solucionario serway 5 edicion en ingles todos los capitulos completo
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Capítulo 35 (5th edition) con soluciones serway volumen 2 edicion 5,
Capítulo 35 (5th edition) con soluciones la naturaleza de la luz y las leyes de la optica geometrica,Capítulo 35 la naturaleza de la luz solucionario,Capítulo 35 la naturaleza de la luz ejemplos,Capítulo 35 la naturaleza de la luz ejercicios,Capítulo 35 las leyes de la optica geometrica,Capítulo 35 solucionario serway
11. 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°
13. 326 Chapter 35 Solutions
*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
15. 328 Chapter 35 Solutions
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
17. 330 Chapter 35 Solutions
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°
19. 332 Chapter 35 Solutions
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°
21. 334 Chapter 35 Solutions
*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°
23. 336 Chapter 35 Solutions
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
25. 338 Chapter 35 Solutions
*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°
27. 340 Chapter 35 Solutions
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
29. 342 Chapter 35 Solutions
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°)]
31. 344 Chapter 35 Solutions
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
33. 346 Chapter 35 Solutions
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