Essential University Physics Global Edition

Richard Wolfson

Chapter 31

Images and Optical Instruments - all with Video Answers

Educators

Chapter Questions

Problem 1

Is it possible to capture both real and virtual images with a camera?

Jacob Shpiece

Jacob Shpiece

Numerade Educator

Problem 2

Are there any similarities between a magnifier and a pair of reading glasses? Are there any differences?

Chitra Gondi

Numerade Educator

Problem 3

If you're handed a converging lens, what can you do to estimate its focal length quickly?

Mayukh Banik

Numerade Educator

Problem 4

Is there any limit to the temperature you can achieve by focusing sunlight? (Hint: Think about the second law of thermodynamics.)

Mayukh Banik

Numerade Educator

Problem 5

If you placed a screen at the location of a virtual image, would the image appear on the screen? Why or why not?

Mayukh Banik

Numerade Educator

Problem 6

If you look into the bowl of a metal spoon, you see yourself upside down. Flip the spoon so you're looking at the back side, and now you're right-side up. Explain.

Mayukh Banik

Numerade Educator

Problem 7

Is the image on a movie screen real or virtual? How do you know?

Mayukh Banik

Numerade Educator

Problem 8

Why do the faces of some people look narrower through the glasses they wear? Explain.

Chitra Gondi

Numerade Educator

Problem 9

When viewed through a refracting telescope, stars appear to be surrounded by blurry, rainbow-colored fringes. What could be the explanation?

Jacob Shpiece

Numerade Educator

Problem 10

The refractive index of the human cornea is about $1.4$. If you can see clearly in air, why can't you see clearly underwater? Why do goggles help?

Mayukh Banik

Numerade Educator

Problem 11

A shoe store uses small floor-level mirrors to let customers view prospective purchases. At what angle should such a mirror be inclined so that a person standing $50 \mathrm{~cm}$ from the mirror with eyes $140 \mathrm{~cm}$ off the floor can see her feet?

Mayukh Banik

Numerade Educator

Problem 12

A candle is on the axis of a $15-\mathrm{cm}$-focal-length concave mirror, $36 \mathrm{~cm}$ from the mirror. (a) Where is its image? (b) How do the image and object sizes compare? (c) Is the image real or virtual?

Mayukh Banik

Numerade Educator

Problem 13

An object is five focal lengths from a concave mirror. (a) How do the object and image heights compare? (b) Is the image upright or inverted?

Mayukh Banik

Numerade Educator

Problem 14

A virtual image is located $40 \mathrm{~cm}$ behind a concave mirror with focal length $18 \mathrm{~cm}$. (a) Where is the object? (b) By how much is the image magnified?

Mayukh Banik

Numerade Educator

Problem 15

(a) Where on the axis of a concave mirror would you place an object to get a half-sized image? (b) Where will the image be located? (c) Will the image be real or virtual?

Chitra Gondi

Numerade Educator

Problem 16

A lightbulb is $56 \mathrm{~cm}$ from a convex lens. Its image appears on a screen $31 \mathrm{~cm}$ from the lens, on the other side. Find (a) the lens's focal length and (b) how much the image is enlarged or reduced.

Mayukh Banik

Numerade Educator

Problem 17

By what factor is the image magnified for an object $1.3$ focal lengths from a converging lens? Is the image upright or inverted?

Jacob Shpiece

Numerade Educator

Problem 18

A lens with $50-\mathrm{cm}$ focal length produces a real image the same size as the object. How far from the lens are image and object?

Mayukh Banik

Numerade Educator

Problem 19

By holding a magnifying glass $24 \mathrm{~cm}$ from your desk lamp, you can focus an image of the lamp's bulb on a wall $1.6 \mathrm{~m}$ from the lamp. What's the focal length of your magnifying glass?

Chitra Gondi

Numerade Educator

Problem 20

You're writing specifications for a new line of magnifying glasses that have double-convex lenses with equal $25-\mathrm{cm}$ curvature radii, made from glass with $n=1.54$. What do you list for the focal length?

Chitra Gondi

Numerade Educator

Problem 21

You're standing in a wading pool and your feet appear to be $33 \mathrm{~cm}$ below the surface. How deep is the pool?

Chitra Gondi

Numerade Educator

Problem 22

Looking straight down from air into a clear liquid, the bottom of the liquid appears at an apparent depth $h^{\prime}$ that's smaller than the actual depth $h$. Find an expression for $h^{\prime}$ in terms of $h$ and the liquid's refractive index $n$.

Chitra Gondi

Numerade Educator

Problem 23

A tiny insect is trapped $1.0 \mathrm{~mm}$ from the center of a spherical dewdrop $4.0 \mathrm{~mm}$ in diameter. As you look straight into the drop, what's the insect's apparent distance from the drop's surface?

Mayukh Banik

Numerade Educator

Problem 24

You're underwater, looking through a spherical air bubble (Fig. 31.35). What's its actual diameter if it appears, along your line of sight, to be $1.5 \mathrm{~cm}$ in diameter?

Mayukh Banik

Numerade Educator

Problem 25

You have to hold a book $40 \mathrm{~cm}$ from your eyes for the print to be in focus. What power lens would correct your farsightedness?

Chitra Gondi

Numerade Educator

Problem 26

What focal length should you specify if you want a magnifying glass with angular magnification 3.2?

Mayukh Banik

Numerade Educator

Problem 27

You're an optometrist helping a nearsighted patient who claims he can't see clearly beyond $84 \mathrm{~cm}$. Prescribe a lens that will put the images of distant objects at $84 \mathrm{~cm}$, giving your patient clear vision at all distances beyond the normal near point.

Chitra Gondi

Numerade Educator

Problem 28

A particular eye has a focal length of $2.0 \mathrm{~cm}$ instead of the $2.2 \mathrm{~cm}$ that would put a sharply focused image on the retina. (a) Is this eye nearsighted or farsighted? (b) Specify the diopter measure of the corrective lens that's needed.

Mayukh Banik

Numerade Educator

Problem 29

A compound microscope has objective and eyepiece focal lengths of $5.9 \mathrm{~mm}$ and $1.9 \mathrm{~cm}$, respectively. If the lenses are $9.7 \mathrm{~cm}$ apart, what is the instrument's magnification?

Chitra Gondi

Numerade Educator

Problem 30

Another of the technicians in Fig. $31.11$ is standing $2.46 \mathrm{~m}$ in front of the mirror. (a) Find the location of the technician's image, (b) determine its magnification, and (c) tell whether it's real or virtual, upright or inverted.

Julie Farhm

Numerade Educator

Problem 31

Another technician in Fig. 31.11 looks in the mirror and sees an upright image of herself that's $4.83$ times her actual height. (a) How far in front of the mirror is the technician located? (b) How far from the mirror is her image and (c) is it in front of or behind the mirror?

Julie Farhm

Numerade Educator

Problem 32

You're helping with a regular resurfacing of the primary mirror on one of the two Keck telescopes. The $10-\mathrm{m}-$ diameter mirror has focal length $17.5 \mathrm{~m}$. You're looking at the mirror from $38.6 \mathrm{~m}$ in front of it. (a) Find the location of your image, (b) determine its magnification, and (c) tell whether it's real or virtual, upright or inverted.

Julie Farhm

Numerade Educator

Problem 33

Standing $64.3 \mathrm{~m}$ in front of the primary mirror of the Thirty Meter Telescope (TMT), you see a real, inverted image of yourself that's $8.00 \mathrm{~m}$ closer to the mirror than you are. Find (a) the TMT primary mirror's focal length and (b) the size of the image relative to your actual size.

Julie Farhm

Numerade Educator

Problem 34

(a) Rework Example $31.4$ to find the apparent distance to a fish that's located $12.0 \mathrm{~cm}$ from the far wall of the aquarium. Assume the cat is looking in on a line that includes the fish and the center of the cylindrical aquarium. (b) In Example $31.4$, the fish appeared closer than its actual distance. Is that the case here?

Julie Farhm

Numerade Educator

Problem 35

You're looking into a glass cylinder with refractive index $1.55$. A spot of dust on the far side appears to be $18.2 \mathrm{~cm}$ from the front edge. What's the cylinder's actual diameter?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 36

You're a glaciologist researching the impact of climate change. On the surface of a glacier you're studying the wind has eroded a groove that forms a half cylinder $86.2 \mathrm{~cm}$ in diameter. Looking into this groove you see an air bubble that appears to be at a depth of $75.0 \mathrm{~cm}$ inside the ice. How far inside the ice is the bubble actually located? Hint: Consult Table $30.1$ for the refractive index of ice.

Julie Farhm

Numerade Educator

Problem 37

You're visiting an aquarium that's the inverse of the situation in Example $31.4$ : Aquarium visitors occupy an air-filled cylindrical viewing room $6.00 \mathrm{~m}$ in diameter surrounded by the water of the aquarium, separated by a layer of strong glass that's thin enough for the refractive effect on light passing through the glass to be negligible. You're standing $1.25 \mathrm{~m}$ from the glass. A fish swims up to the glass and looks straight out at you. To the fish, how far away do you appear to be?

Julie Farhm

Numerade Educator

Problem 38

(a) Find the focal length of a concave mirror if an object placed $38.4 \mathrm{~cm}$ in front of the mirror has a real image $55.7 \mathrm{~cm}$ from the mirror. (b) Where and what type will the image be if the object is moved to a point $16.0 \mathrm{~cm}$ from the mirror?

Mayukh Banik

Numerade Educator

Problem 39

A 14-mm-high object is $11 \mathrm{~cm}$ from a concave mirror with focal length $16 \mathrm{~cm}$. (a) Where is the image, (b) how high is it, and (c) what type is it?

Chitra Gondi

Numerade Educator

Problem 40

Repeat Problem 31 for a convex mirror, assuming all numbers stay the same.

Mayukh Banik

Numerade Educator

Problem 41

An object's image in a $25-\mathrm{cm}$-focal-length concave mirror is upright and magnified by a factor of $2.2$. Where is the object?

Chitra Gondi

Numerade Educator

Problem 42

You're asked to design a concave mirror that will produce a virtual image, enlarged $1.9$ times, of an object $19 \mathrm{~cm}$ from the mirror. What do you specify for the mirror's curvature radius?

Chitra Gondi

Numerade Educator

Problem 43

Viewed from Earth, the Moon subtends an angle of $0.52^{\circ}$ in the sky. What will be the physical size of the Moon's image formed by the primary mirror of the European Extremely Large Telescope, whose focal length is $34.5 \mathrm{~m}$ ? Hint: For objects at astronomical distances, $1 / s$ is essentially zero.

Kristela Garcia

Kristela Garcia

Numerade Educator

Problem 44

At what two distances from a 40 -cm-focal-length concave mirror could you place an object to get an image $1.5$ times the object's size?

Julie Farhm

Numerade Educator

Problem 45

How far from a page should you hold a lens with $29-\mathrm{cm}$ focal length in order to see the print magnified $2.2$ times?

Julie Farhm

Numerade Educator

Problem 46

A converging lens has focal length $4.8 \mathrm{~cm} .$ A $1.5-\mathrm{cm}$-high arrow is located $11 \mathrm{~cm}$ from the lens with its lowest point $4.0 \mathrm{~mm}$ above the lens axis. Make a full-scale ray-tracing diagram to locate both ends of the image. Confirm using the lens equation.

Julie Farhm

Numerade Educator

Problem 47

A lens has focal length $f=40 \mathrm{~cm}$. Find the type and height of the image produced when a $2.6$-cm-high object is placed at distances (a) $f+10 \mathrm{~cm}$ and (b) $f-10 \mathrm{~cm}$.

Mayukh Banik

Numerade Educator

Problem 48

How far apart are the object and image produced by a converging lens with $35.5-\mathrm{cm}$ focal length when the object is (a) $44 \mathrm{~cm}$ and (b) $25 \mathrm{~cm}$ from the lens?

Julie Farhm

Numerade Educator

Problem 49

A candle and a screen are $73 \mathrm{~cm}$ apart. Find two points between candle and screen where you could put a convex lens with $17-\mathrm{cm}$ focal length to give a sharp image of the candle on the screen.

Julie Farhm

Numerade Educator

Problem 50

The cornea of the human eye has refractive index $1.38$, while the eye's lens has a graduated index in the range $1.38$ to $1.40$; use $1.39$ for this problem. For the aqueous humor between cornea and lens, $n=1.34$. Find the angle through which light is deflected at the first surface of (a) the cornea and (b) the lens, if it's incident at $20^{\circ}$ to the normal at each surface. Your result shows that the cornea is the dominant refractive element in the eye.

Julie Farhm

Numerade Educator

Problem 51

How far from a $23-\mathrm{cm}$-focal-length lens should you place an object to get an upright image magnified $2.5$ times?

Jacob Shpiece

Numerade Educator

Problem 52

An object and its lens-produced real image are $2.4 \mathrm{~m}$ apart. If the lens has $55-\mathrm{cm}$ focal length, what are the possible values for the object distance and magnification?

Mayukh Banik

Numerade Educator

Problem 53

An object is $70 \mathrm{~cm}$ from a plano-convex lens whose curved side has curvature radius $29.5 \mathrm{~cm}$. The refractive index of the lens is 1.63. Where is the image, and what type is it?

Julie Farhm

Numerade Educator

Problem 54

Use Equation $31.6$ to show that an object at the center of a glass sphere will appear to be its actual distance-one radius-from the edge. Draw a ray diagram showing why this makes sense.

Mayukh Banik

Numerade Educator

Problem 55

Consider a generalization of Example $31.4$ : There's a cylinder of diameter $D$ made from a transparent material with refractive index $n$. A small object is embedded in the cylinder, a distance $b D$ from the front edge, where $b$ is a number between 0 and 1 with $b=0$ being just inside the front edge, $b=\frac{1}{2}$ being at the center, and $b=1$ being just inside the back edge. The cylinder is surrounded by air with refractive index 1. (a) Find an expression for the apparent distance to the object from the front edge, as seen by an observer looking in the front edge. (b) What does your result say about an object located at the center of the cylinder?

Robert Leedy

Numerade Educator

Problem 56

A magnifier for reading is in the form of a glass half cylinder of radius $R$ and refractive index $n$. To use it, you place the flat side on the page you're trying to read, and several lines of text appear magnified. Find an expression for the magnification this device provides. Hint: With the object (the text) right at the surface of the magnifier, you can't treat the half cylinder as a thin lens. Instead, consult Fig. $31.36$, which shows an arrow as a small object being magnified. Assume the arrow's length $h$ is small enough that the horizontal distance from the arrowhead to the curved surface in Fig. $31.36$ is approximately the radius $R$, and that the angles $\alpha$ and $\gamma$ are therefore small enough that you can use the small-angle approxi sines and tangents.

Daniel Alva

Numerade Educator

Problem 57

Two specks of dirt are trapped in a crystal ball, one at the center and the other halfway to the surface. If you peer into the ball on a line joining the two specks, the outer one appears to be only onethird of the way to the other. Find the refractive index of the ball.

Julie Farhm

Numerade Educator

Problem 58

A contact lens is in the shape of a convex meniscus (see Fig. 31.25). The inner surface is curved to fit the eye, with curvature radius $7.80 \mathrm{~mm}$. The lens is made from plastic with refractive index $n=1.56$. If it has a $44.4-\mathrm{cm}$ focal length, what's the curvature radius of its outer surface?

Mayukh Banik

Numerade Educator

Problem 59

For what refractive index would the focal length of a planoconvex lens be equal to the curvature radius of its one curved surface?

Chitra Gondi

Numerade Educator

Problem 60

An object is $28 \mathrm{~cm}$ from a double-convex lens with $n=1.5$ and curvature radii $35 \mathrm{~cm}$ and $55 \mathrm{~cm}$. Where is the image, and what type is it?

Mayukh Banik

Numerade Educator

Problem 61

You're an optician who's been asked to design a new replacement lens for cataract patients. The lens must be $5.5 \mathrm{~mm}$ in diameter, with focal length $17 \mathrm{~mm}$, and it can't be thicker than $0.8 \mathrm{~mm}$. For the lens material, you have a choice of plastic with refractive index $1.49$ or more expensive silicone with $n=1.58$. Which material do you choose, and why?

Mayukh Banik

Numerade Educator

Problem 62

A double-convex lens with equal $28.5-\mathrm{cm}$ curvature radii is made from glass with refractive indices $n_{\text {red }}=1.512$ and $n_{\text {violet }}=1.547$. If a point source of white light is located on the lens axis at $75.0 \mathrm{~cm}$ from the lens, over what distance will its visible image be smeared?

Mayukh Banik

Numerade Educator

Problem 63

An object placed $17.5 \mathrm{~cm}$ from a convex lens of glass with $n=1.524$ forms a virtual image twice the object's size. If the lens is replaced with an identically shaped one made of diamond, (a) what type of image will appear and (b) what will be its magnification?

Mayukh Banik

Numerade Educator

Problem 64

You're taking a photography class, working with a camera whose zoom lens covers the focal length range $40 \mathrm{~mm} 113 \mathrm{~mm}$. Your in structor asks you to compare the sizes of the images of a distant object when photographed at the two zoom extremes. Your answer?

Mayukh Banik

Numerade Educator

Problem 65

A camera can normally focus as close as $65 \mathrm{~cm}$, but it has provisions for mounting additional lenses just in front of the main lens to provide close-up capability. What type and power of auxiliary lens will allow the camera to focus as close as $22 \mathrm{~cm}$ ?

Mayukh Banik

Numerade Educator

Problem 66

A 261-power compound microscope has a 3.8-mm-focal-length objective lens. If the distance from objective to eyepiece is $9.3 \mathrm{~cm}$, what should be the focal length of the eyepiece?

Mayukh Banik

Numerade Educator

Problem 67

To the unaided eye, Jupiter has an angular diameter of 50 arcseconds. What will its angular size be when viewed through a $1.25-\mathrm{m}$-focal-length refracting telescope with a 44-mm-focal-length eyepiece?

Julie Farhm

Numerade Educator

Problem 68

A Cassegrain telescope like that shown in Fig. $31.34 b$ has a primary mirror with $1.0$ - $\mathrm{m}$ focal length, and its convex secondary mirror is located $0.85 \mathrm{~m}$ from the primary. What should be the focal length of the secondary in order to put the final image $0.12 \mathrm{~m}$ behind the front surface of the primary mirror?

Mayukh Banik

Numerade Educator

Problem 69

You stand with your nose $8.0 \mathrm{~cm}$ from the surface of a reflecting ball, and your nose's image appears three-quarters full size. What's the ball's diameter?

Julie Farhm

Numerade Educator

Problem 70

A contact lens prescription calls for $+2.25$-diopter lenses with inner curvature radius $8.6 \mathrm{~mm}$ to fit the patient's cornea. (a) If the lenses are plastic with $n=1.56$, what should be the outer curvature radius? (b) With these lenses, the patient comfortably reads a newspaper $30 \mathrm{~cm}$ from her eyes. Where's the image as viewed through the lenses?

Mayukh Banik

Numerade Educator

Problem 71

Show that placing a 1-diopter lens in front of a 2-diopter lens gives the equivalent of a single 3 -diopter lens (i.e., the powers of closely spaced lenses add).

Mayukh Banik

Numerade Educator

Problem 72

Derive an expression for the thickness $t$ of a plano-convex lens with diameter $d$, focal length $f$, and refractive index $n$.

Mayukh Banik

Numerade Educator

Problem 73

Show that identical objects placed equal distances on either side of the focal point of a concave mirror or converging lens produce images of equal size. Are the images of the same type?

Julie Farhm

Numerade Educator

Problem 74

Generalize the derivation of the lensmaker's formula (Equation $31.7$ ) to show that a lens of refractive index $n_{\text {lens }}$ in an external medium with index $n_{\text {ext }}$ has focal length given by
$$
\frac{1}{f}=\left(\frac{n_{\text {lens }}}{n_{\text {ext }}}-1\right)\left(\frac{1}{R_{1}}-\frac{1}{R_{2}}\right)
$$

Mayukh Banik

Numerade Educator

Problem 75

Draw a diagram like Fig. $31.10$, but showing a ray from the arrowhead through the center of curvature. Using the fact that this ray reflects back on itself, draw similar triangles with object and image as their vertical sides, and show that the center of curvature is twice as far from the mirror as the focal point-that is, $R=2 f$, with $R$ the curvature radius.

Mayukh Banik

Numerade Educator

Problem 76

Galileo's first telescope used the arrangement shown in Fig. $31.37$, with a double-concave eyepiece slightly before the focus of the objective lens. Use ray tracing to show that this design gives an upright image, which makes the Galilean telescope useful in terrestrial observing.

Mayukh Banik

Numerade Educator

Problem 77

The maximum magnification of a simple magnifier occurs with the image at the $25-\mathrm{cm}$ near point. Show that the angular magnification is $m=1+(25 \mathrm{~cm} / f)$, where $f$ is the focal length.

Mayukh Banik

Numerade Educator

Problem 78

Chromatic aberration results from variation of the refractive index with wavelength. Starting with the lensmaker's formula, find an expression for the fractional change $d f / f$ in the focal length of a thin lens in terms of the change $d n$ in refractive index.

Mayukh Banik

Numerade Educator

Problem 79

For visible wavelengths, the refractive index of the polycarbonate plastic widely used in eyeglasses is given approximately by $n(\lambda)=b+c / \lambda^{2}$, where $b=1.55$ and $c=11,500 \mathrm{~nm}^{2}$. (a) Find an expression for the change in refractive index $d n$ corresponding to a small wavelength change $d \lambda$. (b) Use the results of part (a) and of Problem 78 to determine the variation $d f$ in focal length for a $+2.25$-diopter polycarbonate lens over a wavelength range of $10.0 \mathrm{~nm}$ centered on $589 \mathrm{~nm}$.

Mayukh Banik

Numerade Educator

Problem 80

The table below shows measurements of magnification versus object distance for a lens. Determine a quantity that, when you plot object distance against it, should give a straight line. Make your plot, establish a best-fit line, and use your line to find the focal length of the lens.
$$
\begin{array}{|l|c|c|c|c|c|}
\hline \text { Object distance, } s(\mathrm{~cm}) & 10.1 & 29.2 & 51.6 & 78.3 & 98.9 \\
\hline \text { Magnification, } M & 1.31 & 4.77 & -4.38 & -1.27 & -0.724 \\
\hline
\end{array}
$$

Jacob Shpiece

Numerade Educator

Problem 81

Zooming your camera's lens for telephoto shots increases the focal length. With no change in the lens area, this will
a. increase the f-ratio and increase the lens speed.
b. decrease the f-ratio and decrease the lens speed.
c. increase the f-ratio and decrease the lens speed.
d. not change the f-ratio or the lens speed.

Mayukh Banik

Numerade Educator

Problem 82

Increasing the f-ratio from $2.8$ to $5.6$
a. decreases the light admitted by a factor of 2 .
b. decreases the light admitted by a factor of $4 .$
c. increases the light admitted by a factor of 2 .
d. increases the light admitted by a factor of $4 .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 83

You're given two lenses with different diameters. Knowing nothing else, you can conclude that
a. the larger lens is faster.
b. the smaller lens has the shorter focal length.
c. the smaller lens suffers less spherical aberration.
d. none of the above

Mahendra Kumar

Numerade Educator

Problem 84

If a lens suffers from spherical aberration, stopping down will
a. worsen the focus.
b. improve the focus.
c. not affect the focus.

Mahendra Kumar

Numerade Educator