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University Physics with Modern Physics 12e

Hugh D. Young, Roger A. Freeman

Chapter 33

The Nature and Propagation of Light

Educators


Problem 1

Two plane mirrors intersect at right angles. A laser beam strikes the first of them at a point 11.5 $\mathrm{cm}$ from their point of intersection, as shown in Fig. 33.38 For what angle of incidence at the first mirror will this ray strike the midpoint of the second mirror (which is 28.0 $\mathrm{cm}$ long) after reflecting from the first mirror?

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Problem 2

Three plane mirrors intersect at right angles. A beam of laser light strikes the first of them at an angle $\theta$ with respect to the normal (Fig. 33.39$)$ . (a) Show that when this ray is reflected off of the other two mirrors and crosses the original ray, the angle $\alpha$ between these two rays will be $\alpha=180^{\circ}-2 \theta .$ (b) For what angle $\theta$ will the two rays be perpendicular when they cross?

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Problem 3

A beam of light has a wavelength of 650 nm in vacuum. (a) What is the speed of this light in a liquid whose index of refraction at this wavelength is 1.477 (b) What is the wavelength of these waves in the liquid?

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Problem 4

Light with a frequency of $5.80 \times 10^{14} \mathrm{Hz}$ travels in a block of glass that has an index of refraction of $1.52 .$ What is the wave-length of the light (a) in vacuum and (b) in the glass?

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Problem 5

A light beam travels at $1.94 \times 10^{8} \mathrm{m} / \mathrm{s}$ in quartz. The wave-length of the light in quartz is 355 $\mathrm{nm}$ . (a) What is the index of refraction of quartz at this wavelength? (b) If this same light travels through air, what is its wavelength there?

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Problem 6

Light of a certain frequency has a wavelength of 438 $\mathrm{nm}$ in water. What is the wavelength of this light in benzene?

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Problem 7

A parallel beam of light in air makes an angle of $47.5^{\circ}$ with the surface of a glass plate having a refractive index of 1.66 . (a) What is the angle between the reflected part of the beam and the surface of the glass? (b) What is the angle between the refracted beam and the surface of the glass?

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Problem 8

Using a fast-pulsed laser and electronic timing circuitry, you find that light travels 2.50 $\mathrm{m}$ within a plastic rod in 11.5 $\mathrm{ns}$ . What is the refractive index of the plastic?

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Problem 9

Light traveling in air is incident on the surface of a block of plastic at an angle of $62.7^{\circ}$ to the normal and is bent so that it makes a $48.1^{\circ}$ angle with the normal in the plastic. Find the speed of light in the plastic.

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Problem 10

(a) A tank containing methanol has walls 2.50 $\mathrm{cm}$ thick made of glass of refractive index 1.550 . Light from the outside air strikes the glass at a $41.3^{\circ}$ angle with the normal to the glass. Find the angle the light makes with the normal in the methanol. (b) The tank is emptied and refilled with an unknown liquid. If light incident at the same angle as in part (a) enters the liquid in the tank at an angle of $20.2^{\circ}$ from the normal, what is the refractive index of the unknown liquid?

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Problem 11

(a) Light passes through three parallel slabs of different thicknesses and refractive indexes. The light is incident in the first slab and finally refracts into the third slab. Show that the middle slab has no effect on the final direction of the light. That is, show that the direction of the light in the third slab is the same as if the light had passed directly from the first slab into the third slab. Generalize this result to a stack of $N$ slabs. What determines the final direction of the light in the last slab?

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Problem 12

A horizontal, parallel-sided plate of glass having a refractive index of 1.52 is in contact with the surface of water in a tank. A ray coming from above in air makes an angle of incidence of $35.0^{\circ}$ with the normal to the top surface of the glass. (a) What angle does the ray refracted into the water make with the normal to the surface? (b) What is the dependence of this angle on the refractive index of the glass?

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Problem 13

In a material having an index of refraction $n$ , a light ray has frequency $f,$ wavelength $\lambda,$ and speed $v$ . What are the frequency, wavelength, and speed of this light (a) in vacuum and (b) in a material having refractive index $n^{\prime} ?$ In each case, express your answers in terms of only $f, \lambda, v, n,$ and $n^{\prime} .$

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Problem 14

Prove that a ray of light reflected from a plane mirror rotates through an angle of 2$\theta$ when the mirror rotates through an angle $\theta$ about an axis perpendicular to the plane of incidence.

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Problem 15

A ray of light is incident on a plane surface separating two sheets of glass with refractive indexes 1.70 and 1.58 . The angle of incidence is $62.0^{\circ}$ , and the ray originates in the glass with $n=1.70 .$ Compute the angle of refraction.

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Problem 16

In Example 33.1 the water - glass interface is horizontal. If instead this interface were tilted $15.0^{\circ}$ above the horizontal, with the right side higher than the left side, what would be the angle from the vertical of the ray in the glass? (The ray in the water still makes an angle of $60.0^{\circ}$ with the vertical.)

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Problem 17

Light Pipe. Light enters a solid pipe made of plastic having an index of refraction of $1.60 .$ The light travels parallel to the upper part of the pipe (Fig, 33.40$)$ . You want to cut the face $A B$ so that all the light will reflect back into the pipe after it first strikes that face. (a) What is the largest that $\theta$ can be if the pipe is in air? (b) If the pipe is immersed in water of refractive index 1.33 , what is the largest that $\theta$ can be?

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Problem 18

A beam of light is traveling inside a solid glass cube having index of refraction $1.53 .$ It strikes the surface of the cube from the inside. (a) If the cube is in air, at what minimum angle with the normal inside the glass will this light not enter the air at this surface? (b) What would be the minimum angle in part (a) if the cube were immersed in water?

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Problem 19

The critical angle for total internal reflection at a liquid-air interface is $425^{\circ} .$ (a) If a ray of light traveling in the liquid has an angle of incidence at the interface of $35.0^{\circ}$ , what angle does the refracted ray in the air make with the normal? (b) If a ray of light traveling in air has an angle of incidence at the interface of $35.0^{\circ},$ what angle does the refracted ray in the liquid make with the normal?

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Problem 20

At the very end of Wagner's series of operas Ring of the Nibelung, Brinnhilde takes the golden ring from the finger of the dead Siegfried and throws it into the Rhine, where it sinks to the bottom of the river. Assuming that the ring is small enough compared to the depth of the river to be treated as a point and that the Rhine is 10.0 $\mathrm{m}$ deep where the ring goes in, what is the area of the largest circle at the surface of the water over which light from the ring could escape from the water?

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Problem 21

A ray of light is traveling in a glass cube that is totally immersed in water. You find that if the ray is incident on the glass-water interface at an angle to the normal larger than $48.7^{\circ}$ , no light is refracted into the water. What is the refractive index of the glass?

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Problem 22

Light is incident along the normal on face $A B$ of a glass prism of refractive index 1.52 , as shown in Fig. 33.41 . Find the largest value the angle $\alpha$ can have without any light refracted out of the prism at face $A C$ if (a) the prism is immersed in air and (b) the prism is immersed in water.

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Problem 23

A ray of light in diamond (index of refraction 2.42 ) is incident on an interface with air. What is the largest angle the ray can make with the normal and not be totally reflected back into the diamond?

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Problem 24

A beam of light strikes a sheet of glass at an angle of $57.0^{\circ}$ with the normal in air. You observe that red light makes an angle of $38.1^{\circ}$ with the normal in the glass, while violet light makes a $36.7^{\circ}$ angle. (a) What are the indexes of refraction of this glass for these colors of light? (b) What are the speeds of red and violet light in the glass?

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Problem 25

A beam of unpolarized light of intensity $I_{0}$ passes through a series of ideal polarizing filters with their polarizing directions turned to various angles as shown in Fig. 33.42 (a) What is the light intensity (in terms of $I_{0} )$ at points $A, B,$ and $C ?$ (b) If we remove the middle filter, what will be the light intensity at point $C ?$

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Problem 26

Light traveling in water strikes a glass plate at an angle of incidence of $53.0^{\circ}$ : part of the beam is reflected and part is refracted. If the refiected and refracted portions make an angle of $90.0^{\circ}$ with each other, what is the index of refraction of the glass?

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Problem 27

A parallel beam of unpolarized light in air is incident at an angle of $54.5^{\circ}$ (with respect to the normal) on a plane glass surface. The reflected beam is completely linearly polarized. (a) What is the refractive index of the glass? (b) What is the angle of refraction of the transmitted beam?

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Problem 28

Light of original intensity $I_{0}$ passes through two ideal polarizing filters having their polarizing axes oriented as shown in Fig. 33.43 . You want to adjust the angle $\phi$ so that the intensity at point $P$ is equal to $I_{0} / 10$ . (a) If the original light is unpolarized, what should $\phi$ be? ( $b$ ) If the original light is linearly polarized in the same direction as the polarizing axis of the first polarizer the light reaches, what should $\phi$ be?

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Problem 29

A beam of polarized light passes through a polarizing filter. When the angle between the polarizing axis of the filter and the direction of polarization of the light is $\theta,$ the intensity of the emerging beam is $I .$ If you now want the intensity to be $I / 2$ , what should be the angle (in terms of $\theta )$ between the polarizing angle of the filter and the original direction of polarization of the light?

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Problem 30

The refractive index of a certain glass is $1.66 .$ For what incident angle is light reflected from the surface of this glass completely polarized if the glass is immersed in (a) air and (b) water?

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Problem 31

Unpolarized light of intensity 20.0 $\mathrm{W} / \mathrm{cm}^{2}$ is incident on two polarizing filters. The axis of the first filter is at an angle of $25.0^{\circ}$ counterclockwise from the vertical (viewed in the direction the light is traveling), and the axis of the second filter is at $62.0^{\circ}$ counterclockwise from the vertical. What is the intensity of the light after it has passed through the second polarizer?

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Problem 32

A polarizer and an analyzer are oriented so that the maximum amount of light is transmitted. To what fraction of its maximum value is the intensity of the transmitted light reduced when the analyzer is rotated through (a) $22.5^{\circ} ;(\mathrm{b}) 45.0^{\circ} ;(\mathrm{c}) 67.5^{\circ} ?$

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Problem 33

Three Polarizing filters. Three polarizing filters are stacked with the polarizing axes of the second and third at $45.0^{\circ}$ and $90.0^{\circ}$ , respectively, with that of the first. (a) If unpolarized light of intensity $I_{0}$ is incident on the stack, find the intensity and state of polarization of light emerging from each filter $(b)$ If the second filter is removed, what is the intensity of the light emerging from each remaining filter?

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Problem 34

Three polarizing filters are stacked, with the polarizing axis of the second and third filters at $23.0^{\circ}$ and 62.0 ' respectively, to that of the first. If unpolarized light is incident on the stack, the light has intensity 75.0 $\mathrm{W} / \mathrm{cm}^{2}$ after it passes through the stack. If the incident intensity is kept constant, what is the intensity of the light after it has passed through the stack if the second polarizer is removed?

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Problem 35

A beam of white light passes through a uniform thickness of air. If the intensity of the scattered light in the middle of the green part of the visible spectrum is $I$ , find the intensity (in terms of $I )$ of scattered light in the middle of (a) the red part of the spectrum and $(b)$ the violet part of the spectrum. Consult Table $32.1 .$

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Problem 36

Bending Around Corners. Traveling particles do not bend around corners, but waves do. To see why, suppose that a plane wave front strikes the edge of a sharp object traveling perpendicular to the surface (Fig. 33.44$)$ . Use Huygens's principle to show that this wave will bend around the upper edge of the object. (Note: This effect, called diffraction, can easily be seen for water waves, but it also occurs for light, as you will see in Chapters 35 and 36 . However due to the very short wavelength of visible light, it is not so apparent in daily life.)

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Problem 37

The Corner Reflector. An inside comer of a cube is lined with mirrors to make a corner reflector (see Example 33.3 in Section 33.2 ). A ray of light is reflected successively from each of three mutually perpendicular mirrors; show that its final direction is always exactly opposite to its initial direction.

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Problem 38

A light beam is directed parallel to the axis of a hollow cylindrical tube. When the tube contains only air, it takes the light 8.72 ns to travel the length of the tube, but when the tube is filled with a transparent jelly, it takes the light 2.04 ns longer to travel its length. What is the refractive index of this jelly?

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Problem 39

Light traveling in a material of refractive index $n_{1}$ is incident at angle $\theta_{1}$ with respect to the normal at the interface with a slab of material that has parallel faces and refractive index $n_{2}$ . After the light passes through this material, it is refracted into a material with refractive index $n_{3}$ and in this third material it makes an angle of $\theta_{3}$ with the normal. (a) Find $\theta_{3}$ in terms of $\theta_{1}$ and the refractive indexes of the materials. (b) The ray in the third material is now reversed, so that it is incident on the $n_{3}$ to-n_{2} interface with the angle $\theta_{3}$ found in part (a). Show that when the light refracts into the material with refractive index $n_{1}$ , the angle it makes with the normal is angle $\theta_{1}$ . This shows that the refracted ray is reversible. (c) Are reflected rays also reversible? Explain.

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Problem 40

In a physics lab, light with wavelength 490 $\mathrm{nm}$ travels in air from a laser to a photocell in 17.0 $\mathrm{ns}$ . When a slab of glass 0.840 $\mathrm{m}$ thick is placed in the light beam, with the beam incident along the normal to the parallel faces of the slab, it takes the light 21.2 $\mathrm{ns}$ to travel from the laser to the photocell. What is the wavelength of the light in the glass?

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Problem 41

A ray of light is incident in air on a block of a transparent solid whose index of refraction is $n$ . If $n=1.38$ , what is the largest angle of incidence $\theta_{a}$ for which total internal reflection will occur at the vertical face (point $A$ shown in $\mathrm{Fig} .33 .45 ) ?$

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Problem 42

A light ray in air strikes the right-angle prism shown in Fig. 33.46. This ray consists of two different wavelengths. When it emerges at face $A B,$ it has been split into two different rays that diverge from each other by $8.50^{\circ}$ . Find the index of refraction of the prism for each of the two wavelengths.

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Problem 43

A quarter-wave plate converts linearly polarized light to circularly polarized light. Prove that a quarter-wave plate also converts circularly polarized light to linearly polarized light.

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Problem 44

A glass plate 2.50 $\mathrm{mm}$ thick, with an index of refraction of $1.40,$ is placed between a point source of light with wavelength 540 $\mathrm{nm}$ (in vacuum) and a screen. The distance from source to screen is 1.80 $\mathrm{cm}$ . How many wavelengths are there between the source and the screen?

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Problem 45

Old photographic plates were made of glass with a light-sensitive emulsion on the front surface. This emulsion was some what transparent. When a bright point source is focused on the front of the plate, the developed photograph will show a halo around the image of the spot. If the glass plate is 3.10 $\mathrm{mm}$ thick and the halos have an inner radius of $5.34 \mathrm{mm},$ what is the index of refraction of the glass? (Hint: Light from the spot on the front surface is scattered in all directions by the emulsion. Some of it is then totally reflected at the back surface of the plate and returns to the front surface.)

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Problem 46

After a long day of driving you take a late-night swim in a motel swimming pool. When you go to your room, you realize that you have lost your room key in the pool. You borrow a powerful flashlight and walk around the pool, shining the light into it. The light shines on the key, which is lying on the bottom of the pool, when the flashlight is held 1.2 $\mathrm{m}$ above the water surface and is directed at the surface a borizontal distance of 1.5 $\mathrm{m}$ from the edge (Fig. 33.47$)$ . If the water here is 4.0 in deep, how far is the key from the edge of the pool?

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Problem 47

You sight along the rim of a glass with vertical sides so that the top rim is lined up with the opposite edge of the bottom (Fig. 33.48 $\mathrm{a}$ ). The glass is a thin-walled, hollow cylinder 16.0 $\mathrm{cm}$ high with a top and bottom of the glass diameter of $8.0 \mathrm{cm} .$ While you keep your eye in the same position, a friend fills the glass with a transparent liquid, and you then see a dime that is lying at the center of the bottom of the glass (Fig. 33. 48 ). What is the index of refraction of the liquid?

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Problem 48

A beaker with a mirrored bottom is filled with a liquid whose index of refraction is $1.63 .$ A light beam strikes the top surface of the liquid at an angle of $42.5^{\circ}$ from the normal. At what angle from the normal will the beam exit from the liquid after traveling down through the liquid, reflecting from the mirrored bottom, and returning to the surface?

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Problem 49

A thin layer of ice $(n=1.309)$ floats on the surface of water $(n=1.333)$ in a bucket. A ray of light from the bottom of the bucket travels upward through the water. (a) What is the largest angle with respect to the normal that the ray can make at the ice-water interface and still pass out into the air above the ice? (b) What is this angle after the ice melts?

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Problem 50

A $45^{\circ}-45^{\circ}-90^{\circ}$ prism is immersed in water. A ray of light is incident normally on one of its shorter faces. What is the minimum index of refraction that the prism must have if this ray is to be totally reflected within the glass at the long face of the prism?

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Problem 51

The prism shown in Fig. 33.49 has a refractive index of $1.66,$ and the angles $A$ are $25.0^{\circ} .$ Two light rays $m$ and $n$ are parallel as they enter the prism. What is the angle between them after they emerge?

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Problem 52

Light is incident normally on the short face of a $30^{\circ}-60^{\circ}-90^{\circ}$ prism (Fig. 33.50$)$ . A drop of liquid is placed on the hypotenuse of the prism. If the index of the prism is 1.62 , find the maximum index that the liquid may have if the light is to be totally reflected.

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Problem 53

A horizontal cylindrical tank 2.20 $\mathrm{m}$ in diameter is half full of water. The space above the water is filled with a pressurized gas of unknown refractive index. A small laser can move along the curved bottom of the water and aims a light beam toward the center of the water surface (Fig. 33.51$)$ . You observe that when the laser has moved a distance $S=1.09 \mathrm{m}$ or more (measured along the curved surface) from the lowest point in the water, no light enters the gas. (a) What is the index of refraction of the gas? (b) How long does it take the light beam to travel from the laser to the rim of the tank when (i) $S>1.09 \mathrm{m}$ and (ii) $S<1.09 \mathrm{m} ?$

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Problem 54

A large cube of glass a metal reflector on one face and water on an adjoining face (Fig. 33.52$)$ . A light beam strikes the reflector, as shown. You observe that as you gradually increase the angle of the light beam, if $\theta \geq 59.2^{\circ}$ no light enters the water. What is the speed of light in this glass?

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Problem 55

When the sun is either rising or setting and appears to be just on the horizon, it is in fact below the horizon. The explanation for this seeming paradox is that light from the sun bends slightly when entering the earth's atmosphere, as shown in Fig. 33.53 . Since our perception is based on the idea that light travels in straight lines, we perceive the light to be coming from an apparent position that is an angle $\delta$ above the sun's true position. (a) Make the simplifying assumptions that the atmosphere has uniform density, and hence uniform index of refraction $n$ , and extends to a height $h$ above the earth's surface, at which point it abruptly stops. Show that the angle 8 is given by
$$
\delta=\arcsin \left(\frac{n R}{R+h}\right)-\arcsin \left(\frac{R}{R+h}\right)
$$
where $R=6378 \mathrm{km}$ is the radius of the earth. (b) Calculate $\delta$ using $n=1.0003$ and $h=20 \mathrm{km}$ . How does this compare to the angular radius of the sun, which is about one quarter of a degree? (In actuality a light ray from the sun bends gradually, not abruptly, since the density and refractive index of the atmosphere change gradually with altitude.)

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Problem 56

Fermat's Principle of Least Time. A ray of light traveling with speed $c$ leaves point 1 shown in Fig. 33.54 and is reflected to point 2. The ray strikes the reflecting surface a horizontal distance $x$ fron point $1 .$ (a) Show that the time $t$ required for the light to travel from 1 to 2 is
$$
t=\frac{\sqrt{y_{1}^{2}+x^{2}}+\sqrt{y_{2}^{2}+(l-x)^{2}}}{c}
$$
(b) Take the derivative of $t$ with respect to $x$ . Set the derivative equal to zero to show that this time reaches its minimum value when $\theta_{1}=\theta_{2},$ which is the law of reflection and corresponds to the actual path taken by the light. This is an example of Fermat's principle of least time, which states that among all possible paths between two points, the one actually taken by a ray of light is that for which the time of travel is a minimum. (In fact, there are some cases in which the time is a maximum rather than a minimum.)

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Problem 57

A ray of light goes from point $A$ in a medium in which the speed of light is $v_{1}$ to point $B$ in a medium in which the speed is $v_{2}$ (Fig. 33.55$)$ . The ray strikes the interface a horizontal distance $x$ to the right of point $A$ . (a) Show that the time required for the light to go from $A$ to $B$ is
$$
t=\frac{\sqrt{h_{1}^{2}+x^{2}}}{v_{1}}+\frac{\sqrt{h_{2}^{2}+(l-x)^{2}}}{v_{2}}
$$
(b) Take the derivative of $t$ with respect to $x$ . Set this derivative equal to zero to show that this time reaches its minimum value when $n_{1} \sin \theta_{1}=n_{2} \sin \theta_{2}$ . This is Snell's law, and corresponds to the actual path taken by the light. This is another example of Fermat's principle of least time (see Problem 33.56 ).

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Problem 58

Light is incident in air at an angle $\theta_{a}$ (Fig. 33.56 ) on the upper surface of a transparent plate, the surfaces of the plate being plane and parallel to each other (a) Prove that $\theta_{a}=\theta_{a}^{\prime}$ . (b) Show that this is true for any number of different parallel plates. (c) Prove that the lateral displacement $d$ of the emergent beam is given by the relationship
$$
d=t \frac{\sin \left(\theta_{a}-\theta_{b}^{\prime}\right)}{\cos \theta_{b}^{\prime}}
$$
where $t$ is the thickness of the plate. (d) A ray of light is incident at an angle of $66.0^{\circ}$ on one surface of a glass plate 2.40 $\mathrm{cm}$ thick with an index of refraction 1.80 . The medium on either side of the plate is air. Find the lateral displacement between the incident and emergent rays.

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Problem 59

Light traveling downward is incident on a horizontal film of thickness $t,$ as shown in Fig. 33.57 . The incident ray splits into two rays, $A$ and $B$ . Ray A reflects from the top of the film. Ray B reflects from the bottom of the film and then refracts back into the material that is above the film. If the film has parallel faces, show that rays $A$ and $B$ end up parallel to each other.

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Problem 60

A thin beam of white light is directed at a flat sheet of silicate flint glass at an angle of $20.0^{\circ}$ to the surface of the sheet. Due to dispersion in the glass, the beam is spread out as shown in a spectrum in Fig. 33.58 . The refractive index of silicate flint glass versus wavelength is graphed in Fig. 33.18 . (a) The rays $a$ and $b$ shown in Fig, 33.58 correspond to the extremes of the visible spectrum. Which corresponds to red and which to violet? Explain your reasoning. (b) For what thickness $d$ of the glass sheet will the spectrum be 1.0 $\mathrm{mm}$ wide, as shown (see Problem 33.58 )?

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Problem 61

Angle of Deviation. The incident angle $\theta_{a}$ shown in Fig. 33.59 is chosen so that the light passes symmetrically through the prism, which has refractive index $n$ and apex angle $A$ . (a) Show that the angle of deviation $\delta$ the angle between the initial and final directions of the ray) is given by
$$
\sin \frac{A+\delta}{2}=n \sin \frac{A}{2}
$$
(When the light passes through symmetrically, as shown, the angle of deviation is a minimum. (b) Use the result of part (a) to find the angle of deviation for a ray of light passing symmetrically through a prism having three equal angles $\left(A=60.0^{\circ}\right)$ and $n=1.52 .$ (c) A certain glass has a refractive index of 1.61 for red light $(700 \mathrm{nm})$ and 1.66 for violet light $(400 \mathrm{nm}) .$ If both colors pass through symmetrically, as described in part (a), and if $A=60.0$ , find the difference between the angles of deviation for the two colors.

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Problem 62

A beam of unpolarized sunlight strikes the vertical plastic wall of a water tank at an unknown angle. Some of the light reflects from the wall and enters the water (Fig. 33.60 ). The refractive index of the plastic wall is $1.61 .$ If the light that has been reflected from the wall into the water is observed to be completely polarized, what angle does this beam make with the normal inside the water?

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Problem 63

A beam of light traveling horizontally is made of an unpolarized component with intensity $I_{0}$ and a polarized component with intensity $I_{p}$ . The plane of polarization of the polarized component is oriented at an angle of $\theta$ with respect to the vertical. The data in the table give the intensity measured through a polarizer with an orientation of $\phi$ with respect to the vertical. (a) What is the orientation of the polarized component? (That is, what is the angle (b) (b) What are the values of $I_{0}$ and $I_{p} ?$
TABLE CANNOT COPY

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Problem 64

A certain birefringent material has indexes of refraction $n_{1}$ and $n_{2}$ for the two perpendicular components of lincarly polarized light passing through it. The corresponding wavelengths are $\lambda_{1}=\lambda_{0} / n_{1}$ and $\lambda_{0} / n_{2}$ , where $\lambda_{0}$ is the wavelength in vacuum. (a) If the crystal is to function as a quarter-wave plate, the number of wavelengths of each component within the material must differ by $\frac{1}{4}$ . Show that the minimum thickness for a quarter-wave plate is
$$
d=\frac{\lambda_{0}}{4\left(n_{1}-n_{2}\right)}
$$
(b) Find the minimum thickness of a quarter-wave plate made of siderite $\left(\mathrm{FeO} \cdot \mathrm{CO}_{2}\right)$ if the indexes of refraction are $n_{1}=1.875$ and $n_{2}=1.635$ and the wavelength in vacuum is $\lambda_{0}=589 \mathrm{nm}$ .

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Problem 65

Consider two vibrations of equal amplitude and frequency but differing in phase, one along the $x$ -axis,
$$
x=a \sin (\omega t-\alpha)
$$
and the other along the $y$ -axis,
$$
y=a \sin (\omega t-\beta)
$$
These can be written as follows:
$$
\begin{array}{l}{\frac{x}{a}=\sin \omega t \cos \alpha-\cos \omega t \sin \alpha} \\ {\frac{y}{a}=\sin \omega t \cos \beta-\cos \omega t \sin \beta}\end{array}
$$
(a) Multiply $\mathrm{Eq}$ . (1) by $\sin \beta$ and $\mathbf{E q}$ (2) by sin $\alpha,$ and then subtract the resulting equations. (b) Multiply Eq. (1) by cos $\beta$ and Eq. $(2)$ by cos $\alpha,$ and then subtract the resulting equations. (c) Square and add the results of parts (a) and (b). (d) Derive the equation $x^{2}+y^{2}-2 x y \cos \delta=a^{2} \sin ^{2} \delta,$ where $\delta=\alpha-\beta$ . (e) Use the above result to justify each of the diagrams in Fig. 33.61 (next page). In the figure, the angle given is the phase difference between two simple harmonic motions of the same frequency and amplitude, one horizontal (along the $x$ -axis) and the other vertical (along the $y$ -axis). The figure thus shows the resultant motion from the superposition of the two perpendicular harmonic motions.

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Problem 66

A rainbow is produced by the reflection of sunlight by spherical drops of water in the air. Figure 33.62 shows a ray that refracts into a drop at point $A$ , is reflected from the back surface of the drop at point $B$ , and refracts back into the air at point $C .$ The angles of incidence and refraction, $\theta_{a}$ and $\theta_{b}$ , are shown at points $A$ and $C,$ and the angles of incidence and reflection, $\theta_{a}$ and $\theta_{r}$ are shown at point $B$ (a) Show that $\theta_{a}^{B}=\theta_{b}^{A}, \theta_{a}^{c}=\theta_{b}^{A},$ and $\theta_{b}^{c}=\theta_{a}^{A}$ (b) Show that the angle in radians between the ray before it enters the drop at $A$ and after it exits at $C$ (the total angular deflection of the ray) is $\Delta=2 \theta_{a}^{A}-4 \theta_{b}^{A}+\pi .$ (Hint Find the angular deflections that occur at $A, B,$ and $C,$ and add them to get $\Delta . )(\mathrm{c})$ Use Snell's law to write $\Delta$ in terms of $\theta_{a}^{A}$ and $n,$ the refractive index of the water in the drop. (d) A rainbow will form when the angular deflection $\Delta$ is stationary in the incident angle $\theta_{a}^{A}-$ that is, when $d \Delta / d \theta_{n}^{A}=0$ If this condition is satisfied, all the rays with incident angles close to $\theta_{a}^{A}$ will be sent back in the same direction, producing a bright zone in the sky. Let $\theta_{1}$ be the value of $\theta_{a}^{A}$ for which this occurs. Show that $\cos ^{2} \theta_{1}=\frac{1}{3}\left(n^{2}-1\right)$ . (Hint: You may find the derivative formula $d(\arcsin u(x)) / d x=\left(1-u^{2}\right)^{-1 / 2}(d u / d x)$ helpful. (e) The index of refraction in water is 1.342 for violet light and 1.330 for red light. Use the results of parts $(\mathrm{c})$ and $(\mathrm{d})$ to find $\theta_{1}$ and $\Delta$ for violet and red fight. Do your results agree with the angles shown in Fig. 33.20 $\mathrm{d} ?$ When you view the rainbow, which color, red or violet, is higher above the horizon?

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Problem 67

A secondary rainbow is formed when the incident light undergoes two internal reflections in a spherical drop of water as shown in Fig, 33.200 . (See Challenge Problem $33.66 .$ ) (a) In terms of the incident angle $\theta_{a}^{A}$ and the refractive index $n$ of the drop, what is the angular deflection $\Delta$ of the ray? That is, what is the angle between the ray before it enters the drop and after it exits? (b) What is the incident angle $\theta_{2}$ for which the derivative of $\Delta$ with respect to the incident angle $\theta_{a}^{A}$ is zero? (c) The indexes of refraction for red and violet light in water are given in part (e) of Challenge Problem 33.66 . Use the results of parts (a) and $(b)$ to find $\theta_{2}$ and $\Delta$ for violet and red light. Do your results agree with the angles shown in Fig 33.20$e ?$ When you view a secondary rainbow, is red or violet higher above the horizon? Explain.

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