University Physics with Modern Physics

Wolfgang Bauer, Gary D. Westfall

Chapter 32

Geometric Optics - all with Video Answers

Educators

Chapter Questions

Problem 1

Legend says that Archimedes set the Roman fleet on fire as it was invading Syracuse. Archimedes created a huge _____ mirror and used it to focus the Sun's rays on the Roman vessels.
a) plane
b) parabolic diverging
c) parabolic converging

Ajay Singhal

Ajay Singhal

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) There is no condition under which this can occur.
d) You observe the pool at an angle of $48.2^{\circ}$.

Ajay Singhal

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 $D$ away from you. To what distance should you set the range of focus on the camera?
a) $D$
b) $2 D$
c) $D / 2$
d) $4 D$

Ajay Singhal

Numerade Educator

Problem 6

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

Ajay Singhal

Numerade Educator

Problem 7

A small object is placed in front of a converging mirror with radius $R=7.50 \mathrm{~cm}$ so that the image distance equals the object distance. How far is this object from the mirror?
a) $2.50 \mathrm{~cm}$
b) $5.00 \mathrm{~cm}$
c) $7.50 \mathrm{~cm}$
d) $10.0 \mathrm{~cm}$
e) $15.0 \mathrm{~cm}$

Narayan Hari

Numerade Educator

Problem 8

Sunlight strikes a piece of glass at an angle of incidence of $\theta_{i}=33.4^{\circ} .$ What is the difference between the angle of refraction of a red light ray $(\lambda=660.0 \mathrm{nm})$ and that of a violet light ray $(\lambda=410.0 \mathrm{nm}) ?$ The glass's index of refraction is $n=1.520$ for red light and $n=1.538$ for violet light.
a) $0.03^{\circ}$
b) $0.12^{\circ}$
c) $0.19^{\circ}$
d) $0.26^{\circ}$
e) $0.82^{\circ}$

Melissa Walsh

Numerade Educator

Problem 9

The image of yourself that you see in a bathroom mirror is
a) left-right inverted.
b) front-to-back inverted.
c) bottom-to-top inverted.
d) not inverted.

Ajay Singhal

Numerade Educator

Problem 10

Where do you have to place an object in front of a concave mirror with focal length $f$ for the image to be the same size as the object?
a) at $d_{\mathrm{o}}=0.5 f$
b) at $d_{\mathrm{o}}=f$
c) at $d_{\mathrm{o}}=2 f$
d) at $d_{\mathrm{o}}=2.5 f$
e) none of the above

Narayan Hari

Numerade Educator

Problem 11

You are looking down into a swimming pool at an angle of $20^{\circ}$ relative to the vertical, and you see a coin at the bottom of the pool. This coin appears to you to be at
a) a lesser depth than it really is.
b) the same depth as it really is.
c) a greater depth than it really is.

Ajay Singhal

Numerade Educator

Problem 12

You are looking straight down into a swimming pool, that is, at an angle of $0^{\circ}$ relative to the vertical, and you see a coin at the bottom of the pool. This coin appears to you to be at
a) a lesser depth than it really is.
b) the same depth as it really is.
c) a greater depth than it really is.

Ajay Singhal

Numerade Educator

Problem 13

The figure shows the difference between the refractive index profile of a step index fiber and the refractive index profile of a graded index fiber. Applying geometric optics principles, comment on the path followed by a light ray entering each of the optic fibers.

Mkhitar Hobosyan

Mkhitar Hobosyan

Numerade Educator

Problem 14

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, $\mathrm{A}$ and $\mathrm{B}$, that originate at the letter "t" in the text under the Plexiglas and travel toward an observer who is above the Plexiglas, looking down. Draw on the figure the apparent $y$ -position of the letter under the Plexiglas as seen by the observer. (Hint: From where in the Plexiglas do the two rays appear to originate for 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 15

A concave spherical mirror is used to create an image of an object $5.00 \mathrm{~cm}$ tall that is located at position $x=0 \mathrm{~cm},$ which is $20.0 \mathrm{~cm}$ from point $C,$ the center of curvature of the mirror, as shown in the figure. 32.15 A concave spherical mirror is used to create an image of an object $5.00 \mathrm{~cm}$ tall that is located at position $x=0 \mathrm{~cm},$ which is $20.0 \mathrm{~cm}$ from point $C,$ the center of curvature of the mirror, as shown in the figure.

Ankur S

Numerade Educator

Problem 16

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 index of refraction of water is $1.33 .$
b) Would the pool look more or less deep than it actually is if you looked at it from an angle other than vertical? Answer this qualitatively, without using an equation.

Keshav Singh

Numerade Educator

Problem 17

Why does refraction occur? That is, what is the physical reason a wave moves with a different velocity when it passes from one medium into another?

Ajay Singhal

Numerade Educator

Problem 18

Many optical fibers have minimum specified bending radii. Why?

Ajay Singhal

Numerade Educator

Problem 19

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 the student can just 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?

Ajay Singhal

Numerade Educator

Problem 20

State whether the following is true or false and explain your answer: The wavelength of He-Ne laser light in water is less than its wavelength in air. (The index of refraction of water is $1.33 .)$

Narayan Hari

Numerade Educator

Problem 21

Among the instruments Apollo astronauts left on the Moon were reflectors for bouncing 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 in $384,000 \mathrm{~km}$ ), for the study of both celestial mechanics and Earth's plate tectonics. The reflectors consist not of ordinary mirrors, but of arrays of corner cubes, each constructed of three square plane mirrors fixed perpendicular to each other, as adjacent faces of a cube. Why? Explain the functioning and advantages of this design.

Ajay Singhal

Numerade Educator

Problem 22

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 index of refraction $n_{\text {glass }}=1.520$ and 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 the 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?

Ajay Singhal

Numerade Educator

Problem 23

An object is imaged by a converging spherical mirror as shown in the figure. Suppose a black cloth is put between the object and the mirror so that it covers everything above the optic axis. How will the image be affected?

Ajay Singhal

Numerade Educator

Problem 24

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

Ajay Singhal

Numerade Educator

Problem 25

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 26

A solar furnace uses a large parabolic mirror (such mirrors can be several stories high) 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 photosphere of the Sun) in a solar furnace? If so, how? If not, why not?

Ajay Singhal

Numerade Educator

Problem 27

A person sits $1.00 \mathrm{~m}$ in front of a plane mirror. What is the location of the image?

Ajay Singhal

Numerade Educator

Problem 28

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

Ajay Singhal

Numerade Educator

Problem 29

A person stands at a point $P$ relative to two plane mirrors oriented at $90.0^{\circ}$, as shown in the figure. How far away from each other do the person's images appear to be?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 30

Even the best mirrors absorb or transmit some of the light incident on them. The highest-quality mirrors might reflect $99.997 \%$ of the incident light. 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 was switched off.

Narayan Hari

Numerade Educator

Problem 31

The radius of curvature of a convex mirror is $-25.0 \mathrm{~cm}$. What is its focal length?

Narayan Hari

Numerade Educator

Problem 32

A concave spherical mirror is used to create an image of an object $5.00 \mathrm{~cm}$ tall that is located at position $x=0 \mathrm{~cm},$ which is $20.0 \mathrm{~cm}$ from point $C$, the center of curvature of the mirror, as shown in the figure. The magnitude of the radius of curvature of 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 $h_{\mathrm{i}}$ of the image? Is the image upright (pointing up) or inverted (pointing down)? Is it real or virtual?

Ajay Singhal

Numerade Educator

Problem 33

Convex mirrors are often used as sideview mirrors on cars. Many such mirrors display the warning "Objects in mirror are closer than they appear.” Assume that a convex sideview mirror has a radius of curvature of $14.0 \mathrm{~m}$ and that a car 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 the magnification for this convex mirror.

Narayan Hari

Numerade Educator

Problem 34

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

Narayan Hari

Numerade Educator

Problem 35

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 36

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 produced by them in combination?

Ajay Singhal

Numerade Educator

Problem 37

The shape of an elliptical mirror is described by the curve $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,$ with semimajor axis $a$ and semiminor axis $b .$ The foci of this ellipse are at the 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 that passes through one focus is reflected through the other. "Whispering galleries" make use of this phenomenon for reflecting sound waves.

Keshav Singh

Numerade Educator

Problem 38

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

Narayan Hari

Numerade Educator

Problem 39

An optical fiber with an index of refraction of 1.50 is used to transport light of wavelength $400 . \mathrm{nm}$. What is the critical angle for transporting light through this fiber without loss? If the fiber is immersed in water? In oil?

Narayan Hari

Numerade Educator

Problem 40

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 will each of the following characteristics be?
a) speed
b) frequency
c) wavelength
d) color

Narayan Hari

Numerade Educator

Problem 41

A light ray is incident from water, whose index of refraction is $1.33,$ on a plate of glass whose index of refraction is $1.73 .$ What angle of incidence will result in fully polarized reflected light?

Narayan Hari

Numerade Educator

Problem 42

Suppose you are standing at the bottom of a swimming pool whose surface is perfectly calm. Looking up, you see a circular window to the "outer world." If your eyes are approximately $2.00 \mathrm{~m}$ beneath the surface, what is the diameter of this circular window?

Narayan Hari

Numerade Educator

Problem 43

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

Mkhitar Hobosyan

Mkhitar Hobosyan

Numerade Educator

Problem 44

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

Narayan Hari

Numerade Educator

Problem 45

In a step index fiber, the index of refraction undergoes a discontinuity (jump) at the core-cladding boundary, as shown in the figure. Infrared light with wavelength $1550 \mathrm{nm}$ propagates through such a fiber by total internal reflection at the corecladding boundary. The index of refraction of the core for the infrared light is $n_{\text {core }}=1.48$. If the maximum angle, $\alpha_{\max },$ at which light can enter the fiber with no light lost 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.

Ajay Singhal

Numerade Educator

Problem 46

Refer to Figure 32.51 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 47

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

Ajay Singhal

Numerade Educator

Problem 48

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

Ajay Singhal

Numerade Educator

Problem 49

Suppose your height is $2.00 \mathrm{~m}$ and you are standing $50.0 \mathrm{~cm}$ 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?

Ajay Singhal

Numerade Educator

Problem 50

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 51

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 image size you want to double so you can see it better. At what locations could you put the object? Where will the images be for these object locations, and will they be real or virtual?

Ajay Singhal

Numerade Educator

Problem 52

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 53

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

Narayan Hari

Numerade Educator

Problem 54

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 55

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

Narayan Hari

Numerade Educator

Problem 56

What kind of image, virtual or real, is 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 of the distance to the center of
curvature, and
c) closer than half of the distance to the center of curvature?

Ajay Singhal

Numerade Educator

Problem 57

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

Ajay Singhal

Numerade Educator

Problem 58

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

Narayan Hari

Numerade Educator

Problem 59

How deep does a point in the middle of a 3.00 -m-deep pool appear to be to a person standing beside the pool at a position $2.00 \mathrm{~m}$ horizontally from the point? Take the index of refraction for the liquid in the pool to be 1.30 and that for air to be $1.00 .$

Ajay Singhal

Numerade Educator

Problem 60

What is the smallest incident angle $\theta_{i}$ at which the light beam shown in the figure will undergo total internal reflection in the prism, if the index of refraction of the prism for light of this wavelength is $1.500 ?$

Narayan Hari

Numerade Educator

Problem 61

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 that is 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?

AD

Andrew Davis

Numerade Educator

Problem 62

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 results of parts (a) through (d)?

Keshav Singh

Numerade Educator

Problem 63

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 64

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 that is $100.0 \mathrm{~m}$ in diameter and has 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

Problem 65

The origin of a coordinate system is placed at the center of curvature of a spherical mirror with radius of curvature $R=58.1 \mathrm{~cm}$ (see the figure). An object is placed at $x_{\mathrm{o}}=19.7 \mathrm{~cm} .$ What is the $x$ -coordinate of the image?

Narayan Hari

Numerade Educator

Problem 66

The origin of a coordinate system is placed at the center of curvature of a spherical mirror with radius of curvature $R=59.3 \mathrm{~cm}$ (see the figure). An object is placed at $x_{\mathrm{o}}=39.5 \mathrm{~cm} .$ What is the mirror's magnification?

Narayan Hari

Numerade Educator

Problem 67

The origin of a coordinate system is placed at the center of a spherical mirror with radius of curvature $R$ (see the figure). An object is placed at $x_{\mathrm{o}}=10.1 \mathrm{~cm} .$ The image is formed at $x=-7.405 \mathrm{~cm} .$ What is the value of $R ?$

Narayan Hari

Numerade Educator

Problem 68

The origin of a coordinate system is placed at the center of curvature of a spherical mirror with radius of curvature $R$ (see the figure). An object is placed at $x_{\mathrm{o}}=10.3 \mathrm{~cm} .$ The image is formed at $x=-7.548 \mathrm{~cm} .$ What is the focal length of the mirror?

Narayan Hari

Numerade Educator

Problem 69

A layer of methyl alcohol, with index of refraction $1.329,$ rests on a block of ice, with index of refraction $1.310 .$ A ray of light passes through the methyl alcohol at an angle of $\varphi_{1}=61.07^{\circ}$ relative to the alcohol-ice boundary. What is the angle $\varphi_{2}$ relative to the boundary at which the ray passes through the ice?

Narayan Hari

Numerade Educator

Problem 70

A layer of a transparent material rests on a block of fused quartz, whose index of refraction is $1.460 .$ A ray of light passes through the unknown material at an angle of $\varphi_{1}=63.65^{\circ}$ relative to the boundary between the materials and is refracted at an angle of $\varphi_{2}=70.26^{\circ}$ relative to the boundary. What is the index of refraction of the unknown material?

Narayan Hari

Numerade Educator

Problem 71

A layer of carbon dioxide, with index of refraction 1.00045 , rests on a block of ice, with index of refraction $1.310 .$ A ray of light passes through the carbon dioxide at an angle of $\varphi_{1}$ relative to the boundary between the materials and then passes through the ice at an angle of $\varphi_{2}=72.06^{\circ}$ relative to the boundary. What is the value of $\varphi_{1}$ ?

Narayan Hari

Numerade Educator

Problem 72

A layer of water, with index of refraction 1.333 , rests on a block of an unknown transparent material. A ray of light passes through the water at an angle of $\varphi_{1}=68.77^{\circ}$ relative to the boundary between the materials and then passes through the unknown material at an angle of $\varphi_{2}=72.98^{\circ}$ relative to the boundary. What is the speed of light in the unknown material?

Narayan Hari

Numerade Educator