University Physics with Modern Physics

Wolfgang Bauer, Gary D. Westfall

Chapter 32

Geometric Optics - all with Video Answers

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Chapter Questions

Problem 1

32.1 Legend says that Archimedes set the Roman fleet on fire as it was invading Syracuse. Archimedes created a huge ______ mirror, and he focused the Sun's rays on the Roman vessels.
a) plane
c) parabolic focusing
b) parabolic diverging

Ankur S

Ankur S

Numerade Educator

Problem 2

Which of the following interface combinations has the smallest critical angle?
a) light traveling from ice to diamond
b) light traveling from quartz to lucite
c) light traveling from diamond to glass
d) light traveling from lucite to diamond
e) light traveling from lucite to quartz

Ajay Singhal

Numerade Educator

Problem 3

For specular reflection of a light ray, the angle of incidence
a) must be equal to the angle of reflection.
b) is always less than the angle of reflection.
c) is always greater than the angle of reflection.
d) is equal to $90^{\circ}$ - the angle of reflection.
e) may be greater than, less than, or equal to the angle of reflection.

Ajay Singhal

Numerade Educator

Problem 4

Standing by a pool filled with water, under what condition will you see a reflection of the scenery on the opposite side through total internal reflection of the light from the scenery?
a) Your eyes are level with the water.
b) You observe the pool at an angle of $41.8^{\circ}$
c) Under no condition.
d) You observe the pool at an angle of $48.2^{\circ}$

Ankur S

Numerade Educator

Problem 5

You are using a mirror and a camera to make a self portrait. You focus the camera on yourself through the mirror. The mirror is a distance $\mathrm{D}$ away from you. To what distance should you set the range of focus on the camera?
a) $D$
b) $2 \mathrm{D}$
c) $\mathrm{D} / 2$
d) $4 \mathrm{D}$

Ajay Singhal

Numerade Educator

Problem 6

What is the magnification for a plane mirror?
a) +1
c) greater than +1
b) -1
d) not defined for a plane mirror

Ajay Singhal

Numerade Educator

Problem 7

The figure shows the difference between the refraction index profile of a so-called step index fiber vs. the refraction index profile of a so called graded index fiber. Analyzing light propagation through the fiber from a ray-optics perspective, comment on the path followed by a light ray entering each of the two fibers.

Mkhitar Hobosyan

Mkhitar Hobosyan

Numerade Educator

Problem 8

A slab of Plexiglas $2.00 \mathrm{~cm}$ thick with index of refraction 1.51 is placed over a physics textbook. The slab has parallel sides. The text is at height $y=0 .$ Consider two rays of light that are leaving the letter ${ }^{4} \mathrm{t}$ " in the text under the Plexiglas and going toward an observer who is above the Plexiglas, looking down. Draw on the figure the apparent $y$ -position of the text under the Plexiglas as seen by the observer. Hint: From where in the Plexiglas do these rays appear to originate for the observer? Using the two rays $A$ and $\mathrm{B}$ after they have exited the Plexiglas, determine the apparent height $(y$ -position $)$ of the text under the Plexiglas as seen by the observer. You can easily do this experiment yourself. If you do not have a block of glass or Plexiglas, you can try placing a flat-bottomed drinking glass over the text.

Keshav Singh

Numerade Educator

Problem 9

A single concave spherical mirror is used to create an image of a source $5.00 \mathrm{~cm}$ tall that is located at position $x=0 \mathrm{~cm}$ which is $20.0 \mathrm{~cm}$ to the left of Point $\mathrm{C},$ the center of curvature of the mirror, as shown in the figure. The magnitude of the radius of curvature for the mirror is $10.0 \mathrm{~cm}$. With out changing the mirror, how can one reduce the spherical aberration produced by this mirror? Will there be any dis. advantages to your approach to reducing the spherical aberration?

Ankur S

Numerade Educator

Problem 10

If you look at an object at the bottom of a pool, the pool looks less deep than it actually is.
a) From what you have learned, calculate how deep a pool seems to be if it is actually 4 feet deep and you look directly down on it. The refractive index of water is $1.33 .$
b) Would the pool look more or less deep if you look at it from an angle other than vertical? Answer this qualitatively, without using an equation.

Keshav Singh

Numerade Educator

Problem 11

Why does refraction happen? That is, what is the physical reason a wave moves in a new medium with a different velocity than it did in the original medium?

Ankur S

Numerade Educator

Problem 12

Many fiber-optics devices have minimum specified bending angles. Why?

Ankur S

Numerade Educator

Problem 13

A physics student is eying a steel drum, the top part of which has the approximate shape of a concave spherical surface. The surface is sufficiently polished that she can just barely make out the reflection of her finger when she places it above the drum. As she slowly moves her finger toward the surface and then away from it, you ask her what she is doing. She replies that she is estimating the radius of curvature of the drum. How can she do that?

Ankur S

Numerade Educator

Problem 14

Answer as true or false with an explanation for the following: The wavelength of He-Ne laser light in water is less than its wavelength in the air. (The refractive index of water is $1.33 .$

Ankur S

Numerade Educator

Problem 15

Among the instruments Apollo astronauts left on the Moon were reflectors used to bounce laser beams back to Earth. These made it possible to measure the distance from the Earth to the Moon with unprecedented precision (uncertainties of a few centimeters out of $384,000 \mathrm{~km}$ ), for the study both of celestial mechanics and of plate tectonics on Earth. The reflectors consisted not of ordinary mirrors, but of arrays of corner cubes, each consisting of three square plane mirrors fixed perpendicular to each other, as adjacent faces of a cube. Why? Explain the function and advantages of this design.

Ankur S

Numerade Educator

Problem 16

A $45^{\circ}-45^{\circ}-90^{\circ}$ triangular prism can be used to reverse a light beam: The light enters perpendicular to the hypotenuse of the prism, reflects off each leg, and emerges perpendicular to the hypotenuse again. The surfaces of the prism are not silvered. If the prism is made of glass with in dex of refraction $n_{\text {glass }}=1.520$ and the prism is surrounded by air, the light beam will be reflected with a minimum loss of intensity (there are reflection losses as the light enters and leaves the prism).
a) Will this work if the prism is under water, which has index of refraction $n_{\mathrm{H}_{2} \mathrm{O}}=1.333 ?$
b) Such prisms are used, in preference to mirrors, to bend the optical path in quality binoculars. Why?

Ankur S

Numerade Educator

Problem 17

An object is imaged by a converging spherical mirror as shown in Figure $32.19,$ repeated below. Suppose a black cloth is put between the object and the mirror so that it covers everything above the axis of the mirror. How will the image be affected?

Ankur S

Numerade Educator

Problem 18

You are under water in a pond and look up at the smooth surface of the water, noticing the sun in the sky. Is the sun in fact higher in the sky than it appears to you while under water, or is it lower?

Ankur S

Numerade Educator

Problem 19

Holding a spoon in front of your face, convex side toward you, estimate the location of the image and its magnification.

Ajay Singhal

Numerade Educator

Problem 20

A solar furnace uses a large parabolic mirror (mirrors several stories high have been constructed) to focus the light of the Sun to heat a target. A large solar furnace can melt metals. Is it possible to attain temperatures exceeding $6000 \mathrm{~K}$ (the temperature of the photo sphere of the Sun) in a solar furnace? How, or why not?

Ankur S

Numerade Educator

Problem 21

A person sits $1.0 \mathrm{~m}$ in front of a plane mirror. What is the location of the image?

Ajay Singhal

Numerade Educator

Problem 22

A periscope consists of two flat mirrors and is used for viewing objects when an obstacle impedes direct viewing. Suppose that Curious George is looking through a periscope at the Man in the Yellow Hat, whose hat is at $d_{\mathrm{o}}=3.00 \mathrm{~m}$ from the upper mirror, and suppose that the two flat mirrors are separated by a distance $L=0.400 \mathrm{~m}$. What is the distance $D$ of the final image of the yellow hat from the lower mirror?

Keshav Singh

Numerade Educator

Problem 23

A person stands at a point $P$ relative to two plane mirrors oriented at $90^{\circ}$, as shown in the figure? How far away do the person's images appear from each other to the viewer?

Ankur S

Numerade Educator

Problem 24

Even the best mirrors absorb or transmit some of the light incident on them. The highest-quality mirrors might reflect $99.997 \%$ of incident light intensity. Suppose a cubical "room, $3.00 \mathrm{~m}$ on an edge, were constructed with such mirrors for the walls, floor, and ceiling. How slowly would such a room get dark? Estimate the time required for the intensity of light in such a room to fall to $1.00 \%$ of its initial value after the only light source in the room is switched off.

Narayan Hari

Numerade Educator

Problem 25

The radius of curvature of a convex mirror is $-25 \mathrm{~cm} .$ What is its focal length?

Ankur S

Numerade Educator

Problem 26

A single concave spherical mirror is used to create an image of a source $5.00 \mathrm{~cm}$ tall that is located at position $x=0 \mathrm{~cm}$ which is $20.0 \mathrm{~cm}$ to the left of Point $\mathrm{C}$, the center of curvature of the mirror, as shown in the figure. The magnitude of the radius of curvature for the mirror is $10.0 \mathrm{~cm}$. Calculate the position $x_{\mathrm{i}}$ where the image is formed. Use the coordinate system given in the drawing. What is the height $h_{i}$ of the image? Is the image upright or inverted (upright = pointing up, inverted = pointing down)? Is it real or virtual?

Ankur S

Numerade Educator

Problem 27

Convex mirrors are often used in side view mirrors on cars. Many such mirrors display the warning "Objects in mirror are closer than they appear." Assume a convex mirror has a radius of curvature of $14.0 \mathrm{~m}$ and that there is a car that is $11.0 \mathrm{~m}$ behind the mirror. For a flat mirror, the image distance would be $11.0 \mathrm{~m}$ and the magnification would be
1. Find the image distance and magnification for this mirror.

Ankur S

Numerade Educator

Problem 28

A $5.00-\mathrm{cm}$ object is placed $30.0 \mathrm{~cm}$ away from a convex mirror with a focal length of $-10.0 \mathrm{~cm}$. Determine the size, orientation, and position of the image.

Ankur S

Numerade Educator

Problem 29

The magnification of a convex mirror is $0.60 \times$ for an object $2.0 \mathrm{~m}$ from the mirror. What is the focal length of this mirror?

Ajay Singhal

Numerade Educator

Problem 30

An object is located at a distance of $100 . \mathrm{cm}$ from a concave mirror of focal length $20.0 \mathrm{~cm}$. Another concave mirror of focal length $5.00 \mathrm{~cm}$ is located $20.0 \mathrm{~cm}$ in front of the first concave mirror. The reflecting sides of the two mirrors face each other. What is the location of the final image formed by the two mirrors and the total magnification by the combination?

Ankur S

Numerade Educator

Problem 31

The shape of an elliptical mirror is described by the curve $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,$ with semi major axis $a$ and semi minor axis $b$. The foci of this ellipse are at points $(c, 0)$ and $(-c, 0)$ with $c=\left(a^{2}-b^{2}\right)^{1 / 2}$. Show that any light ray in the $x y$ -plane, which passes through one focus, is reflected through the other. "Whispering galleries" make use of this phenomenon with sound waves.

Keshav Singh

Numerade Educator

Problem 32

What is the speed of light in crown glass, whose index of refraction is $1.52 ?$

Narayan Hari

Numerade Educator

Problem 33

An optical fiber with an index of refraction of 1.5 is used to transport light of wavelength $400 \mathrm{nm}$. What is the critical angle for light to transport through this fiber without loss? If the fiber is immersed in water? In oil?

Ankur S

Numerade Educator

Problem 34

A helium-neon laser produces light of wavelength $\lambda_{\mathrm{vac}}=632.8 \mathrm{nm}$ in vacuum. If this light passes into water, with index of refraction $n=1.333,$ what then will be its
a) speed?
c) wavelength?
b) frequency?
d) color?

Ankur S

Numerade Educator

Problem 35

A light ray is incident from water of index of refraction 1.33 on a plate of glass whose index of refraction is 1.73. What is the angle of incidence, to have fully polarized reflected light?

Ankur S

Numerade Educator

Problem 36

Suppose you are standing at the bottom of a swimming pool looking up at the surface, which we assume to be calm. Looking up, you will see a circular window to the "outer world." If your eyes are approximately 2.00 meters beneath the surface, what is the diameter of this circular window?

Ankur S

Numerade Educator

Problem 37

A ray of light of a particular wavelength is incident on an equilateral triangular prism with an index of refraction for this wavelength of $1.23 .$ The ray is parallel to the base of the prism when it approaches the prism. The ray enters the prism at the midpoint of one of its sides, as shown in the figure. What is the direction of the ray when it emerges from the triangular prism?

Mkhitar Hobosyan

Mkhitar Hobosyan

Numerade Educator

Problem 38

A collimated laser beam strikes the left side (A) of a glass block at an angle $20.0^{\circ}$ with respect to horizontal, as shown in the figure. The block has an index of refraction of 1.55 and is surrounded by air with an index of $1.00 .$ The left side of the glass block is vertical $90.0^{\circ}$ from horizontal) while the right side
(B) is $60.0^{\circ}$ from horizontal. Determine the angle $\theta_{\mathrm{BT}}$ with respect to horizontal at which the light exits surface B.

Narayan Hari

Numerade Educator

Problem 39

In a step-index fiber, the index of refraction undergoes a discontinuity (jump) at the core-cladding interface, as shown in the figure. Infrared light with wavelength $1550 \mathrm{nm}$ propagates through such a step index fiber through total internal reflection at the core-cladding interface. The index of refraction for the core at $1550 \mathrm{nm}$ is $n_{\text {core }}=1.48$ If the maximum angle $\alpha_{\max }$ at which light can be coupled into the fiber such that no light will leak into the cladding is $\alpha_{\max }=14.033^{\circ}$,
calculate the percent difference between the index of refraction of the core and the index of refraction of the cladding.

Ankur S

Numerade Educator

Problem 40

Refer to Figure 32.49 and prove that the arc of the primary rainbow represents the $42^{\circ}$ angle from the direction of the sunlight.

Ajay Singhal

Numerade Educator

Problem 41

Use Fermat's Principle to derive the law of reflection.

Ajay Singhal

Numerade Educator

Problem 42

Fermat's Principle, from which geometric optics can be derived, states that light travels by a path that minimizes the time of travel between the points. Consider a light beam that travels a horizontal distance $D$ and a vertical distance $h$, through two large flat slabs of material, with a vertical interface between the materials. One material has a thickness $D / 2$ and index of refraction $n_{1},$ and the second material has a thickness $D / 2$ and index of refraction $n_{2} .$ Determine the equation involving the indices of refraction and angles from horizontal that the light makes at the interface $\left(\theta_{1}\right.$ and $\theta_{2}$ ) which minimize the time for this travel.

Ankur S

Numerade Educator

Problem 43

Suppose your height is $2.0 \mathrm{~m}$ and you are standing $50 \mathrm{~cm}$ in front of a plane mirror.
a) What is the image distance?
b) What is the image height?
c) Is the image inverted or upright?
d) Is the image real or virtual?

Ankur S

Numerade Educator

Problem 44

A light ray of wavelength 700 . $\mathrm{nm}$ traveling in air $\left(n_{1}=1.00\right)$ is incident on a boundary with a liquid $\left(n_{2}=1.63\right) .$
a) What is the frequency of the refracted ray?
b) What is the speed of the refracted ray?
c) What is the wavelength of the refracted ray?

Narayan Hari

Numerade Educator

Problem 45

You have a spherical mirror with a radius of curvature of $+20.0 \mathrm{~cm}$ (so it is concave facing you). You are looking at an object whose size you want to double in the image, so you can see it better. Where should you put the object? Where will the image be, and will it be real or virtual?

Ankur S

Numerade Educator

Problem 46

You are submerged in a swimming pool. What is the maximum angle at which you can see light coming from above the pool surface? That is, what is the angle for total internal reflection from water into air?

Narayan Hari

Numerade Educator

Problem 47

Light hits the surface of water at an incident angle of $30.0^{\circ}$ with respect to the normal line. What is the angle between the reflected ray and the refracted ray?

Ankur S

Numerade Educator

Problem 48

A spherical metallic Christmas tree ornament has a diameter of $8.00 \mathrm{~cm}$. If Saint Nicholas is by the fireplace, $1.56 \mathrm{~m}$ away, where will he see his reflection in the ornament? Is the image real or virtual?

Narayan Hari

Numerade Educator

Problem 49

One of the factors that cause a diamond to sparkle is its relatively small critical angle. Compare the critical angle of diamond in air compared to that of diamond in water.

Ankur S

Numerade Educator

Problem 50

What kinds of images, virtual or real, are formed by a converging mirror when the object is placed a distance away from the mirror that is
a) beyond the center of curvature of the mirror,
b) between the center of curvature and half the center of curvature, and
c) closer than half of the center of curvature.

Ankur S

Numerade Educator

Problem 51

At what angle $\theta$ shown in the diagram must a beam of light enter the water such that the reflected beam makes an angle of $40.0^{\circ}$ with respect to the normal of the water's surface?

Ankur S

Numerade Educator

Problem 52

concave mirror forms a real image twice as large as the object. The object is then moved such that the new real image produced is three times the size of the object. If the image was moved $75 \mathrm{~cm}$ from its initial position, how far was the object moved and what is the focal length of the mirror?

Supratim Pal

Numerade Educator

Problem 53

How deep does a point in the middle of a 3.00 -m-deep pool appear to a person standing outside of it 2.00 meters horizontally from the point? Take the refractive index for the pool to be 1.30 and for air to be $1.00 .$

Ankur S

Numerade Educator

Problem 54

In the figure, what is the smallest incident angle $\theta_{i}$ for the beam of a particular wavelength to undergo total internal reflection at the surface of the prism having an index of refraction for this wavelength of $1.5 ?$

Mkhitar Hobosyan

Mkhitar Hobosyan

Numerade Educator

Problem 55

Reflection and refraction, like all classical features of light and other electromagnetic waves, are governed by the Maxwell equations. The Maxwell equations are time-reversal invariant, which means that any solution of the equations reversed in time is also a solution.
a) Suppose some configuration of electric charge density $\rho,$ current density $\vec{j},$ electric field $\vec{E},$ and magnetic field $\vec{B}$ is a solution of the Maxwell equations. What is the corresponding time-reversed solution?
b) How, then, do "one-way mirrors" work?

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Andrew Davis

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Problem 56

Refer to Example 32.3 and use the numbers provided there. Further, assume that your eyes are at a height of $1.70 \mathrm{~m}$ above the water.
a) Calculate the time it takes for light to travel on the path from the fish to your eyes.
b) Calculate the time light would take on a straight-line path from the fish to your eyes.
c) Calculate the time light would take on a path from the fish vertically upward to the water surface and then straight to your eyes.
d) Calculate the time light would take on the straight-line path from the apparent location of the fish to your eyes.
e) What can you say about Fermat's Principle from the above numbers?

Keshav Singh

Numerade Educator

Problem 57

If you want to construct a liquid mirror of focal length $2.50 \mathrm{~m}$, with what angular velocity do you have to rotate your liquid?

Narayan Hari

Numerade Educator

Problem 58

One proposal for a space-based telescope is to put a large rotating liquid mirror on the Moon. Suppose you want to use a liquid mirror $100.0 \mathrm{~m}$ in diameter, and you want it to have a focal length of $347.5 \mathrm{~m}$. The gravitational acceleration on the Moon is $1.62 \mathrm{~m} / \mathrm{s}^{2}$
a) What angular velocity does your mirror have?
b) What is the linear speed of a point on the perimeter of the mirror?
c) How high above the center is the perimeter of the mirror?

Narayan Hari

Numerade Educator