − n1 
2
′  1.00 − 1.52 
= = = 0.0426;
S1  n2 + n1 
(b) If medium 1 is glass and medium 2 is air,
  1.00 + 1.52 

There is no difference

2 2
S1  1.76 × 107 − 1.00 
′  1.76 × 107 + 1.00 − 2.00 
(c) =  = 
S1  1.76 × 10 + 1.00 
7
 1.76 × 10 + 1.00
7

2
′
S1  2 .00   2 .00 
= 1.00 − ≈ 1.00 − 2 = 1.00 − 2.27 × 10 −7 or 100%
S1  1.76 × 107 + 1.00 
  1.76 × 107 + 1.00 

This suggests he appearance would be very shiny, reflecting practically all incident light .
See, however, the note concluding the solution to problem 35.51.



2
S1  n − 1
′
*35.57 (a) With n1 = 1 and n2 = n, the reflected fractional intensity is = .
S1  n + 1

The remaining intensity must be transmitted:

S2
= 1−
 n − 1
2
=
(n + 1) − (n − 1) = n2 + 2n + 1 − n2 + 2n − 1 =
2 2
4n
S1  n + 1 (n + 1)2 (n + 1)2 (n + 1)2

S2  n − 1
2
4( 2.419)
(b) At entry, = 1− = = 0.828
S1  n + 1 (2.419 + 1)2
S3
At exit, = 0.828
S2

S3  S3   S2 
=     = (0.828) = 0.685 or 68.5%
2
Overall,
S1  S2   S1 

4n
*35.58 Define T = as the transmission coefficient for one
(n + 1)2
encounter with an
interface. For diamond and air, it is 0.828, as in problem
57.

As shown in the figure, the total amount transmitted is

T 2 + T 2 (1 − T ) + T 2 (1 − T ) + T 2 (1 − T )
2 4 6

+ . . . + T 2 (1 − T )
2n
+ ...

We have 1 − T = 1 − 0.828 = 0.172 so the total
transmission is

[
(0.828)2 1 + (0.172)2 + (0.172)4 + (0.172)6 + . . . ]
To sum this series, define F = 1 + (0.172) + (0.172) + (0.172) + . . . .
2 4 6

Note that (0.172) F = (0.172) + (0.172) + (0.172) + . . ., and
2 2 4 6

1 + (0.172) F = 1 + (0.172) + (0.172) + (0.172) + . . . = F .
2 2 4 6

1
1 = F − (0.172) F or F =
2
Then,
1 − (0.172)
2

The overall transmission is then
(0.828)2 = 0.706 or 70.6%
1 − (0.172)
2


1
35.59 n sin 42.0° = sin 90.0° so n= = 1.49
sin 42.0°

sin 18.0°
sin θ1 = n sin 18.0° and sin θ1 =
sin 42.0°

θ1 = 27.5°


Goal Solution
The light beam shown in Figure P35.59 strikes surface 2 at the critical angle. Determine the angle of
incidence θ1 .

G: From the diagram it appears that the angle of incidence is about 40°.

O: We can find θ1 by applying Snell’s law at the first interface where the light is refracted. At surface 2,
knowing that the 42.0° angle of reflection is the critical angle, we can work backwards to find θ1 .

A: Define n1 to be the index of refraction of the surrounding medium and n2 to be that for the prism
material. We can use the critical angle of 42.0° to find the ratio n2 n1 :

n2 sin 42.0° = n1 sin 90.0°

n2 1
So, = = 1.49
n1 sin 42.0°

Call the angle of refraction θ 2 at the surface 1. The ray inside the prism forms a triangle with
surfaces 1 and 2, so the sum of the interior angles of this triangle must be 180°. Thus,

(90.0° −θ 2 ) + 60.0° +(90.0° −42.0°) = 180°
Therefore, θ 2 = 18.0°
Applying Snell’s law at surface 1, n1 sin θ1 = n2 sin 18.0°

sin θ1 = (n2 n1 ) sin θ 2 = (1.49) sin 18.0°

θ1 = 27.5°

L: The result is a bit less than the 40.0° we expected, but this is probably because the figure is not drawn
to scale. This problem was a bit tricky because it required four key concepts (refraction, reflection,
critical angle, and geometry) in order to find the solution. One practical extension of this problem is
to consider what would happen to the exiting light if the angle of incidence were varied slightly.
Would all the light still be reflected off surface 2, or would some light be refracted and pass through
this second surface?



35.60 Light passing the top of the pole makes an angle of
incidence φ1 = 90.0° − θ . It falls on the water surface at
distance
(L − d)
s1 = from the pole,
tan θ

and has an angle of refraction φ2 from (1.00)sin φ1 = n sin φ 2 .
Then s2 = d tan φ2 and the whole shadow length is

L−d   sin φ 1  
s1 + s2 = + d tan  sin −1
tan θ   n  

L−d   cos θ   2.00 m   cos 40.0°  
s1 + s2 = + d tan  sin −1 = + ( 2.00 m ) tan  sin −1 = 3.79 m
tan θ   n   tan 40.0°
   1.33   

35.61 (a) For polystyrene surrounded by air, internal reflection requires

θ 3 = sin −1 
1.00 
= 42.2°
 1.49 

Then from the geometry, θ 2 = 90.0° – θ 3 = 47.8°

From Snell's law, sin θ1 = (1.49) sin 47.8° = 1.10

This has no solution. Therefore, total internal reflection always happens .

θ 3 = sin −1 
1.33 
(b) For polystyrene surrounded by water, = 63.2°
 1.49 

and θ2 = 26.8°

From Snell's law, θ1 = 30.3°

(c) No internal refraction is possible since the beam is initially traveling in a medium of lower
index of refraction.

4
*35.62 δ = θ1 − θ 2 = 10.0° and n1 sin θ1 = n2 sin θ 2 with n1 = 1, n2 =
3

Thus, θ1 = sin −1(n2 sin θ 2 ) = sin −1[n2 sin(θ1 − 10.0°)]


(You can use a calculator to home in on an approximate solution to this equation, testing
different values of θ1 until you find that θ1 = 36.5° . Alternatively, you can solve for θ1
exactly, as shown below.)

4
We are given that sin θ1 = sin(θ1 − 10.0°)
3

3
This is the sine of a difference, so sin θ1 = sin θ1 cos 10.0° − cos θ1 sin 10.0°
4

 3
Rearranging, sin 10.0° cos θ1 = cos 10.0° − sin θ1
 4

sin 10.0°
= tan θ1 and θ1 = tan −1 0.740 = 36.5°
cos 10.0° − 0.750

4.00 cm 2.00 cm
35.63 tan θ1 = and tan θ 2 =
h h

tan 2 θ1 = (2.00 tan θ 2 ) = 4.00 tan 2 θ 2
2

sin 2 θ1 sin 2 θ 2
= 4.00 (1)
(1 − sin θ1 )
2
(1 − sin 2 θ 2 )

Snell's law in this case is: n1 sin θ1 = n2 sin θ 2

sin θ1 = 1.333 sin θ 2

Squaring both sides, sin 2 θ1 = 1.777 sin 2 θ 2 (2)

1.777 sin 2 θ 2 sin 2 θ 2
Substituting (2) into (1), = 4.00
1 − 1.777 sin θ 2
2
1 − sin 2 θ 2

0.444 1
Defining x = sin 2 θ , =
(1 − 1.777x) (1 − x)
Solving for x, 0.444 − 0.444x = 1 − 1.777x and x = 0.417

From x we can solve for θ 2 : θ 2 = sin −1 0.417 = 40.2°

(2.00 cm) (2.00 cm)
Thus, the height is h= = = 2.37 cm
tan θ 2 tan( 40.2°)



35.64 Observe in the sketch that the angle of incidence at point P is γ ,
and using triangle OPQ: sin γ = L / R .

R 2 − L2
Also, cos γ = 1 − sin 2 γ =
R
Applying Snell’s law at point P, (1.00) sin γ = n sin φ

sin γ L
Thus, sin φ = =
n nR

n2 R 2 − L2
and cos φ = 1 − sin 2 φ =
nR

From triangle OPS, φ + (α + 90.0°) + (90.0° − γ ) = 180° or the angle of incidence at point S is
α = γ − φ . Then, applying Snell’s law at point S gives (1.00) sin θ = n sin α = n sin( γ − φ ) , or

 L  n2 R 2 − L2 R 2 − L2  L  
sin θ = n [sin γ cos φ − cos γ sin φ ] = n  − 
 R 

nR R  nR  


θ = sin −1  2  n2 R 2 − L2 − R 2 − L2  
L  2 2
n R − L2 − R 2 − L2 
L
sin θ = 2  and R  
R  

35.65 To derive the law of reflection, locate point O so that the time of
travel from point A to point B will be minimum.

The total light path is L = a sec θ1 + b sec θ 2

 1
The time of travel is t = (a sec θ1 + b sec θ 2 )
 v

If point O is displaced by dx, then

 1
dt = (a sec θ1 tan θ1 dθ1 + b sec θ 2 tan θ 2 dθ 2 ) = 0 (1)
 v

(since for minimum time dt = 0).

Also, c + d = a tan θ1 + b tan θ 2 = constant

so, a sec 2 θ1 dθ1 + b sec 2 θ 2 dθ 2 = 0 (2)

Divide equations (1) and (2) to find θ 1 = θ 2


35.66 As shown in the sketch, the angle of incidence at point A is:

 ( d 2)  −1  1.00 m 
θ = sin −1   = sin   = 30.0°
 R   2.00 m 

If the emerging ray is to be parallel to the incident ray,
the path must be symmetric about the center line CB of
the cylinder . In the isosceles triangle ABC, γ = α and
β = 180° − θ . Therefore, α + β + γ = 180° becomes

θ
2α + 180° − θ = 180° or α = = 15.0°
2

Then, applying Snell’s law at point A , n sin α = (1.00) sin θ

sin θ sin 30.0°
or n= = = 1.93
sin α sin 15.0°

35.67 (a) At the boundary of the air and glass, the critical angle is given
by
1
sin θ c =
n

d4 sin θ c d
Consider the critical ray PBB′ : tan θ c = or =
t cos θ c 4t

2
sin 2 θ c sin 2 θ c  d
Squaring the last equation gives: = =
cos θ c 1 − sin θ c  4t 
2 2

2
 d
n = 1 + ( 4t d)
1 1 2
Since sin θ c = , this becomes = or
n n2 − 1  4t 

4t
(b) Solving for d, d=
n2 − 1

4(0.600 cm )
Thus, if n = 1.52 and t = 0.600 cm , d= = 2.10 cm
(1.52)2 − 1
(c) Since violet light has a larger index of refraction, it will lead to a smaller critical angle
and the inner edge of the white halo will be tinged with violet light.



35.68 From the sketch, observe that the angle of
incidence at point A is the same as the
prism angle θ at point O . Given that
θ = 60.0° , application of Snell’s law at point
A gives

1.50 sin β = 1.00 sin 60.0° or β = 35.3°

From triangle AOB , we calculate the angle
of incidence (and reflection) at point B.

θ + (90.0° − β ) + (90.0° − γ ) = 180° so γ = θ − β = 60.0° − 35.3° = 24.7°

Now, using triangle BCQ : (90.0° − γ ) + (90.0° − δ ) + (90.0° − θ ) = 180°
Thus the angle of incidence at point C is δ = (90.0° − θ ) − γ = 30.0° − 24.7° = 5.30°

Finally, Snell’s law applied at point C gives 1.00 sin φ = 1.50 sin 5.30°

or φ = sin − 1(1.50 sin 5.30°) = 7.96°

35.69 (a) Given that θ1 = 45.0° and θ 2 = 76.0° , Snell’s law at
the first surface gives

n sin α = (1.00) sin 45.0° (1)

Observe that the angle of incidence at the second
surface is β = 90.0° − α . Thus, Snell’s law at the
second surface yields

n sin β = n sin(90.0° − α ) = (1.00) sin 76.0° , or

n cos α = sin 76.0° (2)

sin 45.0°
Dividing Equation (1) by Equation (2), tan α = = 0.729 or α = 36.1°
sin 76.0°

sin 45.0° sin 45.0°
Then, from Equation (1), n= = = 1.20
sin α sin 36.1°

(b) From the sketch, observe that the distance the light travels in the plastic is d = L sin α . Also,
the speed of light in the plastic is v = c n , so the time required to travel through the plastic is

t=
d
=
nL
=
(1.20)(0.500 m) = 3.40 × 10 −9 s = 3.40 ns
v c sin α ( )
3.00 × 10 m s sin 36.1°
8


35.70 sin θ1 sin θ 2 sin θ1 /sin θ 2
0.174 0.131 1.3304
0.342 0.261 1.3129
0.500 0.379 1.3177
0.643 0.480 1.3385
0.766 0.576 1.3289
0.866 0.647 1.3390
0.940 0.711 1.3220
0.985 0.740 1.3315

The straightness of the graph line
demonstrates Snell's proportionality.
The slope of the line is
n = 1.3276 ± 0.01

and n = 1.328 ± 0.8%


Capítulo 35 (5th edition) con soluciones la naturaleza de la luz y las leyes de la optica geometrica

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Capítulo 35 (5th edition) con soluciones la naturaleza de la luz y las leyes de la optica geometrica