College Physics

Jerry D. Wilson, Anthony J. Buffa, Bo Lou

Chapter 22

Reflection and Refraction of Light - all with Video Answers

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Chapter Questions

Problem 1

The angle of incidence of a light ray on a mirrored surface is $30^{\circ} .$ What is the angle between the incident and reflected rays?

Narayan Hari

Narayan Hari

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Problem 2

A beam of light is incident on a plane mirror at an angle of $25^{\circ}$ relative to the normal. What is the angle between the reflected ray and the surface of the mirror?

Narayan Hari

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Problem 3

A beam of light is incident on a plane mirror at an angle $\alpha$ relative to the surface of the mirror. (a) Will the angle between the reflected ray and the normal be (1) $\alpha$,
(2) $90^{\circ}-\alpha,$ or
(3) $2 \alpha ?$ (b) If $\alpha=33^{\circ},$ what is the angle between the reflected ray and the normal?

Narayan Hari

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Problem 4

Two upright plane mirrors touch along one edge, where their planes make an angle of $\alpha$. A beam of light is directed onto one of the mirrors at an angle of incidence of $\beta(\beta<\alpha)$ and is reflected onto the other mirror.
(a) Will the angle of reflection of the beam from the second mirror be $(1) \alpha,(2) \beta,(3) \alpha+\beta,$ or $(4) \alpha-\beta ?$
(b) If $\alpha=60^{\circ}$ and $\beta=40^{\circ},$ what will be the angle of reflection of the beam from the second mirror?

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Problem 5

Two identical plane mirrors of width $w$ are placed a distance $d$ apart with their mirrored surfaces parallel and facing each other. (a) A beam of light is incident at one end of one mirror so that the light just strikes the far end of the other mirror after reflection. Will the angle of incidence be (1) $\sin ^{-1}(w / d),$ (2) $\cos ^{-1}(w / d),$ or (3) $\tan ^{-1}(w / d) ?(\mathrm{~b})$ If $d=50 \mathrm{~cm}$ and $w=25 \mathrm{~cm},$ what is the angle of incidence?

Narayan Hari

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Problem 6

Two people stand $5.0 \mathrm{~m}$ apart and $3.0 \mathrm{~m}$ away from a large plane mirror in a dark room. At what angle of incidence should one of them shine a flashlight on the mirror so that the reflected beam directly strikes the other person?

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Problem 7

A beam of light is incident on a plane mirror at an angle of incidence of $35^{\circ} .$ If the mirror rotates through a small angle of $\theta$, through what angle will the reflected ray rotate?

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Problem 8

Two plane mirrors, $M_{1}$ and $M_{2},$ are placed together as illustrated in $v$ Fig. $22.25 .$ (a) If the angle $\alpha$ between the mirrors is $70^{\circ}$ and the angle of incidence, $\theta_{\mathrm{i}_{1}}$ of a light ray incident on $M_{1}$ is $35^{\circ},$ what is the angle of reflection, $\theta_{\mathrm{r}_{2}}$, from $M_{2} ?$ (b) If $\alpha=115^{\circ}$ and $\theta_{\mathrm{i}_{1}}=60^{\circ},$ what is $\theta_{\mathrm{r}_{2}} ?$

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Problem 9

For the plane mirrors in Fig. $22.25,$ what angles $\alpha$ and $\theta_{\mathrm{i}}$ would allow a ray to be reflected back in the direction from which it came (parallel to the incident ray)?

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Problem 10

The index of refraction in a certain precious transparent stone is $2.35 .$ What is the speed of light in that stone?

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Problem 11

The speed of light in the core of the crystalline lens in a human eye is $2.13 \times 10^{8} \mathrm{~m} / \mathrm{s}$. What is the index of refraction of the core?

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Problem 12

The indices of refraction for zircon and fused quartz can be found in Table 22.1. (a) The speed of light in fused quartz is (1) greater than, (2) less than, (3) the same as the speed of light in zircon. Explain. (b) Compute the ratio of the speed of light in fused quartz to that in zircon.

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Problem 13

A beam of light traveling in air is incident on a transparent plastic material at an angle of incidence of $50^{\circ} .$ The angle of refraction is $35^{\circ} .$ What is the index of refraction of the plastic?

Narayan Hari

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Problem 14

A beam of light enters water from air. (a) Will the angle of refraction be (1) greater than, (2) equal to, or (3) less than the angle of incidence? Explain. (b) If the angle of incidence is $50^{\circ},$ find the angle of refraction.

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Problem 15

Light passes from a crown glass container into water. (a) Will the angle of refraction be (1) greater than, (2) equal to, or (3) less than the angle of incidence? Explain. (b) If the angle of refraction is $20^{\circ},$ what is the angle of incidence?

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Problem 16

The critical angle for a certain type of glass in air is $41.8^{\circ} .$ What is the index of refraction of the glass?

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Problem 17

(a) For total internal reflection to occur, should the light be directed from (1) air to water or (2) water to air? Explain. (b) What is the critical angle of water in air?

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Problem 18

What is the critical angle of a diamond in water?

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Problem 19

A beam of light in air is incident on the surface of a slab of fused quartz. Part of the beam is transmitted into the quartz at an angle of refraction of $30^{\circ}$ relative to a normal to the surface, and part is reflected. What is the angle of reflection?

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Problem 20

A beam of light is incident from air onto a flat piece of polystyrene at an angle of $55^{\circ}$ relative to a normal to the surface. What angle does the refracted ray make with the plane of the surface?

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Problem 21

The laser used in cornea surgery to treat corneal disease is the excimer laser, which emits ultraviolet light at a wavelength of $193 \mathrm{nm}$ in air. The index of refraction of the cornea is $1.376 .$ What are the wavelength and frequency of the light in the cornea?

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Problem 22

Light passes from material A, which has an index of refraction of $\frac{4}{3},$ into material $\mathrm{B}$, which has an index of refraction of $\frac{5}{4}$. (a) The speed of light in material A is (1) greater than, (2) the same as, (3) less than the speed of light in material B. Explain. (b) Find the ratio of the speed of light in material A to the speed of light in material B.

Narayan Hari

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Problem 23

In Exercise $22,$ (a) the wavelength of light in material A is (1) greater than, (2) the same as, (3) less than the wavelength of light in material B. Explain. (b) What is the ratio of the light's wavelength in material A to that in material B?

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Problem 24

The critical angle between two materials is $43^{\circ} .$ If the angle of incidence is $35^{\circ},$ what is the angle of refraction? (Consider that light can travel to the interface from either material.)

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Problem 25

(a) An object immersed in water appears closer to the surface than it actually is. What is the cause of this illusion? (b) Using $>$ Fig. $22.26,$ show that the apparent depth for small angles of refraction is $d^{\prime}=d / n,$ where $n$ is the index of refraction of the water. [Hint: Recall that for small angles, $\tan \theta \approx \sin \theta .]$

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Problem 26

A person looks over the edge of a pool and sees a bottle cap on the bottom directly below, where the depth is $3.2 \mathrm{~m}$. How far below the water surface does the bottle cap appear to be? (See Exercise 25b.)

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Problem 27

What percentage of the actual depth is the apparent depth of an object submerged in oil if the observer is looking almost straight downward? (See Exercise 25b.)

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Problem 28

A light ray in air is incident on a glass plate $10.0 \mathrm{~cm}$ thick at an angle of incidence of $40^{\circ} .$ The glass has an index of refraction of $1.65 .$ The emerging ray on the other side of the plate is parallel to the incident ray, but is laterally displaced. What is the perpendicular distance between the original direction of the ray and the direction of the emerging ray? [Hint: See Example $22.4 .]$

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Problem 29

To a submerged diver looking upward through the water, the altitude of the Sun (the angle between the Sun and the horizon) appears to be $45^{\circ} .$ (a) The actual altitude of the Sun is (1) greater than, (2) the same as, (3) less than $45^{\circ} .$ Explain. (b) What is the Sun's actual altitude?

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Problem 30

At what angle to the surface must a diver submerged in a lake look toward the surface to see the setting Sun just along the horizon?

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Problem 31

A submerged diver shines a light toward the surface of a body of water at angles of incidence of $40^{\circ}$ and $50^{\circ}$. Can a person on the shore see a beam of light emerging from the surface in either case? Justify your answer mathematically.

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Problem 32

Light in air is incident on a transparent material. It is found that the angle of reflection is twice the angle of refraction. What is the range of the index of refraction of the material?

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Problem 33

A beam of light is to undergo total internal reflection through a $45^{\circ}-90^{\circ}-45^{\circ}$ prism ( $\mathbf{v}$ Fig. 22.27 ). (a) Will this arrangement depend on (1) the index of refraction of the prism, (2) the index of refraction of the surrounding medium, or (3) the indices of refraction of both? Explain. (b) Calculate the minimum index of refraction of the prism if the surrounding medium is air. Repeat if it is water.

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Problem 34

A $45^{\circ}-90^{\circ}-45^{\circ}$ prism (Fig. 22.27 ) is made of a material with an index of refraction of $1.85 .$ Can the prism be used to deflect a beam of light by $90^{\circ}(\mathrm{a})$ in air? (b) What about in water?

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Problem 35

A coin lies on the bottom of a pool under $1.5 \mathrm{~m}$ of water and $0.90 \mathrm{~m}$ from the side wall ( $\mathbf{F}$ Fig. 22.28 ). If a light beam is incident on the water surface at the wall, at what angle $\theta$ relative to the wall must the beam be directed so that it will illuminate the coin?

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Problem 36

A flint glass plate $3.5 \mathrm{~cm}$ thick is placed over a newspaper. How far beneath the top surface of the plate would the print appear to be if you were looking almost vertically downward through the plate? (See Exercise 25b.)

Ajay Singhal

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Problem 37

Yellow-green light of wavelength $550 \mathrm{nm}$ in air is incident on the surface of a flat piece of crown glass at an angle of $40^{\circ} .$ (a) What is the angle of refraction of the light? (b) What is the speed of the light in the glass? (c) What is the wavelength of the light in the glass?

Narayan Hari

Numerade Educator

Problem 38

A light beam traveling upward in a plastic material with an index of refraction of 1.60 is incident on an upper horizontal air interface. (a) At certain angles of incidence, the light is not transmitted into air. The cause of this is (1) reflection, (2) refraction, (3) total internal reflection. Explain. (b) If the angle of incidence is $45^{\circ},$ is some of the beam transmitted into air? (c) Suppose the upper surface of the plastic material is covered with a layer of liquid with an index of refraction of $1.20 .$ What happens in this case?

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Problem 39

$\mathrm{A} 15-\mathrm{cm}$ -deep opaque container is empty except for a single coin resting on its bottom surface. When looking into the container at a viewing angle of $50^{\circ}$ relative to the vertical side of the container, you see nothing on the bottom. When the container is filled with water, you see the coin (from the same viewing angle) on the bottom of, and just beyond, the side of the container. (See Fig. $22.24 .)$ How far is the coin from the side of the container?

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Problem 40

An outdoor circular fish pond has a diameter of $4.00 \mathrm{~m}$ and a uniform full depth of $1.50 \mathrm{~m}$. A fish halfway down in the pond and $0.50 \mathrm{~m}$ from the near side can just see the full height of a 1.80 -m-tall person. How far away from the edge of the pond is the person?

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Problem 41

For total internal reflection to occur inside an optic fiber as shown in $v$ Fig. $22.29,$ the angle $\theta$ must be greater than the critical angle for the fiber-air interface. At the end of the fiber, the incident light undergoes a refraction to enter the fiber. If total internal reflection is to occur for any angle of incidence $\left(\theta_{i}\right)$ outside the end of the fiber, what is the minimum index of refraction of the fiber?

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Problem 42

Two glass prisms are placed together (vFig. 22.30).
(a) If a beam of light strikes the face of one of the prisms at normal incidence as shown, at what angle $\theta$ does the beam emerge from the other prism?
(b) At what angle of incidence would the beam be refracted along the interface of the prisms?

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Problem 43

White light is incident from air onto a transparent material at an angle of incidence of $40^{\circ} .$ The angles of refraction for the red and blue colors are $28.15^{\circ}$ and $27.95^{\circ},$ respectively. What are the indices of refraction for the two colors?

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Problem 44

The index of refraction of crown glass is 1.515 for red light and 1.523 for blue light. (a) If light of both colors is incident on crown glass from air, the blue color will be refracted (1) more, (2) less, or (3) the same amount as the red color. Explain. (b) Find the angle separating rays of the two colors in a piece of crown glass if their angle of incidence is $37^{\circ}$.

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Problem 45

A beam of light with red and blue components of wavelengths $670 \mathrm{nm}$ and $425 \mathrm{nm}$, respectively, strikes a slab of fused quartz at an incident angle of $30^{\circ} .$ On refraction, the different components are separated by an angle of 0.00131 rad. If the index of refraction of the red light is $1.4925,$ what is the index of refraction of the blue light?

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Problem 46

White light passes through a piece of crown glass and strikes an interface with air at an angle of $41.15^{\circ}$ Assume the indices of refraction of crown glass are the same as given in Exercise $44 .$ Which color(s) of light will be refracted out into the air?

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Problem 47

A beam of red light is incident on an equilateral prism as shown in $v$ Fig. $22.31 .$ (a) If the index of refraction of red light of the prism is $1.400,$ at what angle $\theta$ does the beam emerge from the other face of the prism?
(b) Suppose the incident beam were white light. What would be the angular separation of the red and blue components in the emergent beam if the index of refraction of blue light were $1.403 ?$ (c) What would it be if the index of refraction of blue light were $1.405 ?$

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Problem 48

The angle of incidence and angle of refraction along a particular interface between two media are $30^{\circ}$ and $40^{\circ}$, respectively. What is the critical angle for the same interface?

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Problem 49

Light passes from medium A into medium $B$ at an angle of incidence of $30^{\circ} .$ The index of refraction of A is 1.5 times that of B. (a) What is the angle of refraction?
(b) What is the ratio of the speed of light in $\mathrm{B}$ to the speed of light in $\mathrm{A} ?$ (c) What is the ratio of the frequency of light in $B$ to the frequency of light in A? (d) At what angle of incidence would the light be internally reflected?

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Problem 50

The critical angle between two materials is $43^{\circ} .$ If the angle of incidence is twice the angle of refraction, what are the angles of incidence and refraction?

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Problem 51

The critical angle for a glass-air interface is $41.11^{\circ}$ for red light and $41.04^{\circ}$ for blue light. (a) During the time the blue light travels $1.000 \mathrm{~m},$ the red light will travel
(1) more than, (2) less than, (3) exactly $1.000 \mathrm{~m}$. Explain.
(b) Calculate the difference in distance traveled by the two colors.

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Problem 52

In Exercise 47 , if the angle of incidence is too small, light will not emerge from the other side of the prism. How could this happen? Calculate the minimum angle of incidence for the red light so that it does not emerge from the other side of the prism.

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Problem 53

An optic fiber of index of refraction of 1.6 is surrounded by a layer called the cladding layer. The index of refraction of the cladding is $1.3 .$ If the angle of incidence inside the fiber is $40^{\circ},$ will there be light leaking into air? What if the angle of incidence is $50^{\circ} ?$

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