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University Physics with Modern Physics

Hugh D. Young, Roger A. Freeman

Chapter 34

Geometric Optics and Optical Instruments - all with Video Answers

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Chapter Questions

01:25

Problem 1

A candle 4.85 $\mathrm{cm}$ tall is 39.2 $\mathrm{cm}$ to the left of a plane mirror. Where is the image formed by the mirror, and what is the height of this image?

Jason Bane
Jason Bane
Numerade Educator
09:08

Problem 2

The image of a tree just covers the length of a plane mirror 4.00 $\mathrm{cm}$ tall when the mirror is held 35.0 $\mathrm{cm}$ from the eye. The tree is 28.0 $\mathrm{m}$ from the mirror. What is its height?

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
02:56

Problem 3

As shown in Fig. $34.9,$ mirror 1 uses the image $P_{2}^{\prime}$ formed by mirror 2 as an object and forms an image of it. Show that this image is at point $P \}$ in the figure.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
01:10

Problem 4

A concave mirror has a radius of curvature of $34.0 \mathrm{cm} .$ (a) What is its focal length? (b) If the mirror is immersed in water (refractive index $1.33 ),$ what is its focal length?

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
09:09

Problem 5

An object 0.600 $\mathrm{cm}$ tall is placed 16.5 $\mathrm{cm}$ to the left of the vertex of a concave spherical mirror having a radius of curvature of $22.0 \mathrm{cm} .$ (a) Draw a principal-ray diagram showing the formation of the image. (b) Determine the position, size, orientation, and nature (real or virtual) of the image.

Jason Bane
Jason Bane
Numerade Educator
10:04

Problem 6

Repeat Exercise 34.5 for the case in which the mirror is convex.

Ivan Frantz
Ivan Frantz
Numerade Educator
07:41

Problem 7

The diameter of Mars is $6794 \mathrm{km},$ and its minimum distance from the earth is $5.58 \times 10^{7} \mathrm{km} .$ When Mars is at this distance, find the diameter of the image of Mars formed by a spherical, concave, telescope mirror with a focal length of 1.75 $\mathrm{m}$ .

Jason Bane
Jason Bane
Numerade Educator
03:03

Problem 8

An object is 24.0 $\mathrm{cm}$ from the center of a silvered spherical glass Christmas tree ornament 6.00 $\mathrm{cm}$ in diameter. What are the position and magnification of its image?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:40

Problem 9

A coin is placed next to the convex side of a thin spherical glass shell having a radius of curvature of 18.0 $\mathrm{cm}$ . An image of the 1.5 -tall coin is formed 6.00 $\mathrm{cm}$ behind the glass shell.
Where is the coin located? Determine the size, orientation, and nature (real or virtual) of the image.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
04:14

Problem 10

You hold a spherical salad bowl 90 $\mathrm{cm}$ in front of your face with the bottom of the bowl facing you. The salad bowl is made of polished metal with a $35-\mathrm{cm}$ radius of curvature. (a) Where is the image of your 2.0 -cm-tall nose located? (b) What are the image's size, orientation, and nature (real or virtual)?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
06:11

Problem 11

(a) Show that $\mathrm{Eq} .(34.6)$ can be written as $s^{\prime}=s f(s-f)$ and hence that the lateral magnification, given by Eq. $(34.7),$ can be expressed as $m=f(f-s) .$ (b) Use these formulas for $s^{\prime}$ and $m$ to graph $s^{\prime}$ as a function of $s$ for the case $f>0$ (a concave mirror). (c) For what values of $s$ is $s^{\prime}$ positive, so that the image is real? (d) For what values of $s$ is $s$ ' negative, so that the image is virtual? (e) Where is the image if the object is just inside the focal point $(s \text { slightly less than } f) ?(f)$ Where is the image if the object is at infinity? (g) Where is the image if the object is next to the mirror $(s=0) ?(\text { h) Graph } m \text { as a function of } s \text { for the case of a concave }$ mirror. (i) For which values of $s$ is the image erect and larger than the object? (j) For what values of $s$ is the image inverted? (k) For which values of $s$ is the image smaller than the object? (1) What happens to the size of the image when the object is placed at the focal point?

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
06:11

Problem 12

Using the formulas for $s^{\prime}$ and $m$ obtained in part (a) of Exercise $34.11,$ graph $s^{\prime}$ as a function of $s$ , and graph $m$ as a function of $s,$ for the case $f<0$ (a convex mirror), so that $f=-|f| .$ (a) For which values of $s$ is $s^{\prime}$ positive? (b) For what values of $s$ is (d) Where is the image is the image if the object is at infinity? (d) Where is the image if the object is next to the mirror $(s=0) ?$ For which values of $s$ is the image (e) erect; (f) inverted; (g) larger than the object; ( h) smaller than the object?

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
06:28

Problem 13

Dental Mirror. A dentist uses a curved mirror to view teeth on the upper side of the mouth. Suppose she wants an enect image with a magnification of 2.00 when the mirror is 1.25 $\mathrm{cm}$ from a tooth. (Treat this problem as though the object and image lie along a straight line.) (a) What kind of mirror (concave or convex) is needed? Use a ray diagram to decide, without performing any calculations. (b) What must be the focal length and radius of curvature of this mirror?(c) Draw a principal-ray diagram to check your answer in part (b).

Farhanul Hasan
Farhanul Hasan
Numerade Educator
07:49

Problem 14

A spherical, concave, shaving mirror has a radius of curvature of $32.0 \mathrm{cm} .$ (a) What is the magnification of a person's face when it is 12.0 $\mathrm{cm}$ to the left of the vertex of the mirror? (b) Where is the image? Is the image real or virual? (c) Draw a principal-ray diagram showing the formation of the image.

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
02:57

Problem 15

A speck of dirt is embedded 3.50 $\mathrm{cm}$ below the surface of a sheet of ice $(n=1.309) .$ What is its apparent depth when viewed at normal incidence?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
06:45

Problem 16

A tank whose bottom is a mirror is filled with water to a depth of 20.0 $\mathrm{cm}$ . A small fish floats motionless 7.0 $\mathrm{cm}$ under the surface of the water. (a) What is the apparent depth of the fish when viewed at normal incidence? (b) What is the apparent depth of the image of the fish when viewed at normal incidence?

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
05:07

Problem 17

A Spherical Fish Bowl. A small tropical fish is at the center of a water-filled, spherical fish bowl 28.0 $\mathrm{cm}$ in diameter. (a) Find the apparent position and magnification of the fish to an observer outside the bowl. The effect of the thin walls of the bowl may be ignored. (b) A friend advised the owner of the bowl to keep it out of direct sunlight to avoid blinding the fish, which might swim into the focal point of the parallel rays from the sun. Is the focal point actually within the bowl?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
View

Problem 18

The left end of a long glass rod 6.00 $\mathrm{cm}$ in diameter has a convex hemispherical surface 3.00 $\mathrm{cm}$ in radius. The refractive index of the glass is 1.60 . Determine the position of the image if an object is placed in air on the axis of the rod at the following distances to the left of the vertex of the curved end: (a) infinitely far, (b) $12.0 \mathrm{cm} ;(\mathrm{c}) 2.00 \mathrm{cm} .$

Yaqub Khan
Yaqub Khan
Numerade Educator
03:38

Problem 19

The glass rod of Exercise 34.18 is immersed in oil $(n=1.45) .$ An object placed to the left of the rod on the rod's axis is to be imaged 1.20 $\mathrm{m}$ inside the rod. How far from the left end of the rod must the object be located to form the image?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
04:42

Problem 20

The left end of a long glass rod 8.00 $\mathrm{cm}$ in diameter, with an index of refraction 1.60 , is ground and polished to a convex hemispherical surface with a radius of 4.00 $\mathrm{cm}$ . An object in the form of an arrow 1.50 $\mathrm{mm}$ tall, at right angles to the axis of the rod, is located on the axis 24.0 $\mathrm{cm}$ to the left of the vertex of the convex surface. Find the position and height of the image of the arrow formed by paraxial rays incident on the convex surface. Is the image erect or inverted?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:43

Problem 21

Repeat Exercise 34.20 for the case in which the end of the rod is ground to a concave hemispherical surface with radius $4.00 \mathrm{cm} .$

Bruce Edelman
Bruce Edelman
Numerade Educator
06:26

Problem 22

The glass rod of Exercise 34.21 is immersed in a liquid. An object 14.0 $\mathrm{cm}$ from the vertex of the left end of the rod and on its axis is imaged at a point 9.00 $\mathrm{cm}$ from the vertex inside the liquid. What is the index of refraction of the liquid?

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
04:55

Problem 23

An insect 3.75 $\mathrm{mm}$ tall is placed 22.5 $\mathrm{cm}$ to the leff of a thin planoconvex lens. The left surface of this lens is flat, the right surface has a radius of curvature of magnitude $13.0 \mathrm{cm},$ and the index of refraction of the lens material is $1.70 .$ (a) Calculate the location and size of the image this lens forms of the insect. Is it real or virtual? Erect or inverted? (b) Repcat part (a) if the lens is reversed.

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
08:34

Problem 24

A lens forms an image of an object. The object is 16.0 $\mathrm{cm}$ from the lens. The image is 12.0 $\mathrm{cm}$ from the lens on the same side as the object. (a) What is the focal length of the lens? Is the lens converging or diverging? (b) If the object is 8.50 $\mathrm{mm}$ tall, how tall is the image? Is it erect or inverted?(c) Draw a principal-ray diagram.

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
04:31

Problem 25

A converging meniscus lens (see Fig. 34.32 ) with a refractive index of 1.52 has spherical surfaces whose radii are 7.00 $\mathrm{cm}$ and $4.00 \mathrm{cm} .$ What is the position of the image if an object is placed 24.0 $\mathrm{cm}$ to the left of the lens? What is the magnification?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
01:41

Problem 26

A converging lens with a focal length of 90.0 $\mathrm{cm}$ forms an image of a 3.20 -cm-tall real object that is to the left of the lens. The image is 4.50 $\mathrm{cm}$ tall and inverted. Where are the object and image located in relation to the lens? Is the image real or virtual?

Farhanul Hasan
Farhanul Hasan
Numerade Educator
06:01

Problem 27

A converging lens forms an image of an 8.00 -mm-tall real object. The image is 12.0 $\mathrm{cm}$ to the left of the lens, 3.40 $\mathrm{cm}$ tall, and erect. What is the focal length of the lens? Where is the object located?

Meghan Miholics
Meghan Miholics
Numerade Educator
09:44

Problem 28

A photographic slide is to the left of a lens. The lens projects an image of the slide onto a wall 6.00 $\mathrm{m}$ to the right of the slide. The image is 80.0 times the size of the slide. (a) How far is the slide from the lens? (b) Is the image erect or inverted? (c) What is the focal length of the lens? (d) Is the lens converging or diverging?

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
01:34

Problem 29

A double-convex thin lens has surfaces with equal radii of curvature of magnitude $2.50 \mathrm{cm} .$ Looking through this lens, you observe that it forms an image of a very distant tree at a distance of 1.87 $\mathrm{cm}$ from the lens. What is the index of refraction of the lens?

Farhanul Hasan
Farhanul Hasan
Numerade Educator
02:25

Problem 30

Six lenses in air are shown in Fig. 34.32 . Each lens is made of a material with index of refraction $n>1 .$ Considering each lens individually, imagine that light enters the lens from the left. Show that the three lenses shown in Fig. 34.32 a have positive focal lengths and hence are converging lenses. In addition, show that the three lenses in Fig. 34.32 $\mathrm{b}$ have negative focal lengths and hence are diverging lenses.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:49

Problem 31

Exercises 34.11 and 34.12 deal with spherical mirrors. (a) Show that the equations for $s^{\prime}$ and $m$ derived in part (a) of Exercise 34.11 also apply to a thin lens. (b) A concave mirror is used in Exercise 34.11 . Repeat these exercises for a converging lens. Are there any differences in the results when the mirror is replaced by a lens? Explain. (c) A convex mirror is used in Exercise 34.12 . Repeat these exercises for a diverging lens. Are there any differences in the results when the mirror is replaced by a lens? Explain.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
09:32

Problem 32

A converging lens with a focal length of 12.0 $\mathrm{cm}$ forms a virtual image 8.00 $\mathrm{mm}$ tall, 17.0 $\mathrm{cm}$ to the right of the lens. Determine the position and size of the object. Is the image erect or inverted? Are the object and image on the same side or opposite sides of the lens? Draw a principal-ray diagram for this situation.

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
00:00

Problem 33

Repeat Exercise 34.32 for the case in which the lens is diverging, with a focal length of $-48.0 \mathrm{cm} .$

Bruce Edelman
Bruce Edelman
Numerade Educator
07:17

Problem 34

An object is 16.0 $\mathrm{cm}$ to the left of a lens. The lens forms an image 36.0 $\mathrm{cm}$ to the right of the lens. (a) What is the focal length of the lens? Is the lens converging or diverging? (b) If the object is 8.00 $\mathrm{mm}$ tall, how tall is the image? Is it erect or inverted? (c) Draw a principal-ray diagram.

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
01:27

Problem 35

A camera lens has a focal length of 200 $\mathrm{mm}$ . How far from the lens should the subject for the photo be if the lens is 20.4 $\mathrm{cm}$ from the film?

Dading Chen
Dading Chen
Numerade Educator
03:03

Problem 36

When a camera is focused, the lens is moved away from or toward the film. If you take a picture of your friend, who is standing 3.90 m from the lens, using a camera with a lens with a $85-\mathrm{mm}$ focal length, how far from the film is the lens? Will the whole image of your friend, who is 175 $\mathrm{cm}$ tall, fit on film that is $24 \times 36 \mathrm{mm} ?$

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:03

Problem 37

Figure 34.41 shows photographs of the same scene taken with the same camera with lenses of different focal length. If the object is 200 $\mathrm{m}$ from the lens, what is the magnitude of the lateral magnification for a lens of focal length (a) $28 \mathrm{mm} ;(\mathrm{b}) 105 \mathrm{mm}$ ; (c) 300 $\mathrm{mm}$ ?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
05:38

Problem 38

A photographer takes a photograph of a Boeing 747 airliner (length 70.7 $\mathrm{m} )$ when it is flying directly overbead an altitude of 9.50 $\mathrm{km}$ . The lens has a focal length of 5.00 $\mathrm{m}$ . How long is the image of the airliner on the film?

Bethany Campbell
Bethany Campbell
Numerade Educator
05:48

Problem 39

Choosing a Camera Lens. The picture size on ordinary $35-\mathrm{mm}$ camera film is $24 \mathrm{mm} \times 36 \mathrm{mm}$ . Focal lengths of lenses available for $35-\mathrm{mm}$ cameras typically include $28,35,50$ (the "normal" lens), $85,100,135,200,$ and $300 \mathrm{mm},$ among others. Which of these lenses should be used to photograph the following objects, assuming that the object is to fill most of the picture area? (a) a building 240 $\mathrm{m}$ tall and 160 $\mathrm{m}$ wide at a distance of 600 $\mathrm{m}$ , and (b) a mobile home 9.6 $\mathrm{m}$ in length at a distance of 40.0 $\mathrm{m}$ .

Sheh Lit Chang
Sheh Lit Chang
University of Washington
08:18

Problem 40

Zoom Lens. Consider the simple model of the zoom lens shown in Fig. 34.43 $\mathrm{a}$ . The converging lens has focal length $f_{1}=12 \mathrm{cm},$ and the diverging lens has focal length $f_{2}=-12 \mathrm{cm} .$ The lenses are separated by 4 $\mathrm{cm}$ as shown in Fig. 34.43 $\mathrm{a}$ . (a) For a distant object, where is the image of the converging lens? (b) The image of the converging lens serves as the object for the diverging lens. What is the object distance for the diverging lens? (c) Where is the final image? Compare your answer to Fig. 34.43 $\mathrm{a}$ (d) Repeat parts $(a),(b),$ and $(c)$ for the situation shown in Fig. $34.43 b,$ in which the lenses are separated by $8 \mathrm{cm} .$

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:35

Problem 41

A camera lens has a focal length of 180.0 $\mathrm{mm}$ and an aperture diameter of 16.36 $\mathrm{mm}$ (a) What is the $f$ -number of the lens? (b) If the correct exposure of a certain scene is $\frac{1}{30} \mathrm{s}$ at $f / 11$ , what is the correct exposure at $f / 2.8 ?$

Bruce Edelman
Bruce Edelman
Numerade Educator
02:06

Problem 42

Recall that the intensity of light reaching film in a camera is proportional to the effective area of the lens. Camera $A$ has a lens with an aperture diameter of 8.00 $\mathrm{mm}$ . It photographs an object using the correct exposure time of $\frac{1}{30} \mathrm{s}$ . What exposure time should be used with camera $B$ in photographing the same object with the same film if this camera has a lens with an aperture diameter of 23.1 $\mathrm{mm} ?$

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:02

Problem 43

Photography. $\quad$ A $35-\mathrm{mm}$ camera has a standard lens with focal length 50 $\mathrm{mm}$ and can focus on objects between 45 $\mathrm{cm}$ and infinity. (a) Is the lens for such a camera a concave or a convex lens? (b) The camera is focused by rotating the lens, which moves
it on the camera body and changes its distance from the film. In what range of distances between the lens and the film plane must the lens move to focus properly over the 45 $\mathrm{cm}$ to infinity range?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
04:15

Problem 44

You wish to project the image of a slide on a screen 9.00 $\mathrm{m}$ from the lens of a slide projector. (a) If the slide is placed 15.0 $\mathrm{cm}$ from the lens, what focal length lens is required? (b) If the dimensions of the picture on a $35-\mathrm{mm}$ color slide are $24 \mathrm{mm} \times 36 \mathrm{mm},$ what is the minimum size of the projector screen required to accommodate the image?

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
04:46

Problem 45

(a) Where is the near point of an eye for which a contact lens with a power of $+2.75$ diopters is prescribed? (b) Where is the far point of an eye for which a contact lens with a power of $-1.30$ diopters is prescribed for distant vision?

Bruce Edelman
Bruce Edelman
Numerade Educator
05:15

Problem 46

Curvature of the Cornea. In a simplified model of the human eye, the aqueous and vitreous humors and the lens all have a refractive index of $1.40,$ and all the refraction occurs at the cornea, whose vertex is 2.60 $\mathrm{cm}$ from the retina. What should be the radius of curvature of the cornea such that the image of an object 40.0 $\mathrm{cm}$ from the comea's vertex is focused on the retina?

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
04:07

Problem 47

Corrective Lenses. Determine the power of the corrective contact lenses required by (a) a hyperopic eye whose near point is at 60.0 $\mathrm{cm}$ and $(b)$ a myopic eye whose far point is at $60.0 \mathrm{cm} .$

Dading Chen
Dading Chen
Numerade Educator
02:58

Problem 48

A thin lens with a focal length of 6.00 $\mathrm{cm}$ is used as a simple magnifier. (a) What angular magnification is obtainable with the lens if the object is at the focal point? (b) When an object is examined through the lens, how close can it be brought to the lens? Assume that the image viewed by the eye is at the near point, 25.0 $\mathrm{cm}$ from the eye, and that the lens is very close to the eye.

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
03:29

Problem 49

The focal length of a simple magnifier is 8.00 $\mathrm{cm} .$ Assume the magnifier is a thin lens placed very close to the eye. (a) How far in front of the magnifier should an object be placed if the image is formed at the observer's near point, 25.0 $\mathrm{cm}$ in front of her eye? (b) If the object is 1.00 $\mathrm{mm}$ high, what is the height of its image formed by the magnifier?

WM
William Mead
Numerade Educator
01:54

Problem 50

You want to view an insect 2.00 $\mathrm{mm}$ in length through a magnifier. If the insect is to be at the focal point of the magnifier. what focal length will give the image of the insect an angular size of 0.025 radian?

Farhanul Hasan
Farhanul Hasan
Numerade Educator
01:27

Problem 51

You are examining an ant with a magnifying lens that has focal length 5.00 $\mathrm{cm}$ . If the image of the ant appears 25.0 $\mathrm{cm}$ from the lens, how far is the ant from the lens? On which side of the lens is the image located?

Farhanul Hasan
Farhanul Hasan
Numerade Educator
04:08

Problem 52

Resolution of a Microscope. The image formed by a microscope objective with a focal length of 5.00 $\mathrm{mm}$ is 160 $\mathrm{mm}$ from its second focal point. The eyepiece has a focal length of 26.0 $\mathrm{mm}$ . (a) What is the angular magnification of the microscope? (b) The unaided eye can distinguish two points at its near point as separate if they are about 0.10 $\mathrm{mm}$ apart. What is the minimum separation between two points that can be observed (or resolved) through this microscope?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
02:05

Problem 53

The focal length of the eyepiece of a certain microscope is 18.0 $\mathrm{mm}$ . The focal length of the objective is 8.00 $\mathrm{mm}$ . The distance between objective and eyepiece is $19.7 \mathrm{cm} .$ The final image formed by the eyepiece is at infinity. Treat all lenses as thin. (a) What is the distance from the objective to the object being viewed? (b) What is the magnitude of the linear magnification produced by the objective? (c) What is the overall angular magnification of the microscope?

Farhanul Hasan
Farhanul Hasan
Numerade Educator
03:42

Problem 54

A certain microscope is provided with objectives that have focal lengths of $16 \mathrm{mm}, 4 \mathrm{mm},$ and 1.9 $\mathrm{mm}$ and with eyepieces that have angular magnifications of $5 \times$ and $10 \times .$ Each objective forms an image 120 $\mathrm{mm}$ beyond its second focal point. Determine (a) the largest overall angular magnification obtainable and (b) the least overall angular magnification obtainable.

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
03:57

Problem 55

The Yerkes refracting telescope of the University of Chicago has an objective 1.02 $\mathrm{m}$ in diameter with an $f$ -number of $19.0 .$ (This is the largest-diameter refracting telescope in the world.) What is its focal length?

Dading Chen
Dading Chen
Numerade Educator
01:47

Problem 56

The eyepiece of a refracting telescope (see Fig. 34.53 ) has a focal length of 9.00 $\mathrm{cm}$ . The distance between objective and eye-piece is 1.80 $\mathrm{m}$ and the final image is at infinity. What is the angular magnification of the telescope?

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
05:42

Problem 57

A telescope is constructed from two lenses with focal lengths of 95.0 $\mathrm{cm}$ and $15.0 \mathrm{cm},$ the $95.0-\mathrm{cm}$ lens being used as the objective. Both the object being viewed and the final image are at infinity. (a) Find the angular magnification for the telescope. (b) Find the height of the image formed by the objective of a building 60.0 $\mathrm{m}$ tall, 3.00 $\mathrm{km}$ away. (c) What is the angular size of the final image as viewed by an eye very close to the eyepiece?

Keshav Singh
Keshav Singh
Numerade Educator
01:43

Problem 58

Saturn is viewed through the Lick Observatory refracting telescope (objective focal length 18 $\mathrm{m} )$ . If the diameter of the image of Satum produced by the objective is $1.7 \mathrm{mm},$ what angle does Saturn subtend from when viewed from earth?

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
02:51

Problem 59

A reflecting telescope (Fig. 34.55 a) is to be made by using a spherical mirror with a radius of curvature of 1.30 $\mathrm{m}$ and an eye-piece with a focal length of 1.10 $\mathrm{cm}$ . The final image is at infinity. (a) What should the distance between the eyepiece and the mirror vertex be if the object is taken to be at infinity? (b) What will the angular magnification be?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:34

Problem 60

A Cassegrain telescope is a refiecting telescope that uses two mirrors, the secondary mirror focusing the image through a hole in the primary mirror (similar to that shown in Fig. 34.55 ). You wish to focus the image of a distant galaxy onto the detector shown in the figure. If the primary mirror has a focal length of $2.5 \mathrm{m},$ the secondary mirror has a focal length of $-1.5 \mathrm{m}$ and the distance from the vertex of the primary mirror to the detector is 15 $\mathrm{cm}$ . What should be the distance between the vertices of the two mirrors?

Banhishikha Sinha
Banhishikha Sinha
Numerade Educator
01:55

Problem 61

If you run away from a plane mirror at 2.40 $\mathrm{m} / \mathrm{s}$ , at what speed does your image move away from you?

Farhanul Hasan
Farhanul Hasan
Numerade Educator
02:23

Problem 62

An object is placed between two plane mirrors arranged at right angles to each other at a distance $d_{1}$ from the surface of one mirror and a distance $d_{2}$ from the other. (a) How many images are formed? Show the location of the images in a diagram. (b) Draw the paths of rays from the object to the eye of an observer.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:27

Problem 63

What is the size of the smallest vertical plane mirror in which a woman of height $h$ can see her full-length image?

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
05:05

Problem 64

A light bulb is 4.00 $\mathrm{m}$ from a wall. You are to use a concave mirror to project an image of the bulb on the wall, with the image 2.25 times the size of the object. How far should the mirror be from the wall? What should its radius of curvature be?

Vishal Gupta
Vishal Gupta
Numerade Educator
06:06

Problem 65

A concave mirror is to form an image of the filament of a headlight lamp on a screen 8.00 $\mathrm{m}$ from the mirror. The filament is 6.00 $\mathrm{mm}$ tall, and the image is to be 36.0 $\mathrm{cm}$ tall. (a) How far in front of the vertex of the mirror should the filament be placed? (b) What should be the radius of curvature of the mirror?

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
03:05

Problem 66

Rear-View Mirror. A mirror on the passenger side of your car is convex and has a radius of curvature with magnitude 18.0 $\mathrm{cm}$ (a) Another car is seen in this side mirror and is 13.0 $\mathrm{m}$ behind the mirror. If this car is 1.5 $\mathrm{m}$ tall, what is the height of the image? (b) The mirror has a warning attached that objects viewed in it are closer than they appear. Why is this so?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
04:30

Problem 67

Suppose the lamp filament shown in Example 34.1 (Section 34.2) is moved to a position 8.0 $\mathrm{cm}$ in front of the mirror. (a) Where is the image located now? Is it real or virtual? (b) What is the height of the image? Is it erect or inverted? In Example 34.1 , the filament is 10.0 $\mathrm{cm}$ in front of the mirror, and an image of the filament is formed on a wall 3.00 $\mathrm{m}$ from the mirror. If the filament is 8.0 $\mathrm{cm}$ from the mirror, can a wall be placed so that an image is formed onit? If so, where should the wall be placed? If not, why not?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
02:58

Problem 68

Where must you place an object in front of a concave mirror with radius $R$ so that the image is erect and $2_{2}^{1}$ times the size of the object? Where is the image?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
20:13

Problem 69

Virtual Object. If the light incident from the left onto a convex mirror does not diverge from an object point but instead converges toward a point at a (negative) distance $s$ to the right of the mirror, this point is called a virtual object. (a) For a convex mirror having a radius of curvature of $24.0 \mathrm{cm},$ for what range of virtual-object positions is a real image formed? (b) What is the orientation of this real image? (c) Draw a principal-ray diagram showing the formation of such an image.

Matthew Kegley
Matthew Kegley
Numerade Educator
02:12

Problem 70

A layer of benzene $(n=1.50) 2.60 \mathrm{cm}$ deep floats on water $(n-1.33)$ that is 6.50 $\mathrm{cm}$ deep. What is the apparent distance from the upper benzene surface to the bottom of the water layer when it is viewed at normal incidence?

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
03:46

Problem 71

Sketch the various possible thin lenses that can be obtained by combining two surfaces whose radii of curvature are 4.00 $\mathrm{cm}$ and 8.00 $\mathrm{cm}$ in absolute magnitude. Which are converging and which are diverging? Find the focal length of each if the surfaces are made of glass with index of refraction 1.60 .

Vishal Gupta
Vishal Gupta
Numerade Educator
05:12

Problem 72

Figure 34.56 shows a small plant near a thin lens. The ray shown is one of the principal rays for the lens. Each square is 2.0 $\mathrm{cm}$ along the horizontal direction, but the vertical direction is not to
the same scale. Use information from the diagram to answer the following questions: (a) Using only the ray shown, decide what type of lens (converging or diverging) this is. (b) What is the focal length of the lens? (c) Locate the image by drawing the other two principal rays. (d) Calculate where the image should be, and compare this result with the graphical solution in part (c).

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
07:45

Problem 73

You are in your car driving on a highway at 25 $\mathrm{m} / \mathrm{s}$ when you glance in the passenger side mirror (a convex mirror with radius of curvature 150 $\mathrm{cm}$ ) and notice a truck approaching. If the image of the truck is approaching the vertex of the mirror at a speed of 1.5 $\mathrm{m} / \mathrm{s}$ when the truck is 2.0 $\mathrm{m}$ away, what is the speed of the truck relative to the highway?

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
01:34

Problem 74

A microscope is focused on the upper surface of a glass plate. A second plate is then placed over the first. To focus on the bottom surface of the second plate, the microscope must be raised 0.780 $\mathrm{mm}$ . To focus on the upper surface, it must be raised another 2.50 $\mathrm{mm}$ . Find the index of refraction of the second plate.

Narayan Hari
Narayan Hari
Numerade Educator
06:17

Problem 75

Three-Dimensional Image. The longitudinal magnification is defined as $m^{\prime}=d s^{\prime} / d s .$ It relates the longitudinal dimension of a small object to the longitudinal dimension of its image. (a) Show that for a spherical mirror, $m^{\prime}=-m^{2}$ . What is the significance of the fact that $m^{\prime}$ is always negative? (b) A wire frame in the form of a small cube 1.00 $\mathrm{mm}$ on a side is placed with its center on the axis of a concave mirror with radius of curvature 150.0 $\mathrm{cm}$ . The sides of the cube are all either parallel or perpendicular to the axis. The cube face toward the mirror is 200.0 $\mathrm{cm}$ to the left of the mirror vertex. Find (i) the location of the image of this face and of the opposite face of the cube; (ii) the lateral and longitudinal magnifications; (iii) the shape and dimensions of each of the six faces of the image.

Dominador Tan
Dominador Tan
Numerade Educator
03:58

Problem 76

Refer to Problem 34.75 . Show that the longitudinal magnification $m^{\prime}$ for refraction at a spherical surface is given by
$$m^{\prime}=-\frac{n_{b}}{n_{a}} m^{2}$$

Averell Hause
Averell Hause
Carnegie Mellon University
03:33

Problem 77

Pinhole Camera. A pinhole camera is just a rectangular box with a tiny hole in one face. The film is on the face opposite this hole, and that is where the image is formed. The camera forms an image without a lens, (a) Make a clear ray diagram to show how a pinhole camera can form an image on the film without using a lens. (Hint: Put an object outside the hole, and then draw rays passing through the hole to the opposite side of the box.) (b) A certain pinhole camera is a box that is 25 $\mathrm{cm}$ square and 20.0 $\mathrm{cm}$ deep, with the hole in the middle of one of the $25 \mathrm{cm} \times 25 \mathrm{cm}$ faces. If this camera is used to photograph a fierce chicken that is 18 $\mathrm{cm}$ high and 1.5 $\mathrm{m}$ in front of the camera, how large is the image of this bird on the film? What is the magnification of this camera?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
12:40

Problem 78

A Glass Rod. Both ends of a glass rod with index of refraction 1.60 are ground and polished to convex hemispherical surfaces. The radius of curvature at the left end is 6.00 $\mathrm{cm}$ , and the radius of curvature at the right end is 12.0 $\mathrm{cm}$ . The length of the rod between vertices is $40.0 \mathrm{cm} .$ The object for the surface at the left end is an arrow that lies 23.0 $\mathrm{cm}$ to the left of the vertex of this surface. The arrow is 1.50 $\mathrm{mm}$ tall and at right angles to the axis. (a) What constitutes the object for the surface at the right end of the rod? (b) What is the object distance for this surface? (c) Is the object for this surface real or virtual? (Hint: See Problem $34.69 . )(\mathrm{d})$ What is the position of the final image? (e) Is the final image real or virtual? Is it erect or inverted with respect to the original object? (f) What is the height of the final image?

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
10:18

Problem 79

The rod in Problem 34.78 is shortened to a distance of 25.0 $\mathrm{cm}$ between its vertices; the curvatures of its ends remain the same. As in Problem 34.78 , the object for the surface at the left end is an arrow that lies 23.0 $\mathrm{cm}$ to the left of the vertex of this surface. The arrow is 1.50 $\mathrm{mm}$ tall and at right angles to the axis. (a) What is the object distance for the surface at the right end of the rod? (b) Is the object for this surface real or virtual? (c) What is the position of the final image? (d) Is the final image real or vitual? Is it erect or inverted with respect to the original object? (e) What is the height of the final image?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
06:07

Problem 80

Figure 34.57 shows an object and its image formed by a thin lens, (a) What is the focal length of the lens, and what type of lens (converging or diverging) is it? (b) What is the height of the image? Is it real or virtual?

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
06:30

Problem 81

Figure 34.58 shows an object and its image formed by a thin lens. (a) What is the focal length of the lens, and what type of lens (converging or diverging) is it? (b) What is the height of the image? Is it real or virtual?

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
04:32

Problem 82

A transparent rod 30.0 $\mathrm{cm}$ long is cut flat at one end and rounded to a hemispherical surface of radius 10.0 $\mathrm{cm}$ at the other end. A small object is embedded within the rod along its axis and halfway between its ends, 15.0 $\mathrm{cm}$ from the flat end and 15.0 $\mathrm{cm}$ from the vertex of the curved end. When viewed from the flat end of the rod, the apparent depth of the object is 9.50 $\mathrm{cm}$ from the flat end. What is its apparent depth when viewed from the curved end?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:14

Problem 83

A solid glass hemisphere of radius 12.0 $\mathrm{cm}$ and index of refraction $n=1.50$ is placed with its flat face downward on a table. A parallel beam of light with a circular cross section 3.80 $\mathrm{mm}$ in diameter travels straight down and enters the hemisphere at the center of its curved surface. (a) What is the diameter of the circle of light formed on the table? (b) How does your result depend on the radius of the hemisphere?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:15

Problem 84

A thick-walled wine goblet sitting on a table can be considered to be a hollow glass sphere with an outer radius of 4.00 cm and an inner radius of 3.40 $\mathrm{cm}$ . The index of refraction of the goblet glass is $1.50 .(a)$ A beam of parallel light rays enters the side of the empty goblet along a horizontal radius. Where, if anywhere, will an image be formed? ( $b$ ) The goblet is filled with white wine $(n=1.37)$ . Where is the image formed?

Anand Jangid
Anand Jangid
Numerade Educator
09:31

Problem 85

Focus of the Eye. The comea of the eye has a radius of curvature of approximately $0.50 \mathrm{cm},$ and the aqueous humor behind it has an index of refraction of $1.35 .$ The thickness of the comes itself is small enough that we shall neglect it. The depth of a typical human eye is around 25 $\mathrm{mm}$ (a) What would have to be the radius of curvature of the cornea so that it alone would focus the image of a distant mountain on the retina, which is at the back of the eye opposite the cornea? (b) If the cornea focused the mountain correcly on the retina as described in part (a), would it also focus the text from a computer screen on the retina if that screen were 25 $\mathrm{cm}$ in front of the eye? If not, where would it focus that text in front of or behind the retina? (c) Given that the cornea has a radius of curvature of about 5.0 $\mathrm{mm}$ , where does it actually focus the mountain? Is this in front of or behind the retina? Does this help you see why
the eye needs help from a lens to complete the task of focusing?

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
04:15

Problem 86

A transparent rod 50.0 $\mathrm{cm}$ long and with a refractive index of 1.60 is cut flat at the right end and rounded to a hemispherical surface with a 15.0 -cm radius at the left end. An object is placed on the axis of the rod 120 $\mathrm{cm}$ to the left of the vertex of the hemispherical end. (a) What is the position of the final image? (b) What is its magnification?

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
02:07

Problem 87

What should be the index of refraction of a transparent sphere in order for paraxial rays from an infinitely distant object to be brought to a focus at the vertex of the surface opposite the point of incidence?

Bruce Edelman
Bruce Edelman
Numerade Educator
04:38

Problem 88

A glass rod with a refractive index of 1.55 is ground and polished at both ends to hemispherical surfaces with radii of 6.00 $\mathrm{cm}$ . When an object is placed on the axis of the rod, 25.0 $\mathrm{cm}$ to the left of the left-hand end, the final image is formed 65.0 $\mathrm{cm}$ to the right of the right-hand end. What is the length of the rod measured between the vertices of the two hemispherical surfaces?

Keshav Singh
Keshav Singh
Numerade Educator
05:19

Problem 89

Two thin lenses with focal lengths of magnitude $15.0 \mathrm{cm},$ the first diverging and the second converging, are placed 12.00 $\mathrm{cm}$ apart. An object 4.00 $\mathrm{mm}$ tall is placed 5.00 $\mathrm{cm}$ to the left of the first (diverging) lens. (a) Where is the image formed by the first lens located? (b) How far from the object is the final image formed?(c) Is the final image real or virtual? (d) What is the height of the final image? Is the final image erect or inverted?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
18:26

Problem 90

The radii of curvature of the surfaces of a thin converging meniscus lens are $R_{1}=+12.0 \mathrm{cm}$ and $R_{2}=+28.0 \mathrm{cm} .$ The index of refraction is 1.60 . (a) Compute the position and size of the image of an object in the form of an arrow 5.00 $\mathrm{mm}$ tall, perpendicular to the lens axis, 45.0 $\mathrm{cm}$ to the left of the lens. (b) A second converging lens with the same focal length is placed 3.15 $\mathrm{m}$ to the right of the first. Find the position and size of the final image. Is the final image erect or inverted with respect to the original object? (c) Repeat part (b) except with the second lens 45.0 $\mathrm{cm}$ to the right of the first.

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
05:28

Problem 91

An object to the left of a lens is imaged by the lens on a screen 30.0 $\mathrm{cm}$ to the right of the lens. When the lens is moved 4.00 $\mathrm{cm}$ to the right, the screen must be moved 4.00 $\mathrm{cm}$ to the left to refocus the image. Determine the focal length of the lens.

Bruce Edelman
Bruce Edelman
Numerade Educator
03:03

Problem 92

For refraction at a spherical surface, the first focal length $f$ is defined as the value of $s$ corresponding to $s^{\prime}=\infty,$ as shown in Fig. 34.59 a. The second focal length $f^{\prime}$ is defined as the value of $s^{\prime}$ when $s=\infty,$ as shown in Fig. 34.59 $\mathrm{b}$ . (a) Prove that $n_{a} / n_{b}=f / f^{\prime}$ (b) Prove that the general relationship between object and image distance is
$$\frac{\boldsymbol{f}}{\boldsymbol{s}}+\frac{\boldsymbol{f}^{\prime}}{\boldsymbol{s}^{\prime}}=1$$

Christopher Provencher
Christopher Provencher
Numerade Educator
08:55

Problem 93

A convex mirror and a concave mirror are placed on the same optic axis, separated by a distance $L=0.600 \mathrm{m}$ . The radius of curvature of each mirror has a magnitude of 0.360 $\mathrm{m}$ .
A light source is located a distance $x$ from the concave mirror, as shown in Fig. 34.60 . (a) What distance $x$ will result in the rays from the source returning to the source after reflecting first from the convex mirror and then from the concave mirror? (b) Repeat part (a), but now let the rays reflect first from the concave mirror and then from the convex one.

Bruce Edelman
Bruce Edelman
Numerade Educator
05:38

Problem 94

As shown in Fig. 34.61 the candle is at the center of curvature of the concave mirror, whose focal length is $10.0 \mathrm{cm} .$ The converging lens has a focal length of 32.0 $\mathrm{cm}$ and is 85.0 $\mathrm{cm}$ to the right of the candle. The candle is viewed looking through the lens from the right. The lens forms two images of the candle. The first is formed by light passing directly through the lens. The second image is formed from the light that goes from the candle to the mirror, is reflected, and then passes through the lens. (a) For each of these two images, draw a principal-ray diagram that locates the
image. (b) For each image, answer the following questions: (i) Where is the image? (ii) Is the image real or vitual? (iii) Is the image erect or inverted with respect to the original object?

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
03:57

Problem 95

One end of a long glass rod is ground to a convex hemispherical shape. This glass an index of refraction of $1.55 .$ When a small leaf is placed 20.0 $\mathrm{cm}$ in front of the center of the hemisphere along the optic axis, an image is formed inside the glass 9.12 $\mathrm{cm}$ from the spherical surface. Where would the image be formed if the glass were now immersed in water (refractive index 1.33$)$ but nothing else were changed?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
09:10

Problem 96

Two Lenses in Contact. (a) Prove that when two thin lenses with focal lengths $f_{1}$ and $f_{2}$ are placed in contact, the focal length $f$ of the combination is given by the relationship
$$\frac{1}{f}=\frac{1}{f_{1}}+\frac{1}{f_{2}}$$
(b) A converging meniscus lens (see Fig. 34.32 a) has an index of refraction of 1.55 and radii of curvature for its surfaces of 4.50 $\mathrm{cm}$ and $9.00 \mathrm{cm} .$ The concave surface is placed upward and filled with carbon tetrachloride $\left(\mathrm{CCl}_{4}\right),$ which has $n=1.46 .$ What is the
focal length of the $\mathrm{CCl}_{4}$ - glass combination?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
05:08

Problem 97

Rays from a lens are converging toward a point image $P$ located to the right of the lens. What thickness $t$ of glass with index of refraction 1.60 must be interposed between the lens and $P$ for the image to be formed at $P^{\prime},$ located 0.30 $\mathrm{cm}$ to the right of $P ?$ The locations of the piece of glass and of points $P$ and $P^{\prime}$ are shown in Fig. 34.62 .

Sheh Lit Chang
Sheh Lit Chang
University of Washington
04:00

Problem 98

A Lens in a Liquid. A lens obeys Snell's law, bending light rays at cach surface an amount determined by the index of refraction of the lens and the index of the medium in which the lens is located. (a) Equation ( 34.19 ) assumes that the lens is surrounded by air. Consider instead a thin lens immersed in a liquid with refractive index $n / n_{\text { liq }}(\mathrm{b})$ . Prove that the focal length $f^{\prime}$ is then given by Eq. $(34.19)$ with $n$ replaced by $n / n_{\text { liq }}(\mathrm{b})$ A thin lens with index $n$ has focal length $f$ in vacuum. Use the result of part (a) to show that when this lens is immersed in a liquid of index $n / n_{\text { liq }}(\mathrm{b})$ it will have a new focal length given by
$$f^{\prime}=\left[\frac{n_{\operatorname{lig}}(n-1)}{n-n_{\operatorname{liq}}}\right] f$$

Vidhi Bhatt
Vidhi Bhatt
Numerade Educator
03:03

Problem 99

When an object is placed at the proper distance to the left of a converging lens, the image is focused on a screen 30.0 $\mathrm{cm}$ to the right of the lens. A diverging lens is now placed 15.0 $\mathrm{cm}$ to the right of the converging lens, and it is found that the screen must be moved 19.2 $\mathrm{cm}$ farther to the right to obtain a sharp image. What is the focal length of the diverging lens?

Bruce Edelman
Bruce Edelman
Numerade Educator
04:31

Problem 100

A convex spherical mirror with a focal length of magnitude 24.0 $\mathrm{cm}$ is placed 20.0 $\mathrm{cm}$ to the left of a plane mirror. An object 0.250 $\mathrm{cm}$ tall is placed midway between the surface of the plane mirror and the vertex of the spherical mirror. The spherical mirror forms multiple images of the object. Where are the two images of the object formed by the spherical mirror that are closes to the spherical mirror, and how tall is each image?

Bruce Edelman
Bruce Edelman
Numerade Educator
03:28

Problem 101

A glass plate 3.50 $\mathrm{cm}$ thick, with an index of refraction of 1.55 and plane parallel faces, is held with its faces horizontal and its lower face 6.00 $\mathrm{cm}$ above a printed page. Find the position of the image of the page formed by rays making a small angle with the normal to the plate.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
19:10

Problem 102

A symmetric, double-convex, thin lens made of glass with index of refraction 1.52 has a focal length in air of 40.0 $\mathrm{cm}$ . The lens is sealed into an opening in the left hand end of a tank filled with water. At the right-hand end of the tank, opposite the lens, is a plane mirror 90.0 $\mathrm{cm}$ from the lens. The index of refraction of the water is $\frac{4}{3}$ . (a) Find the position of the image formed by the
lens-water-mirror system of a small object outside the tank on the lens axis and 70.0 $\mathrm{cm}$ to the left of the lens. (b) Is the image real or virtual? (c) Is it erect or inverted? (d) If the object has a height of $4.00 \mathrm{mm},$ what is the height of the image?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
05:15

Problem 103

You have a camera with a 35.0 -mm focal length lens and 36.0 -mm-wide film. You wish to take a picture of a 120 -m-long sailboat but find that the image of the boat fills only $\frac{1}{4}$ of the width of the film. (a) How far are you from the boat? (b) How much closer must the boat be to you for its image to fill the width of the film?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:13

Problem 104

An object is placed 18.0 $\mathrm{cm}$ from a screen. (a) At what two points between object and screen may a converging lens with a 3.00 -cm focal length be placed to obtain an image on the screen? (b) What is the magnification of the image for each position of the lens?

Narayan Hari
Narayan Hari
Numerade Educator
04:03

Problem 105

Three thin lenses, each with a focal length of 40.0 cm, are aligned on a common axis; adjacent lenses are separated by 52.0 $\mathrm{cm}$ . Find the position of the image of a small object on the axis, 80.0 $\mathrm{cm}$ to the left of the first lens.

Bruce Edelman
Bruce Edelman
Numerade Educator
02:19

Problem 106

A camera with a 90 -mm-focal-length lens is focused on an object 1.30 $\mathrm{m}$ from the lens. To refocus on an object 6.50 $\mathrm{m}$ from the lens, by how much must the distance between the lens and the film be changed? To refocus on the more distant object, is the lens moved toward or away from the film?

Farhanul Hasan
Farhanul Hasan
Numerade Educator
06:58

Problem 107

The derivation of the expression for angular magnification, Eq. $(34.22)$ , assumed a near point of $25 \mathrm{cm} .$ In fact, the near point changes with age as shown in Table $34.1 .$ In order to achiecve an angular magnification of $2.0 \times$ , what focal length should be used by a person of (a) age $10 ;(b)$ age $30 ;(c)$ age 60$?(\mathrm{d})$ If the lens that gives $M=20$ for a 10 -year-old is used by a 60 -year-old, what angular magnification will the older viewer obtain? (e) Does your answer in part (d) mean that older viewers are able to see more highly magnified images than younger viewers? Explain.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
07:37

Problem 108

Angular Magnification. In deriving Eq. $(34.22)$ for the angular magnification of a magnifier, we assumed that the object is placed at the focal point of the magnifier so that the viritual image is formed at infinity. Suppose instead that the object is placed so that the virtual image appears at an average viewer's near point of $25 \mathrm{cm},$ the closest point at which the viewer can bring an object into focus. (a) Where should the object be placed to achieve this? Give your answer in terms of the magnifier foch length $f .$ (b) What angle $\theta^{\prime}$ will an object of height $y$ subtend at the position found in part (a)? (c) Find the angular magnification $M$ with the object at the position found in part (a). The angle $\theta$ is the same as in Fig. 34.51 $\mathrm{a}$ , since it refers to viewing the object without the magnifier. (d) For a convex lens with $f=+10.0 \mathrm{cm},$ what is the value of $M$ with the object at the position found in part (a)? How many times greater is $M$ in this case than in the case where the image is formed at infinity? (e) In the description of a compound microscope in Section 34.8 , it is stated that in a properly designed instrument, the real image formed by the objective lies just inside the first focal point $F_{1}^{\prime}$ of the eyepiece. What advantages are gained by having the image formed by the objective be just inside $F_{1}^{\prime},$ as opposed to precisely at $F_{1}^{\prime} ?$ What happens if the image formed by the objective is just outside $F_{1}^{\prime} ?$

Sheh Lit Chang
Sheh Lit Chang
University of Washington
02:04

Problem 109

In one form of cataract surgery the person's natural lens, which has become cloudy, is replaced by an artificial lens. The refracting properties of the replacement lens can be chosen so that the person's eye focuses on distant objects. But there is no accommodation, and glasses or contact lenses are needed for close vision. What is the power, in diopters, of the corrective contact lenses that will enable a person who has had such surgery to focus on the page of a book at a distance of 24 $\mathrm{cm} ?$

Bruce Edelman
Bruce Edelman
Numerade Educator
01:13

Problem 110

A Nearsighted Eye. A certain very nearsighted person cannot focus on anything farther than 36.0 $\mathrm{cm}$ from the eye. Consider the simplified model of the eye described in Exercise 34.46 . If the radius of curvature of the comea is 0.75 $\mathrm{cm}$ when the eye is focusing on an object 36.0 $\mathrm{cm}$ from the comea vertex and the indexes of refraction are as described in Exercise $34.46,$ what is the distance from the comea vertex to the retina? What does this tell you about the shape of the nearsighted eye?

Farhanul Hasan
Farhanul Hasan
Numerade Educator
View

Problem 111

Focal Length of a Zoom Lens. Figure 34.63 shows a simple version of a zoom lens. The converging lens has focal length $f_{1},$ and the diverging lens has focal length $f_{2}=-\left|f_{2}\right| .$ The two lenses are separated by a variable distance $d$ that is always less than $f_{1} .$ Also, the magnitude of the focal length of the diverging lens satisfies the inequality $\left|f_{2}\right|>\left(f_{1}-d\right) .$ To determine the effective focal length of the combination lens, consider a bundle of parallel rays of radius $r_{0}$ entering the converging lens. (a) Show that the radius of the ray bundle decreases to $r_{0}^{\prime}=r_{0}\left(f_{1}-d\right) / f_{1}$ at the point that it enters the diverging lens. (b) Show that the final image $I^{\prime}$ is formed a distance $s_{2}^{\prime}=|f|\left(f_{1}-d\right) l\left(\left|f_{2}\right|-f_{1}+d\right)$ to the right of the diverging lens. (c) If the rays that emerge from the diverging lens and reach the final image point are extended backward to the left of the diverging lens, they will eventually expand to the original radius $r_{0}$ at some point $Q$ . The distance from the final image $\boldsymbol{r}^{\prime}$ to the point $Q$ is the effective focal length $f$ of the lens combination; if the combination were replaced by a single
lens of focal length $f$ placed at $Q,$ parallel rays would still be brought to a focus at $I^{\prime} .$ Show that the effective focal length is given by $f=f_{1}|f| /\left(\left|f_{2}\right|-f_{1}+d\right) .$ (d) If $f_{1}=12.0 \mathrm{cm}, f_{2}=$ $-18.0 \mathrm{cm},$ and the separation $d$ is adjustable between 0 and $4.0 \mathrm{cm},$ find the maximum and minimum focal lengths of the combination. What value of $d$ gives $f=30.0 \mathrm{cm} ?$

Lainey Roebuck
Lainey Roebuck
Numerade Educator
06:04

Problem 112

A certain reflecting telescope, constructed as shown in Fig. 34.55 $\mathrm{a}$ , has a spherical mirror with a radius of curvature of 96.0 $\mathrm{cm}$ and an eyepiece with a focal length of $1.20 \mathrm{cm} .$ If the angular magnification has a magnitude of 36 and the object is at infinity, find the position of the eyepiece and the position and nature (real or virtual) of the final image. (Note: $|M|$ is not equal to $\left|f_{1}\right| f_{2} |,$ so the image formed by the eyepiece is not at infinity.)

Sheh Lit Chang
Sheh Lit Chang
University of Washington
03:57

Problem 113

A microscope with an objective of focal length 8.00 $\mathrm{mm}$ and an eyepiece of focal length 7.50 $\mathrm{cm}$ is used to project an image on a screen 2.00 $\mathrm{m}$ from the eyepiece. Let the image distance of the objective be 18.0 $\mathrm{cm}$ . (a) What is the lateral magnification of the image? (b) What is the distance between the objective and the eyepiece?

Farhanul Hasan
Farhanul Hasan
Numerade Educator
03:09

Problem 114

The Galilean Telescope. Figure 34.64 is a diagram of a Galilean telescope, or opera glass, with both the object and its final image at infinity. The image $I$ serves as a virtual object for the eyepiece. The final image is virtual and erect. (a) Prove that the angular magnification is $M=-f_{1} | f_{2} .$ ( b) A Galilean telescope is to be constructed with the same objective lens as in Exercise $34.57 .$ What focal length should the eyepiece have if this telescope is to have the same magnitude of angular magnification as the one in Exercise 34.57$?$ (c) Compare the lengths of the telescopes.

Vipender Yadav
Vipender Yadav
Numerade Educator
27:47

Problem 115

An Object at an Angle. A 16.0 -cm-long pencil is placed at a $45.0^{\circ}$ angle, with its center 15.0 $\mathrm{cm}$ above the optic axis and 45.0 $\mathrm{cm}$ from a lens with a $20.0-\mathrm{cm}$ focal length as shown in Fig. 34.65 . (Note that the figure is not drawn to scale.) Assume that the diameter of the lens is large enough for the paraxial approximation to be valid. (a) Where is the image of the pencil? (Give the location of the images of the points $A, B,$ and $C$ on the object, which are located at the eraser, point, and center of the pencil, respectively.) (b) What is the length of the image (that is, the distance between the images of points $A$ and $B$ ) $(c)$ Show the orientation of the image in a sketch.

Ryan Kutayiah
Ryan Kutayiah
Texas A&M University
07:48

Problem 116

Spherical aberration is a blurring of the image formed by a spherical mirror. It occurs because parallel rays striking the mirror far from the optic axis are focused at a different point than are rays near the axis. This problem is usually minimized by using only the center of a spherical mirror. (a) Show that for a spherical concave mirror, the focus moves toward the mirror as the parallel rays move toward the outer edge of the mirror. (Hint: Derive an analytic expression for the distance from the vertex to the focus of the ray for a particular parallel ray. This expression should be in terms of (i) the radius of curvature $R$ of the mirror and (ii) the angle $\boldsymbol{\theta}$ between the incident ray and a line connecting the center of curvature of the mirror with the point where the ray strikes the mirror (b) What value of $\theta$ produces a 2$\%$ change in the location of the focus compared to the location for $\theta$ very close to zero?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
07:20

Problem 117

(a) For a lens with focal length $f,$ find the smallest distance possible between the object and its real image. (b) Graph the distance between the object and the real image as a function of the distance of the object from the lens. Does your graph agree with the result you found in part (a)?

Bruce Edelman
Bruce Edelman
Numerade Educator
01:01

Problem 118

Two mirrors are placed together as shown in Fig. 34.66 . (a) Show that a point source in front of these mirrors and its two images tie on a circle. (b) Find the center of the circle. (c) In a diagram, show where an observer should stand so as to be able to see both images.

Raj Bala
Raj Bala
Numerade Educator
05:41

Problem 119

People with normal vision cannot focus their eyes under- water if they aren't wearing a face mask or goggles and there is water in contact with their eyes (see Discussion Question Q34.23). (a) Why not? (b) With the simplified model of the eye described in Exercise $34.46,$ what corrective lens (specified by focal length as measured in air) would be needed to enable a person underwater to focus an infinitely distant object? (Be careful- the focal length of a lens underwater is not the same as in air! See Problem 34.98 . Assume that the corrective lens has a refractive index of 1.62 and that the lens is used in eyeglasses, not goggles, so there is water on both sides of the lens. Assume that the eyeglasses are 2.00 $\mathrm{cm}$ in front of the eye.)

Dading Chen
Dading Chen
Numerade Educator