Physics for Scientists and Engineers with Modern Physics

Douglas C. Giancoli

Chapter 32

Light: Reflection and Refraction - all with Video Answers

Educators

+ 6 more educators

Chapter Questions

Problem 1

(I) When you look at yourself in a 60 -cm-tall plane mirror,
you see the same amount of your body whether you are
close to the mirror or far away. (Try it and see.) Use ray
diagrams to show why this should be true.

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 2

(1) Suppose that you want to take a photograph of yourself as
you look at your image in a mirror 2.8 $\mathrm{m}$ away. For what
distance should the camera lens be focused?

Bruce Edelman

Numerade Educator

Problem 3

(II) Two plane mirrors meet at a $135^{\circ}$ angle, Fig. $45 .$ If
light rays strike one mirror at $38^{\circ}$ as shown, at what angle $\phi$ do they leave the second
mirror?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 4

(II) A person whose eyes are 1.64 $\mathrm{m}$ above the floor stands
2.30 $\mathrm{m}$ in front of a vertical plane mirror whose bottom edge
is 38 $\mathrm{cm}$ above the floor, Fig. $46 .$ What is the horizontal
distance $x$ to the base of the wall supporting the mirror of the nearest point on the floor
that can be seen reflected in
the mirror?

Bruce Edelman

Numerade Educator

Problem 5

(II) Show that if two plane mirrors meet at an angle $\phi,$ a
single ray reflected successively from both mirrors is
deflected through an angle of 2$\phi$ independent of the incident angle. Assume $\phi<90^{\circ}$ and that only two reflections, one from each mirror, take place.

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 6

(II) Suppose you are 88 $\mathrm{cm}$ from a plane mirror. What area
of the mirror is used to reflect the rays entering one eye
from a point on the tip of your nose if your pupil diameter
is 4.5 $\mathrm{mm}$ ?

Bruce Edelman

Numerade Educator

Problem 7

(II) Stand up two plane mirrors so they form a $90.0^{\circ}$ angle as in Fig. $47 .$ When you look into this double mirror, you see yourself as others see you, instead of reversed as in a
single mirror. Make a ray diagram to show how this occurs.

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 8

(III) Suppose a third mirror is placed beneath the two shown in Fig. $47,$ so that all three are perpendicular to each other. $(a)$ Show that for such a "corner reflector," any incident ray will return in its original direction after three reflections. (b) What happens if it makes only two reflections?

Bruce Edelman

Numerade Educator

Problem 9

(I) A solar cooker, really a concave mirror pointed at the
Sun, focuses the Sun's rays 18.8 $\mathrm{cm}$ in front of the mirror.
What is the radius of the spherical surface from which the
mirror was made?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 10

(I) How far from a concave mirror (radius 24.0 $\mathrm{cm}$ ) must an
object be placed if its image is to be at infinity

Bruce Edelman

Numerade Educator

Problem 11

(I) When walking toward a concave mirror you notice that
the image flips at a distance of 0.50 $\mathrm{m} .$ What is the radius of
curvature of the mirror?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 12

(II) A small candle is 35 $\mathrm{cm}$ from a concave mirror having a
radius of curvature of 24 $\mathrm{cm} .(a)$ What is the focal length of
the mirror? (b) Where will the image of the candle be
located? (c) Will the image be upright or inverted?

Bruce Edelman

Numerade Educator

Problem 13

(II) You look at yourself in a shiny 9.2 -cm-diameter
Christmas tree ball. If your face is 25.0 $\mathrm{cm}$ away from the
ball's front surface, where is your image? Is it real or
virtual? Is it upright or inverted?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 14

(II) A mirror at an amusement park shows an upright image
of any person who stands 1.7 $\mathrm{m}$ in front of it. If the image is
three times the person's height, what is the radius of
curvature of the mirror? (See Fig. 44.)

Bruce Edelman

Numerade Educator

Problem 15

(II) A mirror at an amusement park shows an upright image
of any person who stands 1.7 m in front of it. If the image is
three times the person's height, what is the radius of
curvature of the mirror? (See Fig. $44 . )$

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 16

(II) A dentist wants a small mirror that, when 2.00 $\mathrm{cm}$ from a
tooth, will produce a $4.0 \times$ upright image. What kind of mirror
must be used and what must its radius of curvature be?

Bruce Edelman

Numerade Educator

Problem 17

(II) Some rearview mirrors produce images of cars to your
rear that are smaller than they would be if the mirror were
flat. Are the mirrors concave or convex? What is a mirror's
radius of curvature if cars 18.0 $\mathrm{m}$ appear 0.33 their
normal size?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 18

(II) You are standing 3.0 $\mathrm{from}$ a convex security mirror in a
store. You estimate the height of your image to be half of your
actual height. Estimate the radius of curvature of the mirror.

Bruce Edelman

Numerade Educator

Problem 19

(II) An object 3.0 $\mathrm{mm}$ high is placed 18 $\mathrm{cm}$ from a convex
mirror of radius of curvature 18 $\mathrm{cm} .(a)$ Show by ray tracing that the image is virtual, and estimate the image distance. (b) Show that the (negative) image distance can be computed
from Eq. 2 using a focal length of $-9.0 \mathrm{cm} .(c)$ Compute the image size, using $\mathrm{Eq} .3 .$
$$\begin{aligned} \frac{1}{d_{\mathrm{o}}} &+\frac{1}{d_{\mathrm{i}}}=\frac{1}{f} \\ m &=\frac{h_{\mathrm{i}}}{h_{\mathrm{o}}}=-\frac{d_{\mathrm{i}}}{d_{\mathrm{o}}} \end{aligned}$$
(II) The image of a distant tree is virtual and very small when viewed in a curved mirror. The image appears to be 16.0 $\mathrm{cm}$ behind the mirror. What kind of mirror is it, and what is its radius of curvature?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 20

(II) Use two techniques, (a) a ray diagram, and (b) the mirror equation, to show that the magnitude of the magnification of a concave mirror is less than 1 if the object is beyond the center of curvature $\mathrm{C}\left(d_{\mathrm{o}}>r\right),$ and is greater than 1 if the object is within $\mathrm{C}\left(d_{\mathrm{o}}< r\right)$

Bruce Edelman

Numerade Educator

Problem 21

(II) Show, using a ray diagram, that the magnification $m$ of a
convex mirror is $m=-d_{1} / d_{0},$ just as for a concave mirror.
$[$ Hint: Consider a ray from the top of the object that reflects
at the center of the mirror.

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 22

(II) Use ray diagrams to show that the mirror equation,
Eq. $2,$ is valid for a convex mirror as long as $f$ is
considered negative.

Bruce Edelman

Numerade Educator

Problem 23

(II) The magnification of a convex mirror is $+0.55 \times$ for
objects 3.2 $\mathrm{m}$ from the mirror. What is the focal length of
this mirror?

BS

Bhupal Shrestha

Numerade Educator

Problem 24

(II) (a) Where should an object be placed in front of a
concave mirror so that it produces an image at the same
location as the object? $(b)$ Is the image real or virtual? $(c)$ Is
the image inverted or upright? (d) What is the magnification of the image?

Bruce Edelman

Numerade Educator

Problem 25

(II) A 4.5 -cm tall object is placed 26 $\mathrm{cm}$ in front of a spherical
mirror. It is desired to produce a virtual image that is upright
and 3.5 $\mathrm{cm}$ tall. (a) What type of mirror should be used?
(b) Where is the image located? (c) What is the focal length of
the mirror? (d) What is the radius of curvature of the mirror?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 26

(II) A shaving or makeup mirror is designed to magnify your face by a factor of 1.35 when your face is placed 20.0 $\mathrm{cm}$ in front of it. (a) What type of mirror is it? $(b)$ Describe the
type of image that it makes of your face. (c) Calculate the required radius of curvature for the mirror.

Bruce Edelman

Numerade Educator

Problem 27

(II) A concave mirror has focal length $f .$ When an object is placed a distance $d_{\mathrm{o}}>f$ from this mirror, a real image with magnification $m$ is formed. $(a)$ Show that $m=f /\left(f-d_{\mathrm{o}}\right)$ (b) Sketch $m$ vs. $d_{0}$ over the range $f < d_{\mathrm{o}} < +\infty$ where $f=0.45 \mathrm{m} .(c)$ For what value of $d_{\mathrm{o}}$ will the real image have the same (lateral) size as the object? $(d)$ To obtain a real image that is much larger than the object, in what general region should the object be placed relative to the mirror?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 28

(II) Let the focal length of a convex mirror be written as $f=-|f| .$ Show that the magnification $m$ of an object a distance $d_{\mathrm{o}}$ from this mirror is given by $m=|f| /\left(d_{\mathrm{o}}+|f|\right) .$ Based on this relation, explain why your nose looks bigger than the rest of your face when looking into a convex mirror.

Bruce Edelman

Numerade Educator

Problem 29

(II) A spherical mirror of focal length $f$ produces an image of an object with magnification $m$ . (a) Show that the object is a distance $d_{\mathrm{o}}=f\left(1-\frac{1}{m}\right)$ from the reflecting side of the mirror. (b) Use the relation in part $(a)$ to show that, no matter where an object is placed in front of a convex mirror, its image will have a magnification in the range $0 \leq m \leq+1$

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 30

(III) An object is placed a distance $r$ in front of a wall, where $r$ exactly equals the radius of curvature of a certain concave mirror. At what distance from the wall should this mirror be placed so that a real image of the object is formed on the wall? What is the magnification of the image?

Bruce Edelman

Numerade Educator

Problem 31

(III) A short thin object (like a short length of wire) of length $\ell$ is placed along the axis of a spherical mirror (perpendicular to the glass surface). Show that its image has length $\ell^{\prime}=m^{2} \ell$ so the longitudinal magnification is equal to $-m^{2}$ where $m$ is the normal "lateral" magnification, Eq. $3 .$ Why the minus sign? [Hint: Find the image positions for both ends of the wire, and assume $\ell$ is very small.]

Dominador Tan

Numerade Educator

Problem 32

(I) The speed of light in ice is $2.29 \times 10^{8} \mathrm{m} / \mathrm{s} .$ What is the index of refraction of ice?

Bruce Edelman

Numerade Educator

Problem 33

(1) What is the speed of light in $(a)$ ethyl alcohol, $(b)$ lucite, (c) crown glass?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 34

(I) Our nearest star (other than the Sun) is 4.2 light years away. That is, it takes 4.2 years for the light to reach Earth. How far away is it in meters?

Bruce Edelman

Numerade Educator

Problem 35

(I) How long does it take light to reach us from the Sun, $1.50 \times 10^{8} \mathrm{km}$ away?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 36

(II) The speed of light in a certain substance is 88$\%$ of its value in water. What is the index of refraction of that substance?

Bruce Edelman

Numerade Educator

Problem 37

(II) Light is emitted from an ordinary lightbulb filament in wave-train bursts of about $10^{-8} \mathrm{s}$ in duration. What is the length in space of such wave trains?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 38

(I) A diver shines a flashlight upward from beneath the water at a $38.5^{\circ}$ angle to the vertical. At what angle does the light leave the water?

Bruce Edelman

Numerade Educator

Problem 39

(I) A flashlight beam strikes the surface of a pane of glass $(n=1.56)$ at a $63^{\circ}$ angle to the normal. What is the angle of refraction?

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 40

(I) Rays of the Sun are seen to make a $33.0^{\circ}$ angle to the vertical beneath the water. At what angle above the horizon is the Sun?

Bruce Edelman

Numerade Educator

Problem 41

(I) Rays of the Sun are seen to make a $33.0^{\circ}$ angle to the vertical beneath the water. At what angle above the horizon is the Sun?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 42

(II) A beam of light in air strikes a slab of glass $(n=1.56)$ and is partially reflected and partially refracted. Determine the angle of incidence if the angle of reflection is twice the angle of refraction.

Bruce Edelman

Numerade Educator

Problem 43

(II) A light beam strikes a $2.0-\mathrm{cm}$ -thick piece of plastic with a refractive index of 1.62 at a $45^{\circ}$ angle. The plastic is on top of a 3.0 -cm-thick piece of glass for which $n=1.47 .$ What is the distance $D$ in Fig. 48$?$

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 44

(II) An aquarium filled with water has flat glass sides whose index of refraction is $1.56 .$ A beam of light from outside the aquarium strikes the glass at a $43.5^{\circ}$ angle to the perpendicular (Fig. 49). What is the angle of this light ray when it enters $(a)$ the glass, and then $(b)$ the water? $(c)$ What would be the refracted angle if the ray entered the water directly?

Bruce Edelman

Numerade Educator

Problem 45

(II) In searching the bottom of a pool at night, a watchman shines a narrow beam of light from his flashlight, 1.3 $\mathrm{m}$ above the water level, onto the surface of the water at a point 2.5 $\mathrm{m}$ from his foot at the edge of the pool (Fig. 50$)$ . Where does the spot of light hit the bottom of the pool, measured from the bottom of the wall beneath his foot, if the pool is 2.1 $\mathrm{m}$ deep?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 46

(II) The block of glass $(n=1.5)$ shown in cross section in Fig. 51 is surrounded by air. A ray of light enters the block at its left-hand face with incident angle $\theta_{1}$ and reemerges into the air from the right-hand face directed parallel to the block's base. Determine $\theta_{1}$ .

Bruce Edelman

Numerade Educator

Problem 47

(II) A laser beam of diameter $d_{1}=3.0 \mathrm{mm}$ in air has an incident angle $\theta_{1}=25^{\circ}$ at a flat air-glass surface. If the index of refraction of the glass is $n=1.5,$ determine the diameter $d_{2}$ of the beam after it enters the glass.

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 48

(II) Light is incident on an equilateral glass prism at a $45.0^{\circ}$ angle to one face, Fig. $52 .$ Calculate the angle at which light emerges from the opposite face. Assume that $n=1.54 .$

Bruce Edelman

Numerade Educator

Problem 49

(II) A triangular prism made of crown glass $(n=1.52)$ with base angles of $30.0^{\circ}$ is surrounded by air. If parallel rays are incident normally on its base as shown in Fig. $53,$ what is the angle $\phi$ between the two emerging rays?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 50

(II) Show in general that for a light beam incident on a uniform layer of transparent material, as in Fig. $24,$ the direction of the emerging beam is parallel to the incident beam, independent of the incident angle $\theta$ . Assume the air on the two sides of the transparent material is the same.

Bruce Edelman

Numerade Educator

Problem 51

(III) A light ray is incident on a flat piece of glass with index of refraction $n$ as in Fig. $24 .$ Show that if the incident angle $\theta$ is small, the emerging ray is displaced a distance $d=t \theta(n-1) / n,$ Where $t$ is the thickness of the glass, $\theta$ is in radians, and $d$ is the perpendicular distance between the incident ray and the (dashed) line of the emerging ray (Fig. 24$)$ .

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 52

(I) By what percent is the speed of blue light $(450 \mathrm{nm})$ less than the speed of red light $(680 \mathrm{nm}),$ in silicate flint glass (see Fig. 28$) ?$

Bruce Edelman

Numerade Educator

Problem 53

(I) A light beam strikes a piece of glass at a $60.00^{\circ}$ incident angle. The beam contains two wavelengths, 450.0 $\mathrm{nm}$ and $700.0 \mathrm{nm},$ for which the index of refraction of the glass is 1.4831 and 1.4754 , respectively. What is the angle between the two refracted beams?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 54

(II) A parallel beam of light containing two wavelengths, $\lambda_{1}=465 \mathrm{nm}$ and $\lambda_{2}=652 \mathrm{nm},$ enters the silicate flint glass of an equilateral prism as shown in Fig. $54 .$ At what angle does each beam leave the prism (give angle with normal to the face)? See Fig. $28 .$

Bruce Edelman

Numerade Educator

Problem 55

(III) A ray of light with wavelength $\lambda$ is incident from air at precisely $60^{\circ} \quad(=\theta)$ on a spherical water drop of radius $r$ and index of refraction $n$ (which depends on $\lambda ) .$ When the ray reemerges into the air from the far side of the drop, it has been deflected an angle $\phi$ from its original direction as shown in Fig. $55 .$ By how much does the value of $\phi$ for violet light $(n=1.341)$ differ from the value for red light $(n=1.330) ?$

Sandeep Desai

Numerade Educator

Problem 56

(III) For visible light, the index of refraction $n$ of glass is roughly $1.5,$ although this value varies by about 1$\%$ across the visible range. Consider a ray of white light incident from air at angle $\theta_{1}$ onto a flat piece of glass. (a) Show that, upon entering the glass, the visible colors contained in this incident ray will be dispersed over a range $\Delta \theta_{2}$ of refracted angles given approximately by
$$\Delta \theta_{2} \approx \frac{\sin \theta_{1}}{\sqrt{n^{2}-\sin ^{2} \theta_{1}}} \frac{\Delta n}{n}$$
[Hint: For $x$ in radians, $(d / d x)\left(\sin ^{-1} x\right)=1 / \sqrt{1-x^{2}} . ]$ (b) If $\theta_{1}=0^{\circ},$ what is $\Delta \theta_{2}$ in degrees? $(c)$ If $\theta_{1}=90^{\circ}$
what is $\Delta \theta_{2}$ in degrees?

Bruce Edelman

Numerade Educator

Problem 57

(I) What is the critical angle for the interface between water
and diamond? To be internally reflected, the light must start in
which material?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 58

I) The critical angle for a certain liquid-air surface is $49.6^{\circ} .$
What is the index of refraction of the liguid?

Bruce Edelman

Numerade Educator

Problem 59

(II) A beam of light is emitted in a pool of water from a
depth of 72.0 $\mathrm{cm} .$ Where must it strike the air-water inter-
face, relative to the spot directly above it, in order that the
light does not exit the water?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 60

(II) A ray of light, after entering a light fiber, reflects at an angle of $14.5^{\circ}$ with the long axis of the fiber, as in Fig. $56 .$ Calculate the distance along the axis of the fiber that the light ray travels between successive reflections off the sides of the fiber. Assume that the fiber has an index of refraction
of 1.55 and is $1.40 \times 10^{-4} \mathrm{m}$ in diameter.

Bruce Edelman

Numerade Educator

Problem 61

(II) A beam of light is emitted 8.0 $\mathrm{cm}$ beneath the surface of a liquid and strikes the surface 7.6 $\mathrm{cm}$ from the point directly above the source. If total internal reflection occurs, what can you say about the index of refraction of the liquid?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 62

(II) Figure 57 shows a liquid-detecting prism device that might be used inside a washing machine or other liquid-containing appliance. If no liquid covers the prism's hypotenuse, total internal reflection of the beam from the light source produces a large signal in the light sensor. If liquid covers the hypotenuse, some light escapes from the prism into the liquid and the light sensor's signal decreases. Thus a large signal from the light sensor indicates the absence of liquid in the reservoir. If this device is designed to detect the presence of water, determine the allowable range for the prism's index of refraction $n .$ Will the device work properly if the prism is constructed from (inexpensive) lucite? For lucite, $n=1.5 .$

Bruce Edelman

Numerade Educator

Problem 63

(II) Two rays $A$ and $B$ travel down a cylindrical optical fiber of diameter $d=75.0 \mu \mathrm{m},$ length $\ell=1.0 \mathrm{km},$ and index of refraction $n_{1}=1.465 .$ Ray A travels a straight path down the fiber's axis, whereas ray $\mathrm{B}$ propagates down the fiber by repeated reflections at the critical angle each time it impinges on the fiber's boundary. Determine the extra time $\Delta t$ it takes for ray $B$ to travel down the entire fiber in comparison with ray A (Fig. $58 ),$ assuming $(a)$ the fiber is surrounded by air, (b) the fiber is surrounded by a cylindrical glass "cladding" with index of refraction $n_{2}=1.460 .$

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 64

(II) $(a)$ What is the minimum index of refraction for a glass or plastic prism to be used in binoculars (Fig. 33$)$ so that total internal reflection occurs at $45^{\circ} ?(b)$ Will binoculars work if their prisms (assume $n=1.58 )$ are immersed in water? (c) What minimum $n$ is needed if the prisms are immersed in water?

Vishal Gupta

Numerade Educator

Problem 65

(III) Suppose a ray strikes the left face of the prism in Fig. 52 at $45.0^{\circ}$ as shown, but is totally internally reflected at the opposite side. If the apex angle (at the top) is $\theta=60.0^{\circ},$ what can you say about the index of refraction of the prism?

Christopher Provencher

Christopher Provencher

Numerade Educator

Problem 66

(III) A beam of light enters the end of an optic fiber as shown in Fig. $59 .$ (a) Show that we can guarantee total internal reflection at the side surface of the material (at point $A ),$ if the index of refraction is greater than about 1.42. In other words, regardless of the angle $\alpha,$ the light beam reflects back into the material at point $\mathrm{A},$ assuming air outside.

Bruce Edelman

Numerade Educator

Problem 67

(II) A 13.0 -cm-thick plane piece of glass $(n=1.58)$ lies on
the surface of a 12.0 -cm-deep pool of water. How far below
the top of the glass does the bottom of the pool seem, as
viewed from directly above?

Prabhat Tyagi

Numerade Educator

Problem 68

(II) A fish is swimming in water inside a thin spherical glass bowl of uniform thickness. Assuming the radius of curvature of the bowl is $28.0 \mathrm{cm},$ locate the image of the fish if the fish is located: $(a)$ at the center of the bowl; $(b) 20.0 \mathrm{cm}$ from the side of the bowl between the observer and the center of the bowl. Assume the fish is small.

Bruce Edelman

Numerade Educator

Problem 69

(III) In Section 8 of "Light: Reflection and Refraction," we derived Eq. 8 for a convex spherical surface with $n_{2}>n_{1}$ Using the same conventions and using diagrams similar to Fig. 37 , show that $\mathrm{Eq} .8$ is valid also for $(a)$ a convex spherical surface with $n_{2}<n_{1},(b)$ a concave spherical surface with $n_{2}>n_{1},$ and $(c)$ a concave spherical surface with $n_{2}<n_{1}$ .

Lainey Roebuck

Numerade Educator

Problem 70

(III) A coin lies at the bottom of a 0.75 -m-deep pool. If a
viewer sees it at a $45^{\circ}$ angle, where is the image of the coin,
relative to the coin? [Hint: The image is found by tracing
back to the intersection of two rays. $]$

Bruce Edelman

Numerade Educator

Problem 71

Two identical concave mirrors are set facing each other 1.0 $\mathrm{m}$ apart. A small lightbulb is placed halfway between the mirrors A small piece of paper placed just to the left of the bulb prevents light from the bulb from directly shining on the left mirror, but light reflected from the right mirror still reaches the left mirror. A good image of the bulb appears on the left side of the piece of paper. What is the focal length of the mirrors?

Sandeep Desai

Numerade Educator

Problem 72

A slab of thickness $D,$ whose two faces are parallel, has index of refraction $n .$ A ray of light incident from air onto one face of the slab at incident angle $\theta_{1}$ splits into two rays A and B. Ray A reflects directly back into the air, while $B$ travels a total distance $\ell$ within the slab before reemerging from the slab's face a distance $d$ from its point of entry (Fig. $60 ) .(a)$ Derive expressions for $\ell$ and $d$ in terms of $D, n,$ and $\theta_{1},$ (b) For normal incidence (i.e., $\theta_{1}=0^{\circ} )$ show that your expressions yield the expected values for $\ell$ and $d$

Bruce Edelman

Numerade Educator

Problem 73

Two plane mirrors are facing each other 2.2 $\mathrm{m}$ apart as in Fig. $61 .$ You stand 1.5 $\mathrm{m}$ away from one of these mirrors and look into it. You will see multiple images of yourself. (a) How far away from you are the first three images of yourself in the mirror in front of you? (b) Are these first three images facing toward you or away from you?

Sandeep Desai

Numerade Educator

Problem 74

We wish to determine the depth of a swimming pool filled with water by measuring the width $(x=5.50 \mathrm{m})$ and then noting that the bottom edge of the pool is just visible at an angle of $13.0^{\circ}$ above the horizontal as shown in Fig. $62 .$ Calculate the depth of the pool.

Bruce Edelman

Numerade Educator

Problem 75

A 1.80 -m-tall person stands 3.80 $\mathrm{m}$ from a convex mirror and notices that he looks precisely half as tall as he does in a plane mirror placed at the same distance. What is the radius of curvature of the convex mirror? (Assume that $\sin \theta \approx \theta )$ IHint: The viewing angle is half.

Prabhat Tyagi

Numerade Educator

Problem 76

The critical angle of a certain piece of plastic in air is $\theta_{C}=39.3^{\circ} .$ What is the critical angle of the same plastic if it is immersed in water?

Bruce Edelman

Numerade Educator

Problem 77

Each student in a physics lab is assigned to find the location where a bright object may be placed in order that a concave mirror, with radius of curvature $r=46 \mathrm{cm},$ will produce an image three times the size of the object. Two students complete the assignment at different times using identical equipment, but when they compare notes later, they discover that their answers for the object distance are not the same. Explain why they do not necessarily need to repeat the lab, and justify your response with a calculation.

Sandeep Desai

Numerade Educator

Problem 78

A kaleidoscope makes symmetric patterns with two plane mirrors having a $60^{\circ}$ angle between them as shown in Fig. $63 .$ Draw the location of the images (some of them images of images) of the object placed between the mirrors.

Bruce Edelman

Numerade Educator

Problem 79

When light passes through a prism, the angle that the refracted ray makes relative to the incident ray is called the deviation angle $\delta,$ Fig. $64 .$ Show that this angle is a minimum when the ray passes through the prism symmetrically, perpendicular to the bisector of the apex angle $\phi,$ and show that the minimum deviation angle, $\delta_{m},$ is related to the prism's index of refraction $n$ by
$$n=\frac{\sin \frac{1}{2}\left(\phi+\delta_{\mathrm{m}}\right)}{\sin \phi / 2}$$
$\left[$Hint. For $\theta$ in radians, $(d / d \theta)\left(\sin ^{-1} \theta\right)=1 / \sqrt{1-\theta^{2}}\right]$

Sam Stansfield

Numerade Educator

Problem 80

If the apex angle of a prism is $\phi=72^{\circ}$ (see Fig, $64 ),$ what is
the minimum incident angle for a ray if it is to emerge from
the opposite side (i.e., not be totally internally reflected),
given $n=1.58 ?$

Bruce Edelman

Numerade Educator

Problem 81

Fermat's principle states that "light travels between two points along the path that requires the least time, as compared to other nearby paths" From Fermat's principle derive (a) the law of reflection $\left(\theta_{i}=\theta_{r}\right)$ and $(b)$ the law of refraction (Snell's law). [Hint: Choose two appropriate points so that a ray between them can undergo reflection or refraction. Draw a rough path for a ray between these points, and write down an expression of the time required for light to travel the arbitrary path chosen. Then take the derivative to find the minimum.

Lainey Roebuck

Numerade Educator

Problem 82

Suppose Fig. 36 shows a cylindrical rod whose end has a
radius of curvature $R=2.0 \mathrm{cm},$ and the rod is immersed in
water with index of refraction of $1.33 .$ The rod has index of
refraction $1.53 .$ Find the location and height of the image of
an object 2.0 $\mathrm{mm}$ high located 23 $\mathrm{cm}$ away from the rod.

Bruce Edelman

Numerade Educator

Problem 83

An optical fiber is a long transparent cylinder of diameter $d$ and index of refraction $n .$ If this fiber is bent sharply, some light hitting the side of the cylinder may escape rather than reflect back into the fiber (Fig. 65 ). What is the smallest radius $r$ at a short bent section for which total internal reflection will be assured ling parallel to the axis of ling parallel to the axis of the fiber?

Sandeep Desai

Numerade Educator

Problem 84

An object is placed 15 $\mathrm{cm}$ from a certain mirror. The image is half the height of the object, inverted, and real. How far is the image from the mirror, and what is the radius of curvature of the mirror?

Bruce Edelman

Numerade Educator

Problem 85

The end faces of a cylindrical glass rod $(n=1.51)$ are perpendicular to the sides. Show that a light ray entering an end face at any angle will be totally internally reflected inside the rod when it strikes the sides. Assume the rod is in air. What if it were in water?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 86

The paint used on highway signs often contains small transparent spheres which provide nighttime illumination of the sign's lettering by retro-reflecting vehicle headlight beams. Consider a light ray from air incident on one such sphere of radius $r$ and index of refraction $n .$ Let $\theta$ be its incident angle, and let the ray follow the path shown in Fig. $66,$ so that the ray exits the sphere in the direction exactly antiparallel to its incoming direction. Considering only rays for which $\sin \theta$ can be approximated as $\theta,$ determine the required value for $n$

Narayan Hari

Numerade Educator

Problem 87

(II) The index of refraction, $n,$ of crown flint glass at different wavelengths $(\lambda)$ of light are given in the Table below.
$$\begin{array}{|c|c|c|c|c|}\hline \lambda(\mathrm{nm}) & {1060} & {546.1} & {365.0} & {312.5} \\ \hline n & {1.50586} & {1.51978} & {1.54251} & {1.5600} \\ \hline\end{array}$$
Make a graph of $n$ versus $\lambda$ . The variation in index of refraction with wavelength is given by the Cauchy equation $n=A+B / \lambda^{2} .$ Make another graph of $n$ versus 1$/ \lambda^{2}$ and determine the constants $A$ and $B$ for the glass by fitting the data with a straight line.

Sandeep Desai

Numerade Educator

Problem 88

(III) Consider a ray of sunlight incident from air on a spherical raindrop of radius $r$ and index of refraction $n .$ Defining $\theta$ to be its incident angle, the ray then follows the path shown in Fig. 67 , exiting the drop at a "scattering angle" $\phi$ compared with its original incoming direction. $(a)$ Show that $\phi=180^{\circ}+2 \theta-4 \sin ^{-1}(\sin \theta / n) .$ (b) The parallel rays of sunlight illuminate a raindrop with rays of all possible incident angles from $0^{\circ}$ to $90^{\circ} .$ Plot $\phi$ vs. $\theta$ in the range $0^{\circ} \leq \theta \leq 90^{\circ},$ in $0.5^{\circ}$ steps, assuming $n=1.33$ as is appropriate for water at visible-light wavelengths. (c) From your plot, you should find that a fairly large fraction of the incident angles have nearly the same scattering angle. Approximately what fraction of the possible incident angles is within roughly $1^{\circ}$ of $\phi=139^{\circ} ?$ [This subset of incident rays is what creates the rainbow. Wavelength- dependent variations in $n$ cause the rainbow to form at slightly different $\phi$ for the various visible colors.]

Bruce Edelman

Numerade Educator