Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 34

Images - all with Video Answers

Educators

Chapter Questions

Problem 1

You look through a camera toward an image of a hummingbird in
a plane mirror. The camera is $4.30 \mathrm{~m}$ in front of the mirror. The bird is
at camera level, $5.00 \mathrm{~m}$ to your right and $3.30 \mathrm{~m}$ from the mirror. What is the distance between the camera and the apparent position of the bird's image in the mirror?

Zachary Warner

Zachary Warner

Numerade Educator

Problem 2

A moth at about eye level is $10 \mathrm{~cm}$ in front of a plane mirror; you are behind the moth, $30 \mathrm{~cm}$ from the mirror. What is the distance between your eyes and the apparent position of the moth's image in the mirror?

Averell Hause

Carnegie Mellon University

Problem 3

In Fig. $34-32$, an isotropic point source of light
$S$ is positioned at distance $d$ from a viewing screen $A$ and the light intensity $I_{P}$ at point $P$ (level with $S$ ) is measured. Then a plane mirror $M$ is placed behind $S$ at distance $d$. By how much is $I_{P}$ multiplied by the presence of the mirror?

Zachary Warner

Numerade Educator

Problem 4

Figure $34-33$ shows an overhead view of a corridor with a plane mirror $M$ mounted at one end. A burglar $B$ sneaks along the corridor directly toward the center of the mirror. If $d=3.0 \mathrm{~m}$, how far from the mirror will she be when the se-
curity guard $S$ can first see her in the mirror?

Naresh Bagrecha

Naresh Bagrecha

Numerade Educator

Problem 5

Figure $34-34$ shows a small lightbulb suspended at distance $d_{1}=250 \mathrm{~cm}$ above the surface of the water in a swimming pool where the water depth is $d_{2}=$ $200 \mathrm{~cm}$. The bottom of the pool is a large mirror. How far below the mir-
ror surface is the image of the bulb? (Hint: Assume that the rays are close to a vertical axis through the bulb, and use the small-angle approximation in which $\sin \theta \approx \tan$

Eduard Sanchez

Numerade Educator

Problem 6

An object is moved along the central axis of a spherical mirror while the lateral magnification $m$ of it is measured. Figure $34-35$ gives $m$ versus object distance $p$ for the range $p_{a}=2.0 \mathrm{~cm}$ to $p_{b}=8.0 \mathrm{~cm} .$ What is $m$
for $p=14.0 \mathrm{~cm} ?$

Averell Hause

Carnegie Mellon University

Problem 7

A concave shaving mirror has a radius of curvature of $35.0 \mathrm{~cm}$. It is positioned so that the (upright) image of a man's face is $2.50$ times the size of the face. How far is the mirror from the face?

Zachary Warner

Numerade Educator

Problem 8

An object is placed against the center of a spherical mirror and then moved $70 \mathrm{~cm}$ from it along the central axis as the image distance $i$ is measured. Figure $34-36$ gives versus object distance $p$ out to $p_{x}=$ $40 \mathrm{~cm}$. What is $i$ for $p=70 \mathrm{~cm} ?$

Sheh Lit Chang

University of Washington

Problem 9

Spherical mirrors. Object $Q$ stands on the central axis of a spher-
ical mirror. For this situation, each problem in Table $34-3$ gives object distance $p_{s}$ (centimeters), the type of mirror, and then the distance (centimeters, without proper sign) between the focal point and the mirror. Find (a) the radius of curvature $r$ (including sign),
(b) the image distance $i$, and (c) the lateral magnification $m$. Also, determine whether the image is (d) real (R) or virtual (V), (e) inverted (I) from object $O$ or noninverted (NI), and (f) on the same side of the mirror as $O$ or on the opposite side.
$$
+18 \quad \text { Concave, } 12
$$

Sheh Lit Chang

University of Washington

Problem 10

Spherical mirrors. Object $Q$ stands on the central axis of a spher-
ical mirror. For this situation, each problem in Table $34-3$ gives object distance $p_{s}$ (centimeters), the type of mirror, and then the distance (centimeters, without proper sign) between the focal point and the mirror. Find (a) the radius of curvature $r$ (including sign),
(b) the image distance $i$, and (c) the lateral magnification $m$. Also, determine whether the image is (d) real (R) or virtual (V), (e) inverted (I) from object $O$ or noninverted (NI), and (f) on the same side of the mirror as $O$ or on the opposite side.
$$
+15 \quad \text { Concave, } 10
$$

Sheh Lit Chang

University of Washington

Problem 11

Spherical mirrors. Object $Q$ stands on the central axis of a spher-
ical mirror. For this situation, each problem in Table $34-3$ gives object distance $p_{s}$ (centimeters), the type of mirror, and then the distance (centimeters, without proper sign) between the focal point and the mirror. Find (a) the radius of curvature $r$ (including sign),
(b) the image distance $i$, and (c) the lateral magnification $m$. Also, determine whether the image is (d) real (R) or virtual (V), (e) inverted (I) from object $O$ or noninverted (NI), and (f) on the same side of the mirror as $O$ or on the opposite side.
$$
\begin{aligned}
&+8.0\\
&\text { Convex. } 10
\end{aligned}
$$

Sheh Lit Chang

University of Washington

Problem 12

Spherical mirrors. Object $Q$ stands on the central axis of a spher-
ical mirror. For this situation, each problem in Table $34-3$ gives object distance $p_{s}$ (centimeters), the type of mirror, and then the distance (centimeters, without proper sign) between the focal point and the mirror. Find (a) the radius of curvature $r$ (including sign),
(b) the image distance $i$, and (c) the lateral magnification $m$. Also, determine whether the image is (d) real (R) or virtual (V), (e) inverted (I) from object $O$ or noninverted (NI), and (f) on the same side of the mirror as $O$ or on the opposite side.
$$
\begin{aligned}
&+24\\
&\text { Concave, } 36
\end{aligned}
$$

Sheh Lit Chang

University of Washington

Problem 13

Spherical mirrors. Object $Q$ stands on the central axis of a spher-
ical mirror. For this situation, each problem in Table $34-3$ gives object distance $p_{s}$ (centimeters), the type of mirror, and then the distance (centimeters, without proper sign) between the focal point and the mirror. Find (a) the radius of curvature $r$ (including sign),
(b) the image distance $i$, and (c) the lateral magnification $m$. Also, determine whether the image is (d) real (R) or virtual (V), (e) inverted (I) from object $O$ or noninverted (NI), and (f) on the same side of the mirror as $O$ or on the opposite side.
$$
\begin{aligned}
&+12\\
&\text { Concave, } 18
\end{aligned}
$$

Sheh Lit Chang

University of Washington

Problem 14

Spherical mirrors. Object $Q$ stands on the central axis of a spher-
ical mirror. For this situation, each problem in Table $34-3$ gives object distance $p_{s}$ (centimeters), the type of mirror, and then the distance (centimeters, without proper sign) between the focal point and the mirror. Find (a) the radius of curvature $r$ (including sign),
(b) the image distance $i$, and (c) the lateral magnification $m$. Also, determine whether the image is (d) real (R) or virtual (V), (e) inverted (I) from object $O$ or noninverted (NI), and (f) on the same side of the mirror as $O$ or on the opposite side.
$$
+22 \quad \text { Convex, } 35
$$

Sheh Lit Chang

University of Washington

Problem 15

Spherical mirrors. Object $Q$ stands on the central axis of a spher-
ical mirror. For this situation, each problem in Table $34-3$ gives object distance $p_{s}$ (centimeters), the type of mirror, and then the distance (centimeters, without proper sign) between the focal point and the mirror. Find (a) the radius of curvature $r$ (including sign),
(b) the image distance $i$, and (c) the lateral magnification $m$. Also, determine whether the image is (d) real (R) or virtual (V), (e) inverted (I) from object $O$ or noninverted (NI), and (f) on the same side of the mirror as $O$ or on the opposite side.
$$
+10 \quad \text { Convex, } 8.0
$$

Sheh Lit Chang

University of Washington

Problem 16

Spherical mirrors. Object $Q$ stands on the central axis of a spher-
ical mirror. For this situation, each problem in Table $34-3$ gives object distance $p_{s}$ (centimeters), the type of mirror, and then the distance (centimeters, without proper sign) between the focal point and the mirror. Find (a) the radius of curvature $r$ (including sign),
(b) the image distance $i$, and (c) the lateral magnification $m$. Also, determine whether the image is (d) real (R) or virtual (V), (e) inverted (I) from object $O$ or noninverted (NI), and (f) on the same side of the mirror as $O$ or on the opposite side.
$$
+17 \quad \text { Convex, } 14
$$

Sheh Lit Chang

University of Washington

Problem 17

More mirrors. Object $O$ stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table $34-4$ refers to (a) the type of mirror,
(b) the focal distance $f,(\mathrm{c})$ the radius of curvature $r,(\mathrm{~d})$ the object distance $p,(\mathrm{e})$ the image distance $i$, and $(\mathrm{f})$ the lateral magnifica-tion $m$. (All distances are in centimeters.) It also refers to whether
(g) the image is real (R) or virtual
(V), (h) inverted (I) or noninverted (NI) from $O$, and (i) on the same side of the mirror as object $O$ or on the opposite side. Fill in the missing information. Where only a sign is missing, answer with the sign.
$$
\begin{aligned}
&17 \text { Concave }\\
&20 \quad+10
\end{aligned}
$$

Sheh Lit Chang

University of Washington

Problem 18

More mirrors. Object $O$ stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table $34-4$ refers to (a) the type of mirror,
(b) the focal distance $f,(\mathrm{c})$ the radius of curvature $r,(\mathrm{~d})$ the object distance $p,(\mathrm{e})$ the image distance $i$, and $(\mathrm{f})$ the lateral magnifica-tion $m$. (All distances are in centimeters.) It also refers to whether
(g) the image is real (R) or virtual
(V), (h) inverted (I) or noninverted (NI) from $O$, and (i) on the same side of the mirror as object $O$ or on the opposite side. Fill in the missing information. Where only a sign is missing, answer with the sign.
$$
\begin{aligned}
&+24\\
&0.5\\
&\text { 1 }\\
&1
\end{aligned}
$$

Sheh Lit Chang

University of Washington

Problem 19

More mirrors. Object $O$ stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table $34-4$ refers to (a) the type of mirror,
(b) the focal distance $f,(\mathrm{c})$ the radius of curvature $r,(\mathrm{~d})$ the object distance $p,(\mathrm{e})$ the image distance $i$, and $(\mathrm{f})$ the lateral magnifica-tion $m$. (All distances are in centimeters.) It also refers to whether
(g) the image is real (R) or virtual
(V), (h) inverted (I) or noninverted (NI) from $O$, and (i) on the same side of the mirror as object $O$ or on the opposite side. Fill in the missing information. Where only a sign is missing, answer with the sign.
$$
\begin{aligned}
&19\\
&-2-40 \quad-10
\end{aligned}
$$

Sheh Lit Chang

University of Washington

Problem 20

More mirrors. Object $O$ stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table $34-4$ refers to (a) the type of mirror,
(b) the focal distance $f,(\mathrm{c})$ the radius of curvature $r,(\mathrm{~d})$ the object distance $p,(\mathrm{e})$ the image distance $i$, and $(\mathrm{f})$ the lateral magnifica-tion $m$. (All distances are in centimeters.) It also refers to whether
(g) the image is real (R) or virtual
(V), (h) inverted (I) or noninverted (NI) from $O$, and (i) on the same side of the mirror as object $O$ or on the opposite side. Fill in the missing information. Where only a sign is missing, answer with the sign.
$$
+40 \quad-0.70
$$

Sheh Lit Chang

University of Washington

Problem 21

More mirrors. Object $O$ stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table $34-4$ refers to (a) the type of mirror,
(b) the focal distance $f,(\mathrm{c})$ the radius of curvature $r,(\mathrm{~d})$ the object distance $p,(\mathrm{e})$ the image distance $i$, and $(\mathrm{f})$ the lateral magnifica-tion $m$. (All distances are in centimeters.) It also refers to whether
(g) the image is real (R) or virtual
(V), (h) inverted (I) or noninverted (NI) from $O$, and (i) on the same side of the mirror as object $O$ or on the opposite side. Fill in the missing information. Where only a sign is missing, answer with the sign.
$$
\begin{array}{lll}
21 & +20 & +30
\end{array}
$$

Sheh Lit Chang

University of Washington

Problem 22

More mirrors. Object $O$ stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table $34-4$ refers to (a) the type of mirror,
(b) the focal distance $f,(\mathrm{c})$ the radius of curvature $r,(\mathrm{~d})$ the object distance $p,(\mathrm{e})$ the image distance $i$, and $(\mathrm{f})$ the lateral magnifica-tion $m$. (All distances are in centimeters.) It also refers to whether
(g) the image is real (R) or virtual
(V), (h) inverted (I) or noninverted (NI) from $O$, and (i) on the same side of the mirror as object $O$ or on the opposite side. Fill in the missing information. Where only a sign is missing, answer with the sign.
$$
\begin{array}{lll}
22 & 20 & +0.10
\end{array}
$$

Sheh Lit Chang

University of Washington

Problem 23

More mirrors. Object $O$ stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table $34-4$ refers to (a) the type of mirror,
(b) the focal distance $f,(\mathrm{c})$ the radius of curvature $r,(\mathrm{~d})$ the object distance $p,(\mathrm{e})$ the image distance $i$, and $(\mathrm{f})$ the lateral magnifica-tion $m$. (All distances are in centimeters.) It also refers to whether
(g) the image is real (R) or virtual
(V), (h) inverted (I) or noninverted (NI) from $O$, and (i) on the same side of the mirror as object $O$ or on the opposite side. Fill in the missing information. Where only a sign is missing, answer with the sign.
$$
\begin{array}{ll}
30 & +0.20
\end{array}
$$

Sheh Lit Chang

University of Washington

Problem 24

More mirrors. Object $O$ stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table $34-4$ refers to (a) the type of mirror,
(b) the focal distance $f,(\mathrm{c})$ the radius of curvature $r,(\mathrm{~d})$ the object distance $p,(\mathrm{e})$ the image distance $i$, and $(\mathrm{f})$ the lateral magnifica-tion $m$. (All distances are in centimeters.) It also refers to whether
(g) the image is real (R) or virtual
(V), (h) inverted (I) or noninverted (NI) from $O$, and (i) on the same side of the mirror as object $O$ or on the opposite side. Fill in the missing information. Where only a sign is missing, answer with the sign.
$$
\begin{aligned}
&+60 \quad-0.50\\
&\begin{aligned}
&=
\end{aligned}
\end{aligned}
$$

Sheh Lit Chang

University of Washington

Problem 26

More mirrors. Object $O$ stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table $34-4$ refers to (a) the type of mirror,
(b) the focal distance $f,(\mathrm{c})$ the radius of curvature $r,(\mathrm{~d})$ the object distance $p,(\mathrm{e})$ the image distance $i$, and $(\mathrm{f})$ the lateral magnifica-tion $m$. (All distances are in centimeters.) It also refers to whether
(g) the image is real (R) or virtual
(V), (h) inverted (I) or noninverted (NI) from $O$, and (i) on the same side of the mirror as object $O$ or on the opposite side. Fill in the missing information. Where only a sign is missing, answer with the sign.
$$
26 \quad 20 \quad+60 \quad \text { Same }
$$

Sheh Lit Chang

University of Washington

Problem 27

More mirrors. Object $O$ stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table $34-4$ refers to (a) the type of mirror,
(b) the focal distance $f,(\mathrm{c})$ the radius of curvature $r,(\mathrm{~d})$ the object distance $p,(\mathrm{e})$ the image distance $i$, and $(\mathrm{f})$ the lateral magnifica-tion $m$. (All distances are in centimeters.) It also refers to whether
(g) the image is real (R) or virtual
(V), (h) inverted (I) or noninverted (NI) from $O$, and (i) on the same side of the mirror as object $O$ or on the opposite side. Fill in the missing information. Where only a sign is missing, answer with the sign.
$$
\begin{array}{lll}
7 & -30 & -15
\end{array}
$$

Sheh Lit Chang

University of Washington

Problem 28

More mirrors. Object $O$ stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table $34-4$ refers to (a) the type of mirror,
(b) the focal distance $f,(\mathrm{c})$ the radius of curvature $r,(\mathrm{~d})$ the object distance $p,(\mathrm{e})$ the image distance $i$, and $(\mathrm{f})$ the lateral magnifica-tion $m$. (All distances are in centimeters.) It also refers to whether
(g) the image is real (R) or virtual
(V), (h) inverted (I) or noninverted (NI) from $O$, and (i) on the same side of the mirror as object $O$ or on the opposite side. Fill in the missing information. Where only a sign is missing, answer with the sign.
$$
\begin{array}{lll}
28 & +10 & +1.0
\end{array}
$$

Sheh Lit Chang

University of Washington

Problem 29

More mirrors. Object $O$ stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table $34-4$ refers to (a) the type of mirror,
(b) the focal distance $f,(\mathrm{c})$ the radius of curvature $r,(\mathrm{~d})$ the object distance $p,(\mathrm{e})$ the image distance $i$, and $(\mathrm{f})$ the lateral magnifica-tion $m$. (All distances are in centimeters.) It also refers to whether
(g) the image is real (R) or virtual
(V), (h) inverted (I) or noninverted (NI) from $O$, and (i) on the same side of the mirror as object $O$ or on the opposite side. Fill in the missing information. Where only a sign is missing, answer with the sign.
$$
\begin{aligned}
&29\\
&8\\
&\begin{aligned}
&0
\end{aligned}\\
&\begin{aligned}
&0
\end{aligned}\\
&1\\
&\begin{array}{lll}
\text { Convex } & 40 & 4.0
\end{array}\\
&\begin{aligned}
&1
\end{aligned}
\end{aligned}
$$

Sheh Lit Chang

University of Washington

Problem 30

Figure $34-37$ gives the lateral magnification $m$ of an object
versus the object distance $p$ from a spherical mirror as the object is moved along the mirror's central axis through a range of values for
$p$. The horizontal scale is set by $p_{s}=10.0 \mathrm{~cm} .$ What is the magnification of the object when the object is $21 \mathrm{~cm}$ from the mirror?

Averell Hause

Carnegie Mellon University

Problem 31

A luminous point is moving at speed $v_{o}$ toward a spherical mirror with radius of curvature $r$, along the central axis of the mirror. Show that the image of this point is moving at speed
$$
v_{I}=-\left(\frac{r}{2 p-r}\right)^{2} v_{O}
$$
where $p$ is the distance of the luminous point from the mirror at any given time. Now assume the mirror is concave, with $r=15 \mathrm{~cm}$,
and let $v_{O}=5.0 \mathrm{~cm} / \mathrm{s}$. Find $v_{l}$ when (b) $p=30 \mathrm{~cm}$ (far outside the focal point), (c) $p=8.0 \mathrm{~cm}$ (just outside the focal point), and
(d) $p=10 \mathrm{~mm}$ (very near the mirror).

Sheh Lit Chang

University of Washington

Problem 32

Spherical refracting surfaces. An object $O$ stands on the central axis of a spherical refracting surface. For this situation, each problem in Table $34-5$ refers to the index of refraction $n_{1}$ where the object is located, (a) the index of refraction $n_{2}$ on the other side of the refracting surface, (b) the object distance $p,(\mathrm{c})$ the radius of curvature $r$ of the surface, and
(d) the image distance $i$. (All distances are in centimeters.) Fill in the missing information, including whether the image is (e) real
(R) or virtual (V) and (f) on the same side of the surface as object $O$ or on the opposite side.
$\mathbf{3 2}$
$\begin{array}{llll}1.0 & 1.5 & +10 & +30\end{array}$

Sheh Lit Chang

University of Washington

Problem 33

Spherical refracting surfaces. An object $O$ stands on the central axis of a spherical refracting surface. For this situation, each problem in Table $34-5$ refers to the index of refraction $n_{1}$ where the object is located, (a) the index of refraction $n_{2}$ on the other side of the refracting surface, (b) the object distance $p,(\mathrm{c})$ the radius of curvature $r$ of the surface, and
(d) the image distance $i$. (All distances are in centimeters.) Fill in the missing information, including whether the image is (e) real
(R) or virtual (V) and (f) on the same side of the surface as object $O$ or on the opposite side.
$\begin{array}{lllll}\mathbf{3 3} & 1.0 & 1.5 & +10 & -13\end{array}$

Sheh Lit Chang

University of Washington

Problem 34

Spherical refracting surfaces. An object $O$ stands on the central axis of a spherical refracting surface. For this situation, each problem in Table $34-5$ refers to the index of refraction $n_{1}$ where the object is located, (a) the index of refraction $n_{2}$ on the other side of the refracting surface, (b) the object distance $p,(\mathrm{c})$ the radius of curvature $r$ of the surface, and
(d) the image distance $i$. (All distances are in centimeters.) Fill in the missing information, including whether the image is (e) real
(R) or virtual (V) and (f) on the same side of the surface as object $O$ or on the opposite side.
$\begin{array}{lllll}\mathbf{3 4} & 1.5 & +100 & -30 & +600\end{array}$

Sheh Lit Chang

University of Washington

Problem 35

Spherical refracting surfaces. An object $O$ stands on the central axis of a spherical refracting surface. For this situation, each problem in Table $34-5$ refers to the index of refraction $n_{1}$ where the object is located, (a) the index of refraction $n_{2}$ on the other side of the refracting surface, (b) the object distance $p,(\mathrm{c})$ the radius of curvature $r$ of the surface, and
(d) the image distance $i$. (All distances are in centimeters.) Fill in the missing information, including whether the image is (e) real
(R) or virtual (V) and (f) on the same side of the surface as object $O$ or on the opposite side.
35
$\begin{array}{llll}1.5 & 1.0 & +70 & +30\end{array}$

Sheh Lit Chang

University of Washington

Problem 36

Spherical refracting surfaces. An object $O$ stands on the central axis of a spherical refracting surface. For this situation, each problem in Table $34-5$ refers to the index of refraction $n_{1}$ where the object is located, (a) the index of refraction $n_{2}$ on the other side of the refracting surface, (b) the object distance $p,(\mathrm{c})$ the radius of curvature $r$ of the surface, and
(d) the image distance $i$. (All distances are in centimeters.) Fill in the missing information, including whether the image is (e) real
(R) or virtual (V) and (f) on the same side of the surface as object $O$ or on the opposite side.
$\begin{array}{lllll}36 & 1.5 & 1.0 & -30 & -7.5\end{array}$

Sheh Lit Chang

University of Washington

Problem 37

Spherical refracting surfaces. An object $O$ stands on the central axis of a spherical refracting surface. For this situation, each problem in Table $34-5$ refers to the index of refraction $n_{1}$ where the object is located, (a) the index of refraction $n_{2}$ on the other side of the refracting surface, (b) the object distance $p,(\mathrm{c})$ the radius of curvature $r$ of the surface, and
(d) the image distance $i$. (All distances are in centimeters.) Fill in the missing information, including whether the image is (e) real
(R) or virtual (V) and (f) on the same side of the surface as object $O$ or on the opposite side.
$\begin{array}{lllll}7 & 1.5 & 1.0 & +10 & -6.0\end{array}$

Sheh Lit Chang

University of Washington

Problem 38

Spherical refracting surfaces. An object $O$ stands on the central axis of a spherical refracting surface. For this situation, each problem in Table $34-5$ refers to the index of refraction $n_{1}$ where the object is located, (a) the index of refraction $n_{2}$ on the other side of the refracting surface, (b) the object distance $p,(\mathrm{c})$ the radius of curvature $r$ of the surface, and
(d) the image distance $i$. (All distances are in centimeters.) Fill in the missing information, including whether the image is (e) real
(R) or virtual (V) and (f) on the same side of the surface as object $O$ or on the opposite side.
\begin{tabular}{lllll}
38 & $1.0$ & $1.5$ & $+30$ & $+600$ \\
\hline
\end{tabular}

Sheh Lit Chang

University of Washington

Problem 39

In Fig. 34-38, a beam of parallel light rays from a laser is incident on a solid transparent sphere of index of refraction $n$. (a) If
a point image is produced at the back of the sphere, what is the index of refraction of the sphere? (b) What index of refraction, if any, will produce a point image at the center of the sphere?

Zachary Warner

Numerade Educator

Problem 40

A glass sphere has radius $R=5.0 \mathrm{~cm}$ and index of refraction 1.6. A paperweight is constructed by slicing through the sphere along a plane that is $2.0 \mathrm{~cm}$
from the center of the sphere, leaving height $h=3.0 \mathrm{~cm}$. The paperweight is placed on a table and viewed from directly above by an observer who is distance $d=8.0 \mathrm{~cm}$ from the tabletop (Fig. 34-39). When viewed through the paperweight, how far away does the tabletop appear to be to the observer?

Sheh Lit Chang

University of Washington

Problem 41

A lens is made of glass having an index of refraction of
1.5. One side of the lens is flat, and the other is convex with a radius of curvature of $20 \mathrm{~cm} .$ (a) Find the focal length of the lens. (b) If an object is placed $40 \mathrm{~cm}$ in front of the lens, where is the image?

Zachary Warner

Numerade Educator

Problem 42

Figure 34-40 gives the lateral magnification $m$ of an object versus the object distance $p$ from a lens as the object is moved along the central axis of the lens through a range of values for $p$ out to $p_{s}=20.0 \mathrm{~cm}$. What is the magnification of the object when the object is $35 \mathrm{~cm}$ from the lens?

Averell Hause

Carnegie Mellon University

Problem 43

A movie camera with a (sin-
gle) lens of focal length $75 \mathrm{~mm}$ takes a picture of a person standing $27 \mathrm{~m}$ away. If the person is $180 \mathrm{~cm}$ tall, what is the height of the image on the film?

Sheh Lit Chang

University of Washington

Problem 44

An object is placed against the center of a thin lens and then moved away from it along the central axis as the image distance $i$ is measured. Figure $34-41$ gives $i$ versus object distance $p$ out to $p_{s}=60 \mathrm{~cm} .$ What is the image distance when $p=100$ $\mathrm{cm} ?$

Eduard Sanchez

Numerade Educator

Problem 45

You produce an image of the Sun on a screen, using a thin lens whose focal length is $20.0 \mathrm{~cm}$. What is the diameter of the image? (See Appendix $\mathrm{C}$ for needed data on the Sun.)

Zachary Warner

Numerade Educator

Problem 46

An object is placed against the center of a thin lens and then moved $70 \mathrm{~cm}$ from it along the central axis as the image distance $\vec{i}$
is measured. Figure $34-42$ gives $i$ versus object distance $p$ out to $p_{s}=40 \mathrm{~cm} .$ What is the image distance when $p=70 \mathrm{~cm}$ ?

Sheh Lit Chang

University of Washington

Problem 47

A double-convex lens is to be made of glass with an index of refraction of $1.5 .$ One surface is to have twice the radius of
curvature of the other and the focal length is to be $60 \mathrm{~mm}$. What is the (a) smaller and (b) larger radius?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 48

An object is moved along the central axis of a thin lens while the lateral magnification $m$ is measured. Figure $34-43$ gives $m$ versus object distance $p$ out to $p_{s}=8.0 \mathrm{~cm} .$ What is the magnification of the object when the object is $14.0 \mathrm{~cm}$ from the lens?

Averell Hause

Carnegie Mellon University

Problem 49

An illuminated slide is held $44 \mathrm{~cm}$ from a screen. How far from the slide must a lens of focal length $11 \mathrm{~cm}$ be placed (between the slide and the screen) to form an image of the slide's picture on the screen?

Sheh Lit Chang

University of Washington

Problem 50

Thin lenses. Object $O$ stands on the central axis of a thin symmetric lens. For this situation, each problem in Table $34-6$ gives object distance $p$ (centimeters), the type of lens (C stands for converging and D for diverging), and then the distance (centimeters, without proper sign) between a focal point and the lens. Find (a) the image distance $i$ and (b) the lateral magnification $m$ of the object, including signs. Also, determine whether the image is (c) real (R) or virtual (V), (d) inverted
(I) from object $O$ or noninverted (NI), and (e) on the same side of the lens as object $O$ or on the opposite side.
$\begin{array}{lll}\mathbf{5 0} & +16 & \text { C, } 4.0\end{array}$

Sheh Lit Chang

University of Washington

Problem 51

Thin lenses. Object $O$ stands on the central axis of a thin symmetric lens. For this situation, each problem in Table $34-6$ gives object distance $p$ (centimeters), the type of lens (C stands for converging and D for diverging), and then the distance (centimeters, without proper sign) between a focal point and the lens. Find (a) the image distance $i$ and (b) the lateral magnification $m$ of the object, including signs. Also, determine whether the image is (c) real (R) or virtual (V), (d) inverted
(I) from object $O$ or noninverted (NI), and (e) on the same side of the lens as object $O$ or on the opposite side.
$\begin{array}{lll}\mathbf{5 1} & +12 & \text { C. 16 }\end{array}$

Sheh Lit Chang

University of Washington

Problem 52

Thin lenses. Object $O$ stands on the central axis of a thin symmetric lens. For this situation, each problem in Table $34-6$ gives object distance $p$ (centimeters), the type of lens (C stands for converging and D for diverging), and then the distance (centimeters, without proper sign) between a focal point and the lens. Find (a) the image distance $i$ and (b) the lateral magnification $m$ of the object, including signs. Also, determine whether the image is (c) real (R) or virtual (V), (d) inverted
(I) from object $O$ or noninverted (NI), and (e) on the same side of the lens as object $O$ or on the opposite side.
$\begin{array}{lll}\mathbf{5 2} & +25 & \text { C, } 35\end{array}$

Sheh Lit Chang

University of Washington

Problem 53

Thin lenses. Object $O$ stands on the central axis of a thin symmetric lens. For this situation, each problem in Table $34-6$ gives object distance $p$ (centimeters), the type of lens (C stands for converging and D for diverging), and then the distance (centimeters, without proper sign) between a focal point and the lens. Find (a) the image distance $i$ and (b) the lateral magnification $m$ of the object, including signs. Also, determine whether the image is (c) real (R) or virtual (V), (d) inverted
(I) from object $O$ or noninverted (NI), and (e) on the same side of the lens as object $O$ or on the opposite side.
$\begin{array}{lll}\mathbf{5 3} & +8.0 & \text { D, 12 }\end{array}$

Sheh Lit Chang

University of Washington

Problem 54

Thin lenses. Object $O$ stands on the central axis of a thin symmetric lens. For this situation, each problem in Table $34-6$ gives object distance $p$ (centimeters), the type of lens (C stands for converging and D for diverging), and then the distance (centimeters, without proper sign) between a focal point and the lens. Find (a) the image distance $i$ and (b) the lateral magnification $m$ of the object, including signs. Also, determine whether the image is (c) real (R) or virtual (V), (d) inverted
(I) from object $O$ or noninverted (NI), and (e) on the same side of the lens as object $O$ or on the opposite side.
$\begin{array}{lll}\mathbf{5 4} & +10 & \text { D, } 6.0\end{array}$

Sheh Lit Chang

University of Washington

Problem 55

Thin lenses. Object $O$ stands on the central axis of a thin symmetric lens. For this situation, each problem in Table $34-6$ gives object distance $p$ (centimeters), the type of lens (C stands for converging and D for diverging), and then the distance (centimeters, without proper sign) between a focal point and the lens. Find (a) the image distance $i$ and (b) the lateral magnification $m$ of the object, including signs. Also, determine whether the image is (c) real (R) or virtual (V), (d) inverted
(I) from object $O$ or noninverted (NI), and (e) on the same side of the lens as object $O$ or on the opposite side.
$\begin{array}{lll}\mathbf{5 5} & +22 & \mathrm{D}, 14\end{array}$

Sheh Lit Chang

University of Washington

Problem 56

Thin lenses. Object $O$ stands on the central axis of a thin symmetric lens. For this situation, each problem in Table $34-6$ gives object distance $p$ (centimeters), the type of lens (C stands for converging and D for diverging), and then the distance (centimeters, without proper sign) between a focal point and the lens. Find (a) the image distance $i$ and (b) the lateral magnification $m$ of the object, including signs. Also, determine whether the image is (c) real (R) or virtual (V), (d) inverted
(I) from object $O$ or noninverted (NI), and (e) on the same side of the lens as object $O$ or on the opposite side.
$\begin{array}{lll}\mathbf{5 6} & +12 & \mathrm{D}, 31\end{array}$

Sheh Lit Chang

University of Washington

Problem 57

Thin lenses. Object $O$ stands on the central axis of a thin symmetric lens. For this situation, each problem in Table $34-6$ gives object distance $p$ (centimeters), the type of lens (C stands for converging and D for diverging), and then the distance (centimeters, without proper sign) between a focal point and the lens. Find (a) the image distance $i$ and (b) the lateral magnification $m$ of the object, including signs. Also, determine whether the image is (c) real (R) or virtual (V), (d) inverted
(I) from object $O$ or noninverted (NI), and (e) on the same side of the lens as object $O$ or on the opposite side.
$\begin{array}{lll}57 & +45 & \text { C, } 20\end{array}$

Sheh Lit Chang

University of Washington

Problem 58

situation, each problem in Table $34-7$ gives object distance $p$, index of refraction $n$ of the lens, radius $r_{1}$ of the nearer lens surface, and radius $r_{2}$ of the farther lens surface. (AII distances are in centimeters.) Find (a) the image distance $i$ and (b) the lateral magnification $m$ of the object, including signs. Also, determine whether the image is (c) real (R) or virtual (V), (d) inverted (I) from object $O$ or noninverted (NI), and (e) on the same side of the lens as object $O$ or on the opposite side.
$\begin{array}{lllll}58 & +29 & 1.65 & +35 & \infty\end{array}$

Sheh Lit Chang

University of Washington

Problem 59

Lenses with given radii. Object $O$ stands in front of a thin lens, on the central axis. For this situation, each problem in Table $34-7$ gives object distance $p$, index of refraction $n$ of the lens, radius $r_{1}$ of the nearer lens surface, and radius $r_{2}$ of the farther lens surface. (AII distances are in centimeters.) Find (a) the image distance $i$ and (b) the lateral magnification $m$ of the object, including signs. Also, determine whether the image is (c) real (R) or virtual (V), (d) inverted (I) from object $O$ or noninverted (NI), and (e) on the same side of the lens as object $O$ or on the opposite side.
$\begin{array}{lllll}\mathbf{5 9} & +75 & 1.55 & +30 & -42\end{array}$

Sheh Lit Chang

University of Washington

Problem 60

Lenses with given radii. Object $O$ stands in front of a thin lens, on the central axis. For this situation, each problem in Table $34-7$ gives object distance $p$, index of refraction $n$ of the lens, radius $r_{1}$ of the nearer lens surface, and radius $r_{2}$ of the farther lens surface. (AII distances are in centimeters.) Find (a) the image distance $i$ and (b) the lateral magnification $m$ of the object, including signs. Also, determine whether the image is (c) real (R) or virtual (V), (d) inverted (I) from object $O$ or noninverted (NI), and (e) on the same side of the lens as object $O$ or on the opposite side.
$\begin{array}{lllll}60 & +6.0 & 1.70 & +10 & -12\end{array}$

Sheh Lit Chang

University of Washington

Problem 61

Lenses with given radii. Object $O$ stands in front of a thin lens, on the central axis. For this situation, each problem in Table $34-7$ gives object distance $p$, index of refraction $n$ of the lens, radius $r_{1}$ of the nearer lens surface, and radius $r_{2}$ of the farther lens surface. (AII distances are in centimeters.) Find (a) the image distance $i$ and (b) the lateral magnification $m$ of the object, including signs. Also, determine whether the image is (c) real (R) or virtual (V), (d) inverted (I) from object $O$ or noninverted (NI), and (e) on the same side of the lens as object $O$ or on the opposite side.
$\mathbf{6 1}$
$\begin{array}{llll}+24 & 1.50 & -15 & -25\end{array}$

Sheh Lit Chang

University of Washington

Problem 62

Lenses with given radii. Object $O$ stands in front of a thin lens, on the central axis. For this situation, each problem in Table $34-7$ gives object distance $p$, index of refraction $n$ of the lens, radius $r_{1}$ of the nearer lens surface, and radius $r_{2}$ of the farther lens surface. (AII distances are in centimeters.) Find (a) the image distance $i$ and (b) the lateral magnification $m$ of the object, including signs. Also, determine whether the image is (c) real (R) or virtual (V), (d) inverted (I) from object $O$ or noninverted (NI), and (e) on the same side of the lens as object $O$ or on the opposite side.
$\begin{array}{lllll}62 & +10 & 1.50 & +30 & -30\end{array}$

Sheh Lit Chang

University of Washington

Problem 63

Lenses with given radii. Object $O$ stands in front of a thin lens, on the central axis. For this situation, each problem in Table $34-7$ gives object distance $p$, index of refraction $n$ of the lens, radius $r_{1}$ of the nearer lens surface, and radius $r_{2}$ of the farther lens surface. (AII distances are in centimeters.) Find (a) the image distance $i$ and (b) the lateral magnification $m$ of the object, including signs. Also, determine whether the image is (c) real (R) or virtual (V), (d) inverted (I) from object $O$ or noninverted (NI), and (e) on the same side of the lens as object $O$ or on the opposite side.
$\begin{array}{lllll}\mathbf{6 3} & +35 & 1.70 & +42 & +33\end{array}$

Sheh Lit Chang

University of Washington

Problem 64

Lenses with given radii. Object $O$ stands in front of a thin lens, on the central axis. For this situation, each problem in Table $34-7$ gives object distance $p$, index of refraction $n$ of the lens, radius $r_{1}$ of the nearer lens surface, and radius $r_{2}$ of the farther lens surface. (AII distances are in centimeters.) Find (a) the image distance $i$ and (b) the lateral magnification $m$ of the object, including signs. Also, determine whether the image is (c) real (R) or virtual (V), (d) inverted (I) from object $O$ or noninverted (NI), and (e) on the same side of the lens as object $O$ or on the opposite side.
$\begin{array}{lllll}\mathbf{6 4} & +10 & 1.50 & -30 & -60\end{array}$

Sheh Lit Chang

University of Washington

Problem 65

Lenses with given radii. Object $O$ stands in front of a thin lens, on the central axis. For this situation, each problem in Table $34-7$ gives object distance $p$, index of refraction $n$ of the lens, radius $r_{1}$ of the nearer lens surface, and radius $r_{2}$ of the farther lens surface. (AII distances are in centimeters.) Find (a) the image distance $i$ and (b) the lateral magnification $m$ of the object, including signs. Also, determine whether the image is (c) real (R) or virtual (V), (d) inverted (I) from object $O$ or noninverted (NI), and (e) on the same side of the lens as object $O$ or on the opposite side.
$\begin{array}{lllll}\mathbf{6 5} & +10 & 1.50 & -30 & +30\end{array}$

Sheh Lit Chang

University of Washington

Problem 66

Lenses with given radii. Object $O$ stands in front of a thin lens, on the central axis. For this situation, each problem in Table $34-7$ gives object distance $p$, index of refraction $n$ of the lens, radius $r_{1}$ of the nearer lens surface, and radius $r_{2}$ of the farther lens surface. (AII distances are in centimeters.) Find (a) the image distance $i$ and (b) the lateral magnification $m$ of the object, including signs. Also, determine whether the image is (c) real (R) or virtual (V), (d) inverted (I) from object $O$ or noninverted (NI), and (e) on the same side of the lens as object $O$ or on the opposite side.
$\begin{array}{lllll}66 & +18 & 1.60 & -27 & +24\end{array}$

Sheh Lit Chang

University of Washington

Problem 67

Lenses with given radii. Object $O$ stands in front of a thin lens, on the central axis. For this situation, each problem in Table $34-7$ gives object distance $p$, index of refraction $n$ of the lens, radius $r_{1}$ of the nearer lens surface, and radius $r_{2}$ of the farther lens surface. (AII distances are in centimeters.) Find (a) the image distance $i$ and (b) the lateral magnification $m$ of the object, including signs. Also, determine whether the image is (c) real (R) or virtual (V), (d) inverted (I) from object $O$ or noninverted (NI), and (e) on the same side of the lens as object $O$ or on the opposite side.
67
$\begin{array}{lllll}7 & +60 & 1.50 & +35 & -35\end{array}$

Sheh Lit Chang

University of Washington

Problem 68

In Fig. 34-44, a real inverted image $I$ of an object $O$ is formed by a particular lens (not shown); the object-image separation is $d=40.0$ $\mathrm{cm}$, measured along the central axis of the lens. The image is just half the size of the object. (a) What kind of lens must be used to produce this
image? (b) How far from the object must the lens be placed? (c) What is the focal length of the lens?

Averell Hause

Carnegie Mellon University

Problem 69

More lenses. Object $O$ stands on the central axis of a thin symmetric lens. For this situation, each problem in Table $34-8$ refers to (a) the lens type, converging (C) or diverging (D), (b) the focal distance $f$, (c) the object distance $p,(\mathrm{~d})$ the image distance $i$, and $(\mathrm{e})$ the lateral magnification $m$. (All distances are in centimeters.) It also refers to whether
(f) the image is real (R) or virtual (V), (g) inverted (I) or noninverted (NI) from $O$, and (h) on the same side of the lens as $O$ or on the opposite side. Fill in the missing information, including the value of $m$ when only an inequality is given. Where only a sign is missing, answer with the sign.
69
$+10 \quad+5.0$

Sheh Lit Chang

University of Washington

Problem 70

More lenses. Object $O$ stands on the central axis of a thin symmetric lens. For this situation, each problem in Table $34-8$ refers to (a) the lens type, converging (C) or diverging (D), (b) the focal distance $f$, (c) the object distance $p,(\mathrm{~d})$ the image distance $i$, and $(\mathrm{e})$ the lateral magnification $m$. (All distances are in centimeters.) It also refers to whether
(f) the image is real (R) or virtual (V), (g) inverted (I) or noninverted (NI) from $O$, and (h) on the same side of the lens as $O$ or on the opposite side. Fill in the missing information, including the value of $m$ when only an inequality is given. Where only a sign is missing, answer with the sign.
$\begin{array}{lllll}\mathbf{7 0} & 20 & +8.0 & <1.0 & \text { NI }\end{array}$

Sheh Lit Chang

University of Washington

Problem 71

More lenses. Object $O$ stands on the central axis of a thin symmetric lens. For this situation, each problem in Table $34-8$ refers to (a) the lens type, converging (C) or diverging (D), (b) the focal distance $f$, (c) the object distance $p,(\mathrm{~d})$ the image distance $i$, and $(\mathrm{e})$ the lateral magnification $m$. (All distances are in centimeters.) It also refers to whether
(f) the image is real (R) or virtual (V), (g) inverted (I) or noninverted (NI) from $O$, and (h) on the same side of the lens as $O$ or on the opposite side. Fill in the missing information, including the value of $m$ when only an inequality is given. Where only a sign is missing, answer with the sign.
$1+16 \quad+0.25$

Sheh Lit Chang

University of Washington

Problem 72

More lenses. Object $O$ stands on the central axis of a thin symmetric lens. For this situation, each problem in Table $34-8$ refers to (a) the lens type, converging (C) or diverging (D), (b) the focal distance $f$, (c) the object distance $p,(\mathrm{~d})$ the image distance $i$, and $(\mathrm{e})$ the lateral magnification $m$. (All distances are in centimeters.) It also refers to whether
(f) the image is real (R) or virtual (V), (g) inverted (I) or noninverted (NI) from $O$, and (h) on the same side of the lens as $O$ or on the opposite side. Fill in the missing information, including the value of $m$ when only an inequality is given. Where only a sign is missing, answer with the sign.
$72 \quad+16 \quad-0.25$

Sheh Lit Chang

University of Washington

Problem 73

More lenses. Object $O$ stands on the central axis of a thin symmetric lens. For this situation, each problem in Table $34-8$ refers to (a) the lens type, converging (C) or diverging (D), (b) the focal distance $f$, (c) the object distance $p,(\mathrm{~d})$ the image distance $i$, and $(\mathrm{e})$ the lateral magnification $m$. (All distances are in centimeters.) It also refers to whether
(f) the image is real (R) or virtual (V), (g) inverted (I) or noninverted (NI) from $O$, and (h) on the same side of the lens as $O$ or on the opposite side. Fill in the missing information, including the value of $m$ when only an inequality is given. Where only a sign is missing, answer with the sign.
$\begin{array}{lll}3 & +10 & -0.50\end{array}$

Sheh Lit Chang

University of Washington

Problem 74

More lenses. Object $O$ stands on the central axis of a thin symmetric lens. For this situation, each problem in Table $34-8$ refers to (a) the lens type, converging (C) or diverging (D), (b) the focal distance $f$, (c) the object distance $p,(\mathrm{~d})$ the image distance $i$, and $(\mathrm{e})$ the lateral magnification $m$. (All distances are in centimeters.) It also refers to whether
(f) the image is real (R) or virtual (V), (g) inverted (I) or noninverted (NI) from $O$, and (h) on the same side of the lens as $O$ or on the opposite side. Fill in the missing information, including the value of $m$ when only an inequality is given. Where only a sign is missing, answer with the sign.
$\begin{array}{ll}74 & \text { C }\end{array}$
10
$+20$

Sheh Lit Chang

University of Washington

Problem 75

More lenses. Object $O$ stands on the central axis of a thin symmetric lens. For this situation, each problem in Table $34-8$ refers to (a) the lens type, converging (C) or diverging (D), (b) the focal distance $f$, (c) the object distance $p,(\mathrm{~d})$ the image distance $i$, and $(\mathrm{e})$ the lateral magnification $m$. (All distances are in centimeters.) It also refers to whether
(f) the image is real (R) or virtual (V), (g) inverted (I) or noninverted (NI) from $O$, and (h) on the same side of the lens as $O$ or on the opposite side. Fill in the missing information, including the value of $m$ when only an inequality is given. Where only a sign is missing, answer with the sign.
$\mathbf{7 5}$
$\begin{array}{lll}10 & +5.0 & <1.0\end{array}$
Same

Sheh Lit Chang

University of Washington

Problem 76

More lenses. Object $O$ stands on the central axis of a thin symmetric lens. For this situation, each problem in Table $34-8$ refers to (a) the lens type, converging (C) or diverging (D), (b) the focal distance $f$, (c) the object distance $p,(\mathrm{~d})$ the image distance $i$, and $(\mathrm{e})$ the lateral magnification $m$. (All distances are in centimeters.) It also refers to whether
(f) the image is real (R) or virtual (V), (g) inverted (I) or noninverted (NI) from $O$, and (h) on the same side of the lens as $O$ or on the opposite side. Fill in the missing information, including the value of $m$ when only an inequality is given. Where only a sign is missing, answer with the sign.
76
$+5.0$
$>1.0$

Sheh Lit Chang

University of Washington

Problem 77

More lenses. Object $O$ stands on the central axis of a thin symmetric lens. For this situation, each problem in Table $34-8$ refers to (a) the lens type, converging (C) or diverging (D), (b) the focal distance $f$, (c) the object distance $p,(\mathrm{~d})$ the image distance $i$, and $(\mathrm{e})$ the lateral magnification $m$. (All distances are in centimeters.) It also refers to whether
(f) the image is real (R) or virtual (V), (g) inverted (I) or noninverted (NI) from $O$, and (h) on the same side of the lens as $O$ or on the opposite side. Fill in the missing information, including the value of $m$ when only an inequality is given. Where only a sign is missing, answer with the sign.
$\begin{array}{lll}77 & +16 & +1.25\end{array}$

Sheh Lit Chang

University of Washington

Problem 78

More lenses. Object $O$ stands on the central axis of a thin symmetric lens. For this situation, each problem in Table $34-8$ refers to (a) the lens type, converging (C) or diverging (D), (b) the focal distance $f$, (c) the object distance $p,(\mathrm{~d})$ the image distance $i$, and $(\mathrm{e})$ the lateral magnification $m$. (All distances are in centimeters.) It also refers to whether
(f) the image is real (R) or virtual (V), (g) inverted (I) or noninverted (NI) from $O$, and (h) on the same side of the lens as $O$ or on the opposite side. Fill in the missing information, including the value of $m$ when only an inequality is given. Where only a sign is missing, answer with the sign.
NI $\begin{array}{lll}78 & +10 & 0.50\end{array}$
0

Sheh Lit Chang

University of Washington

Problem 79

More lenses. Object $O$ stands on the central axis of a thin symmetric lens. For this situation, each problem in Table $34-8$ refers to (a) the lens type, converging (C) or diverging (D), (b) the focal distance $f$, (c) the object distance $p,(\mathrm{~d})$ the image distance $i$, and $(\mathrm{e})$ the lateral magnification $m$. (All distances are in centimeters.) It also refers to whether
(f) the image is real (R) or virtual (V), (g) inverted (I) or noninverted (NI) from $O$, and (h) on the same side of the lens as $O$ or on the opposite side. Fill in the missing information, including the value of $m$ when only an inequality is given. Where only a sign is missing, answer with the sign.
79
$20 \quad+8.0$
$>1.0$

Sheh Lit Chang

University of Washington

Problem 80

Two-lens systems. In Fig. $34-45$, stick figure $O$ (the object) stands on the common central axis of two thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closer to $O$, which is at object distance $p_{1}$. Lens 2 is mounted within the farther boxed region, at distance $d .$ Each problem in Table $34-9$ refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by $\mathrm{C}$ for converging and D for diverging; the number after C or $\mathrm{D}$ is the distance between a lens and either of its focal points (the proper sign of the focal distance is not indicated).
Find (a) the image distance $i_{2}$ for the image produced by lens 2 (the final image produced by the system) and (b) the overall lateral magnification $M$ for the system, including signs. Also, determine whether the final image is (c) real (R) or virtual (V),
(d) inverted
(I) from object $O$ or noninverted (NI), and
(e) on the same side of lens 2 as object $O$ or on the opposite side.
$\begin{array}{lllll}\mathbf{8 0} & +10 & \mathrm{C}, 15 & 10 & \mathrm{C}, 8.0\end{array}$

Eduard Sanchez

Numerade Educator

Problem 81

Two-lens systems. In Fig. $34-45$, stick figure $O$ (the object) stands on the common central axis of two thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closer to $O$, which is at object distance $p_{1}$. Lens 2 is mounted within the farther boxed region, at distance $d .$ Each problem in Table $34-9$ refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by $\mathrm{C}$ for converging and D for diverging; the number after C or $\mathrm{D}$ is the distance between a lens and either of its focal points (the proper sign of the focal distance is not indicated).
Find (a) the image distance $i_{2}$ for the image produced by lens 2 (the final image produced by the system) and (b) the overall lateral magnification $M$ for the system, including signs. Also, determine whether the final image is (c) real (R) or virtual (V),
(d) inverted
(I) from object $O$ or noninverted (NI), and
(e) on the same side of lens 2 as object $O$ or on the opposite side.
$\begin{array}{lllll}\mathbf{8 1} & +12 & \mathrm{C} .8 .0 & 32 & \mathrm{C}, 6.0\end{array}$

Eduard Sanchez

Numerade Educator

Problem 82

Two-lens systems. In Fig. $34-45$, stick figure $O$ (the object) stands on the common central axis of two thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closer to $O$, which is at object distance $p_{1}$. Lens 2 is mounted within the farther boxed region, at distance $d .$ Each problem in Table $34-9$ refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by $\mathrm{C}$ for converging and D for diverging; the number after C or $\mathrm{D}$ is the distance between a lens and either of its focal points (the proper sign of the focal distance is not indicated).
Find (a) the image distance $i_{2}$ for the image produced by lens 2 (the final image produced by the system) and (b) the overall lateral magnification $M$ for the system, including signs. Also, determine whether the final image is (c) real (R) or virtual (V),
(d) inverted
(I) from object $O$ or noninverted (NI), and
(e) on the same side of lens 2 as object $O$ or on the opposite side.
$\mathbf{8 2} \quad+8.0$
$\begin{array}{lll}\text { D. } 6.0 & 12 & \text { C, } 6.0\end{array}$

Eduard Sanchez

Numerade Educator

Problem 83

Two-lens systems. In Fig. $34-45$, stick figure $O$ (the object) stands on the common central axis of two thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closer to $O$, which is at object distance $p_{1}$. Lens 2 is mounted within the farther boxed region, at distance $d .$ Each problem in Table $34-9$ refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by $\mathrm{C}$ for converging and D for diverging; the number after C or $\mathrm{D}$ is the distance between a lens and either of its focal points (the proper sign of the focal distance is not indicated).
Find (a) the image distance $i_{2}$ for the image produced by lens 2 (the final image produced by the system) and (b) the overall lateral magnification $M$ for the system, including signs. Also, determine whether the final image is (c) real (R) or virtual (V),
(d) inverted
(I) from object $O$ or noninverted (NI), and
(e) on the same side of lens 2 as object $O$ or on the opposite side.
$\begin{array}{lllll}\mathbf{8 3} & +20 & \text { C } 9.0 & 8.0 & \text { C }, 5.0\end{array}$

Eduard Sanchez

Numerade Educator

Problem 84

Two-lens systems. In Fig. $34-45$, stick figure $O$ (the object) stands on the common central axis of two thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closer to $O$, which is at object distance $p_{1}$. Lens 2 is mounted within the farther boxed region, at distance $d .$ Each problem in Table $34-9$ refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by $\mathrm{C}$ for converging and D for diverging; the number after C or $\mathrm{D}$ is the distance between a lens and either of its focal points (the proper sign of the focal distance is not indicated).
Find (a) the image distance $i_{2}$ for the image produced by lens 2 (the final image produced by the system) and (b) the overall lateral magnification $M$ for the system, including signs. Also, determine whether the final image is (c) real (R) or virtual (V),
(d) inverted
(I) from object $O$ or noninverted (NI), and
(e) on the same side of lens 2 as object $O$ or on the opposite side.
$\begin{array}{lll}84 & +15 & \text { C. } 12\end{array}$
67
C. 10

Eduard Sanchez

Numerade Educator

Problem 85

Two-lens systems. In Fig. $34-45$, stick figure $O$ (the object) stands on the common central axis of two thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closer to $O$, which is at object distance $p_{1}$. Lens 2 is mounted within the farther boxed region, at distance $d .$ Each problem in Table $34-9$ refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by $\mathrm{C}$ for converging and D for diverging; the number after C or $\mathrm{D}$ is the distance between a lens and either of its focal points (the proper sign of the focal distance is not indicated).
Find (a) the image distance $i_{2}$ for the image produced by lens 2 (the final image produced by the system) and (b) the overall lateral magnification $M$ for the system, including signs. Also, determine whether the final image is (c) real (R) or virtual (V),
(d) inverted
(I) from object $O$ or noninverted (NI), and
(e) on the same side of lens 2 as object $O$ or on the opposite side.
$\begin{array}{lllll}\mathbf{8 5} & +4.0 & \text { C, } 6.0 & 8.0 & \text { D, } 6.0\end{array}$

Eduard Sanchez

Numerade Educator

Problem 86

Two-lens systems. In Fig. $34-45$, stick figure $O$ (the object) stands on the common central axis of two thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closer to $O$, which is at object distance $p_{1}$. Lens 2 is mounted within the farther boxed region, at distance $d .$ Each problem in Table $34-9$ refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by $\mathrm{C}$ for converging and D for diverging; the number after C or $\mathrm{D}$ is the distance between a lens and either of its focal points (the proper sign of the focal distance is not indicated).
Find (a) the image distance $i_{2}$ for the image produced by lens 2 (the final image produced by the system) and (b) the overall lateral magnification $M$ for the system, including signs. Also, determine whether the final image is (c) real (R) or virtual (V),
(d) inverted
(I) from object $O$ or noninverted (NI), and
(e) on the same side of lens 2 as object $O$ or on the opposite side.
$\begin{array}{lllll}86 & +12 & \text { C, } 8.0 & 30 & \text { D, } 8.0\end{array}$

Eduard Sanchez

Numerade Educator

Problem 87

Two-lens systems. In Fig. $34-45$, stick figure $O$ (the object) stands on the common central axis of two thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closer to $O$, which is at object distance $p_{1}$. Lens 2 is mounted within the farther boxed region, at distance $d .$ Each problem in Table $34-9$ refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by $\mathrm{C}$ for converging and D for diverging; the number after C or $\mathrm{D}$ is the distance between a lens and either of its focal points (the proper sign of the focal distance is not indicated).
Find (a) the image distance $i_{2}$ for the image produced by lens 2 (the final image produced by the system) and (b) the overall lateral magnification $M$ for the system, including signs. Also, determine whether the final image is (c) real (R) or virtual (V),
(d) inverted
(I) from object $O$ or noninverted (NI), and
(e) on the same side of lens 2 as object $O$ or on the opposite side.
$\begin{array}{lllll}\mathbf{8 7} & +20 & \mathrm{D}, 12 & 10 & \mathrm{D}, 8.0\end{array}$

Eduard Sanchez

Numerade Educator

Problem 88

If the angular magnification of an astronomical telescope is 36 and the diameter of the objective is $75 \mathrm{~mm}$, what is the minimum diameter of the eyepiece required to collect all the light entering the objective from a distant point source on the telescope axis?

Averell Hause

Carnegie Mellon University

Problem 89

In a microscope of the type shown in Fig. $34-20$, the focal length of the objective is $4.00 \mathrm{~cm}$, and that of the eyepiece is $8.00 \mathrm{~cm} .$ The distance between the lenses is $25.0 \mathrm{~cm} .$ (a) What is the tube length $s ?$ (b) If image $I$ in Fig. $34-20$ is to be just inside focal point $F_{1}^{\prime}$, how far from the objective should the object be? What then are (c) the lateral magnification $m$ of the objective,
(d) the angular magnification $m_{\theta}$ of the eyepiece, and (e) the overall magnification $M$ of the microscope?

Eduard Sanchez

Numerade Educator

Problem 90

Figure $34-46 a$ shows the basic structure of an old film camera. A lens can be moved forward or back to produce an image on film at the back of the camera. For a certain camera, with the distance $i$ between the lens and the film set at $f=5.0 \mathrm{~cm}$, parallel light rays from a very distant object $O$ converge to a point image on the film, as shown. The object is now brought closer, to a distance of $p=100 \mathrm{~cm}$, and the lens-film distance is adjusted so that an inverted real image forms on the film (Fig. $34-46 b$ ). (a) What is the lens-film distance $i$ now? (b) By how much was distance $i$ changed?

Eduard Sanchez

Numerade Educator

Problem 91

Figure $34-47 a$ shows the basic structure of a human eye. Light refracts into the eye through the cornea and is then further redirected by a lens whose shape (and thus ability to focus the light) is controlled by muscles. We can treat the cornea and eye lens as a single effective thin lens (Fig. $34-47 b$ ). A "normal" eye can focus parallel light rays from a distant object $O$ to a point on the retina at the back of the eye, where processing of the visual information begins. As an object is brought close to the eye, however, the muscles must change the shape of the lens so that rays form an inverted real image on the retina (Fig. $34-47 c$ ). (a) Suppose that for the parallel rays of Figs. $34.47 a$ and $b$, the focal length $f$ of the effective thin lens of the eye is $2.50 \mathrm{~cm}$. For an object at distance $p=$ $40.0 \mathrm{~cm}$, what focal length $f^{\prime}$ of the effective lens is required for the object to be seen clearly? (b) Must the eye muscles increase or decrease the radii of curvature of the eye lens to give focal length $f^{\prime} ?$

Eduard Sanchez

Numerade Educator

Problem 92

An object is $10.0 \mathrm{~mm}$ from the objective of a certain compound microscope. The lenses are $300 \mathrm{~mm}$ apart, and the intermediate image is $50.0 \mathrm{~mm}$ from the eyepiece. What overall magnification is produced by the instrument?

Averell Hause

Carnegie Mellon University

Problem 93

Someone with a near point $P_{n}$ of $25 \mathrm{~cm}$ views a thimble through a simple magnifying lens of focal length $10 \mathrm{~cm}$ by placing
the lens near his eye. What is the angular magnification of the thimble if it is positioned so that its image appears at (a) $P_{n}$ and
(b) infinity?

Zachary Warner

Numerade Educator

Problem 94

An object is placed against the center of a spherical mirror and then moved $70 \mathrm{~cm}$ from it along the central axis as the image distance $i$ is measured. Figure 34-48 gives $i$ versus object distance $p$ out to $p_{s}=40$ $\mathrm{cm}$. What is the image distance when the object is $70 \mathrm{~cm}$ from the mirror?

Averell Hause

Carnegie Mellon University

Problem 95

stick figure $O$ (the object) stands on the common central axis of three thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closest to $O$, which is at object distance $p_{1} .$ Lens 2 is mounted within the middle boxed region, at distance $d_{12}$ from lens $1 .$ Lens 3 is mounted in the farthest boxed region, at distance $d_{23}$ from lens $2 .$ Each problem in Table $34-10$ refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by $C$ for converging and $D$ for diverging:
the number after $\mathrm{C}$ or $\mathrm{D}$ is the distance between a lens and either of the focal points (the proper sign of the focal distance is not indicated).
Find (a) the image distance $i_{2}$ for the (final) image produced by lens 3 (the final image produced by the system) and (b) the overall lateral magnification $M$ for the system, including signs. Also, determine whether the final image is (c) real
(R) or virtual (V), (d) inverted (I) from object $O$ or noninverted (NI), and (e) on the same side of lens 3 as object $O$ or on the opposite side.
$\begin{array}{lllllll}95 & +12 & \mathrm{C}, 8.0 & 28 & \mathrm{C}, 6.0 & 8.0 & \mathrm{C}, 6.0\end{array}$

Eduard Sanchez

Numerade Educator

Problem 96

stick figure $O$ (the object) stands on the common central axis of three thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closest to $O$, which is at object distance $p_{1} .$ Lens 2 is mounted within the middle boxed region, at distance $d_{12}$ from lens $1 .$ Lens 3 is mounted in the farthest boxed region, at distance $d_{23}$ from lens $2 .$ Each problem in Table $34-10$ refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by $C$ for converging and $D$ for diverging:
the number after $\mathrm{C}$ or $\mathrm{D}$ is the distance between a lens and either of the focal points (the proper sign of the focal distance is not indicated).
Find (a) the image distance $i_{2}$ for the (final) image produced by lens 3 (the final image produced by the system) and (b) the overall lateral magnification $M$ for the system, including signs. Also, determine whether the final image is (c) real
(R) or virtual (V), (d) inverted (I) from object $O$ or noninverted (NI), and (e) on the same side of lens 3 as object $O$ or on the opposite side.
$\begin{array}{lllllll}96 & +4.0 & \mathrm{D}, 6.0 & 9.6 & \mathrm{C}, 6.0 & 14 & \mathrm{C}, 4.0\end{array}$

Eduard Sanchez

Numerade Educator

Problem 97

stick figure $O$ (the object) stands on the common central axis of three thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closest to $O$, which is at object distance $p_{1} .$ Lens 2 is mounted within the middle boxed region, at distance $d_{12}$ from lens $1 .$ Lens 3 is mounted in the farthest boxed region, at distance $d_{23}$ from lens $2 .$ Each problem in Table $34-10$ refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by $C$ for converging and $D$ for diverging:
the number after $\mathrm{C}$ or $\mathrm{D}$ is the distance between a lens and either of the focal points (the proper sign of the focal distance is not indicated).
Find (a) the image distance $i_{2}$ for the (final) image produced by lens 3 (the final image produced by the system) and (b) the overall lateral magnification $M$ for the system, including signs. Also, determine whether the final image is (c) real
(R) or virtual (V), (d) inverted (I) from object $O$ or noninverted (NI), and (e) on the same side of lens 3 as object $O$ or on the opposite side.
$\begin{array}{lllllll}97 & +18 & \mathrm{C}, 6.0 & 15 & \mathrm{C}, 3.0 & 11 & \mathrm{C}, 3.0\end{array}$

Eduard Sanchez

Numerade Educator

Problem 98

stick figure $O$ (the object) stands on the common central axis of three thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closest to $O$, which is at object distance $p_{1} .$ Lens 2 is mounted within the middle boxed region, at distance $d_{12}$ from lens $1 .$ Lens 3 is mounted in the farthest boxed region, at distance $d_{23}$ from lens $2 .$ Each problem in Table $34-10$ refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by $C$ for converging and $D$ for diverging:
the number after $\mathrm{C}$ or $\mathrm{D}$ is the distance between a lens and either of the focal points (the proper sign of the focal distance is not indicated).
Find (a) the image distance $i_{2}$ for the (final) image produced by lens 3 (the final image produced by the system) and (b) the overall lateral magnification $M$ for the system, including signs. Also, determine whether the final image is (c) real
(R) or virtual (V), (d) inverted (I) from object $O$ or noninverted (NI), and (e) on the same side of lens 3 as object $O$ or on the opposite side.
$\begin{array}{lllllll}98 & +2.0 & \mathrm{C}, 6.0 & 15 & \mathrm{C}, 6.0 & 19 & \mathrm{C}, 5.0\end{array}$

Eduard Sanchez

Numerade Educator

Problem 99

stick figure $O$ (the object) stands on the common central axis of three thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closest to $O$, which is at object distance $p_{1} .$ Lens 2 is mounted within the middle boxed region, at distance $d_{12}$ from lens $1 .$ Lens 3 is mounted in the farthest boxed region, at distance $d_{23}$ from lens $2 .$ Each problem in Table $34-10$ refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by $C$ for converging and $D$ for diverging:
the number after $\mathrm{C}$ or $\mathrm{D}$ is the distance between a lens and either of the focal points (the proper sign of the focal distance is not indicated).
Find (a) the image distance $i_{2}$ for the (final) image produced by lens 3 (the final image produced by the system) and (b) the overall lateral magnification $M$ for the system, including signs. Also, determine whether the final image is (c) real
(R) or virtual (V), (d) inverted (I) from object $O$ or noninverted (NI), and (e) on the same side of lens 3 as object $O$ or on the opposite side.
$\begin{array}{lllllll}99 & +8.0 & \mathrm{D}, 8.0 & 8.0 & \mathrm{D}, 16 & 5.1 & \mathrm{C}, 8.0\end{array}$

Eduard Sanchez

Numerade Educator

Problem 100

stick figure $O$ (the object) stands on the common central axis of three thin, symmetric lenses, which are mounted in the boxed regions. Lens 1 is mounted within the boxed region closest to $O$, which is at object distance $p_{1} .$ Lens 2 is mounted within the middle boxed region, at distance $d_{12}$ from lens $1 .$ Lens 3 is mounted in the farthest boxed region, at distance $d_{23}$ from lens $2 .$ Each problem in Table $34-10$ refers to a different combination of lenses and different values for distances, which are given in centimeters. The type of lens is indicated by $C$ for converging and $D$ for diverging:
the number after $\mathrm{C}$ or $\mathrm{D}$ is the distance between a lens and either of the focal points (the proper sign of the focal distance is not indicated).
Find (a) the image distance $i_{2}$ for the (final) image produced by lens 3 (the final image produced by the system) and (b) the overall lateral magnification $M$ for the system, including signs. Also, determine whether the final image is (c) real
(R) or virtual (V), (d) inverted (I) from object $O$ or noninverted (NI), and (e) on the same side of lens 3 as object $O$ or on the opposite side.
\begin{tabular}{lllllll}
$\mathbf{1 0 0}$ & $+4.0$ & $\mathbf{C}, 6.0$ & $8.0$ & $\mathrm{D}, 4.0$ & $5.7$ & $\mathrm{D}, 12$ \\
\hline
\end{tabular}

Eduard Sanchez

Numerade Educator

Problem 101

The formula $1 / p+1 / i=1 / f$ is called the Gaussian form of the thin-lens formula. Another form of this formula, the Newtonian form, is obtained by considering the distance $x$ from the object to the first focal point and the distance $x^{\prime}$ from the second focal point to the image. Show that $x x^{\prime}=f^{2}$ is the Newtonian form of the thin-lens formula.

Zachary Warner

Numerade Educator

Problem 102

Figure $34-50 a$ is an overhead view of two vertical plane mirrors with an object $O$ placed between them. If you look into the
mirrors, you see multiple images of $O$. You can find them by drawing the reflection in each mirror of the angular region between the mirrors, as is done in Fig. $34-50 b$ for the left-hand mirror. Then draw the reflection of the reflection. Continue this on the left and on the right until the reflections meet or overlap at the rear of the mirrors. Then you can count the number of images of $O .$ How many images of $O$ would you see if $\theta$ is (a) $90^{\circ}$, (b) $45^{\circ}$, and (c) $60^{\circ}$ ? If $\theta=120^{\circ}$, determine the (d) smallest and (e) largest number of images that can be seen, depending on your perspective and the location of $O$. (f) In each situation, draw the image locations and orientations as in Fig. $34-50 b$.

Averell Hause

Carnegie Mellon University

Problem 103

Two thin lenses of focal lengths $f_{1}$ and $f_{2}$ are in contact and share the same central axis. Show that, in image formation, they are equivalent to a single thin lens for which the focal length is $f=f_{1} f_{2} /\left(f_{1}+f_{2}\right)$

Zachary Warner

Numerade Educator

Problem 104

Two plane mirrors are placed parallel to each other and $40 \mathrm{~cm}$ apart. An object is placed $10 \mathrm{~cm}$ from one mirror. Determine the (a) smallest, (b) second smallest, (c) third smallest (occurs twice), and (d) fourth smallest distance between the object and image of the object.

Averell Hause

Carnegie Mellon University

Problem 105

In Fig. $34-51$, a box is somewhere at the left, on the central axis of the thin converging lens. The image $I_{m}$ of the box produced by the plane mirror is $4.00 \mathrm{~cm}$ "inside" the mirror. The lens-mirror separation is $10.0 \mathrm{~cm}$, and the focal length of the lens is $2.00 \mathrm{~cm}$. (a) What is the
distance between the box and the lens? Light reflected by the mirror travels back through the lens, which produces a final image of the box. (b) What is the distance between the lens and that final image?

Sheh Lit Chang

University of Washington

Problem 106

In Fig. $34-52$, an object is placed in front of a converging lens at a distance equal to twice the focal length $f_{1}$ of the lens. On the other side of the lens is a concave mirror of focal length $f_{2}$ separated from the lens by a distance $2\left(f_{1}+f_{2}\right) .$ Light from the object passes rightward through the lens, reflects from the mirror, passes leftward through the lens, and forms a final image of the object. What are (a) the distance between the lens and that final image and (b) the overall lateral magnification $M$ of the object? Is the image (c) real or virtual (if it is virtual, it requires someone looking through the lens toward the mirror), (d) to the left or right of the lens, and (e) inverted or noninverted relative to the object?

Averell Hause

Carnegie Mellon University

Problem 107

A fruit fly of height $H$ sits in front of lens 1 on the central axis through the lens. The lens forms an image of the fly at a distance $d=20 \mathrm{~cm}$ from the fly; the image has the fly's orientation and height $H_{l}=2.0 \mathrm{H}$. What are (a) the focal length $f_{1}$ of the lens and (b) the object distance $p_{1}$ of the fly? The fly then leaves lens 1 and sits in front of lens 2, which also forms an image at $d=20 \mathrm{~cm}$ that has the same orientation as the fly, but now $H_{I}=0.50 \mathrm{H}$. What are (c) $f_{2}$ and (d) $p_{2}$ ?

Eduard Sanchez

Numerade Educator

Problem 108

You grind the lenses shown in Fig. $34-53$ from flat glass disks $(n=$ 1.5) using a machine that can grind a radius of curvature of either $40 \mathrm{~cm}$ or $60 \mathrm{~cm} .$ In a lens where either radius is appropriate, you select the $40 \mathrm{~cm}$ radius. Then you hold each lens in sunshine to form an image of the Sun. What are the (a) focal length $f$ and
(b) image type (real or virtual) for (bi-convex) lens 1, (c) $f$ and (d) image type for (plane-convex) lens $2,(\mathrm{e}) f$ and (f) image type for (meniscus convex) lens $3,(\mathrm{~g}) f$ and $(\mathrm{h})$ image type
for (bi-concave) lens 4, (i) $f$ and (j) image type for (plane-concave) lens 5, and $(\mathrm{k}) f$ and (1) image type for (meniscus concave) lens $6 ?$

Eduard Sanchez

Numerade Educator

Problem 109

In Fig. $34-54$, a fish watcher at point $P$ watches a fish through a glass wall of a fish tank. The watcher is level with the fish; the index of refraction of the glass is $8 / 5$, and that of the water is $4 / 3$. The distances are $d_{1}=8.0 \mathrm{~cm}, d_{2}=3.0 \mathrm{~cm}$, and $d_{3}=$
$6.8 \mathrm{~cm}$. (a) To the fish, how far away does the watcher appear to be? (Hint: The watcher is the object. Light from that object passes
through the wall's outside surface, which acts as a refracting surface. Find the image produced by that surface. Then treat that image as an object whose light passes through the wall's inside surface, which acts as another refracting surface.) (b) To the watcher, how far away does the fish appear to be?

Zachary Warner

Numerade Educator

Problem 110

A goldfish in a spherical fish bowl of radius $R$ is at the level of the center $C$ of the bowl and at distance $R / 2$ from the glass (Fig. 34-55). What magnification of the fish is produced by the water in the bowl for a viewer looking along a line that includes the fish and the center, with the fish on the near side of the center? The index of refraction of the water is $1.33 .$ Neglect the glass wall of the bowl. Assume the viewer looks with one eye. (Hint:
Equation $34-5$ holds, but Eq. $34-6$ does not. You need to work with a ray diagram of the situation and assume that the rays are close to the observer's line of sight $-$ that is, they deviate from that line by only small angles.)

Averell Hause

Carnegie Mellon University

Problem 111

Figure $34-56$ shows a beam expander made with two coaxial converging lenses of focal lengths $f_{1}$ and $f_{2}$ and separation $d=f_{1}+f_{2}$. The device can expand a laser beam while keeping the light rays in the beam parallel to the central axis through the lenses. Suppose a uniform laser beam of width $W_{i}=2.5 \mathrm{~mm}$ and intensity $I_{i}=9.0 \mathrm{~kW} / \mathrm{m}^{2}$ enters a beam expander for which $f_{1}=12.5$ $\mathrm{cm}$ and $f_{2}=30.0 \mathrm{~cm}$. What are (a) $W_{f}$ and (b) $I_{f}$ of the beam leaving the expander? (c) What value of $d$ is needed for the beam expander if lens 1 is replaced with a diverging lens of focal length $f_{1}=-26.0 \mathrm{~cm} ?$

Eduard Sanchez

Numerade Educator

Problem 112

You look down at a coin that lies at the bottom of a pool of liquid of depth $d$ and index of refraction $n$ (Fig. $34-57$ ). Because you view with two eyes, which intercept different rays of light from the coin, you perceive the coin to be where extensions of the intercepted rays cross, at depth $d_{a}$ instead of $d$. Assuming that the intercepted rays in Fig. $34-57$ are close to a vertical axis through the coin, show that $d_{\mathrm{a}}$ $=d / n .$ (Hint: Use the small-angle approximation $\sin \theta \approx \tan \theta \approx \theta .$ )

Eduard Sanchez

Numerade Educator

Problem 113

A pinhole camera has the hole a distance $12 \mathrm{~cm}$ from the film plane, which is a rectangle of height $8.0 \mathrm{~cm}$ and width $6.0 \mathrm{~cm} .$ How far from a painting of dimensions $50 \mathrm{~cm}$ by $50 \mathrm{~cm}$ should the camera be placed so as to get the largest complete image possible on the film plane?

Zachary Warner

Numerade Educator

Problem 114

Light travels from point $A$ to point $B$ via reflection at point $O$ on the surface of a mirror. Without using calculus, show that length $A O B$ is a minimum when the angle of incidence $\theta$ is equal to the angle of reflection $\phi$. (Hint: Consider the image of $A$ in the mirror.)

Averell Hause

Carnegie Mellon University

Problem 115

A point object is $10 \mathrm{~cm}$ away from a plane mirror, and the eye of an observer (with pupil diameter $5.0 \mathrm{~mm}$ ) is $20 \mathrm{~cm}$ away. Assuming the eye and the object to be on the same line perpendicular to the mirror surface, find the area of the mirror used in observing the reflection of the point. (Hint: Adapt Fig. $34-4 .$ )

Zachary Warner

Numerade Educator

Problem 116

Show that the distance between an object and its real image formed by a thin converging lens is always greater than or equal to four times the focal length of the lens.

Averell Hause

Carnegie Mellon University

Problem 117

A luminous object and a screen are a fixed distance $D$ apart.
(a) Show that a converging lens of focal length $f$, placed between object and screen, will form a real image on the screen for two lens positions that are separated by a distance $d=\sqrt{D(D-4 f)}$.
(b) Show that
$$
\left(\frac{D-d}{D+d}\right)^{2}
$$
gives the ratio of the two image sizes for these two positions of the lens.

Zachary Warner

Numerade Educator

Problem 118

An eraser of height $1.0 \mathrm{~cm}$ is placed $10.0 \mathrm{~cm}$ in front of a two-lens system. Lens 1 (nearer the eraser) has focal length $f_{1}=$ $-15 \mathrm{~cm}$, lens 2 has $f_{2}=12 \mathrm{~cm}$, and the lens separation is $d=12 \mathrm{~cm} .$ For the image produced by lens 2, what are (a) the image distance $i_{2}$ (including sign), (b) the image height, (c) the image type (real or virtual), and (d) the image orientation (inverted relative to the eraser or not inverted)?

Eduard Sanchez

Numerade Educator

Problem 119

A peanut is placed $40 \mathrm{~cm}$ in front of a two-lens system:
lens 1 (nearer the peanut) has focal length $f_{1}=+20 \mathrm{~cm}$, lens 2 has $f_{2}=-15 \mathrm{~cm}$, and the lens separation is $d=10 \mathrm{~cm} .$ For the image produced by lens 2, what are (a) the image distance $i_{2}$ (including sign), (b) the image orientation (inverted relative to the peanut or not inverted), and (c) the image type (real or virtual)?
(d) What is the net lateral magnification?

Zachary Warner

Numerade Educator

Problem 120

A coin is placed $20 \mathrm{~cm}$ in front of a two-lens system. Lens 1 (nearer the coin) has focal length $f_{1}=+10 \mathrm{~cm}$, lens 2 has $f_{2}=$ $+12.5 \mathrm{~cm}$, and the lens separation is $d=30 \mathrm{~cm} .$ For the image produced by lens 2, what are (a) the image distance $i_{2}$ (including sign),
(b) the overall lateral magnification, (c) the image type (real or virtual), and (d) the image orientation (inverted relative to the coin or not inverted)?

Averell Hause

Carnegie Mellon University

Problem 121

An object is $20 \mathrm{~cm}$ to the left of a thin diverging lens that has a $30 \mathrm{~cm}$ focal length. (a) What is the image distance $i$ ? (b) Draw a ray diagram showing the image position.

Zachary Warner

Numerade Educator

Problem 122

In Fig $34-58$ a pinecone is at distance $p_{1}=1.0 \mathrm{~m}$ in front of a lens of focal length $f_{1}=0.50 \mathrm{~m} ;$ a flat mirror is at distance $d=2.0 \mathrm{~m}$ behind the lens. Light from the pinecone passes rightward through the lens. reflects from the mirror, passes leftward through the lens, and forms a final image of the pinecone. What
are (a) the distance between the lens and that image and (b) the overall lateral magnification of the pinecone? Is the image (c) real or virtual (if it is virtual, it requires someone looking through the lens toward the mirror), (d) to the left or right of the lens, and
(e) inverted relative to the pinecone or not inverted?

Cyra Jelle Calleja

Cyra Jelle Calleja

Numerade Educator

Problem 123

One end of a long glass rod $(n=1.5)$ is a convex surface of radius $6.0 \mathrm{~cm}$. An object is located in air along the axis of the rod, at a distance of $10 \mathrm{~cm}$ from the convex end. (a) How far apart are the
object and the image formed by the glass rod? (b) Within what range of distances from the end of the rod must the object be located in order to produce a virtual image?

Zachary Warner

Numerade Educator

Problem 124

A short straight object of length $L$ lies along the central axis of a spherical mirror, a distance $p$ from the mirror. (a) Show that its image in the mirror has a length $L^{\prime}$, where
$$
L^{\prime}=L\left(\frac{f}{p-f}\right)^{2}
$$
(Hint: Locate the two ends of the object.) (b) Show that the longitudinal magnification $m^{\prime}\left(=L^{\prime} / L\right)$ is equal to $m^{2}$, where $m$ is the lateral magnification.

Averell Hause

Carnegie Mellon University

Problem 125

Prove that if a plane mirror is rotated through an angle $\alpha$, the reflected beam is rotated through an angle $2 \alpha .$ Show that this result is reasonable for $\alpha=45^{\circ}$.

Zachary Warner

Numerade Educator

Problem 126

An object is $30.0 \mathrm{~cm}$ from a spherical mirror, along the mirror's central axis. The mirror produces an inverted image with a lateral magnification of absolute value $0.500 .$ What is the focal length of the mirror?

Averell Hause

Carnegie Mellon University

Problem 127

A concave mirror has a radius of curvature of $24 \mathrm{~cm}$. How far is an object from the mirror if the image formed is (a) virtual and $3.0$ times the size of the object, (b) real and $3.0$ times the size of the object, and (c) real and $1 / 3$ the size of the object?

Zachary Warner

Numerade Educator

Problem 128

A pepper seed is placed in front of a lens. The lateral magnification of the seed is $+0.300$. The absolute value of the lens's focal length is $40.0 \mathrm{~cm}$. How far from the lens is the image?

Averell Hause

Carnegie Mellon University

Problem 129

The equation $1 / p+1 / i=2 / r$ for spherical mirrors is an approximation that is valid if the image is formed by rays that make only small angles with the central axis. In reality, many of the angles are large, which smears the image a little. You can determine how much. Refer to Fig. $34-22$ and consider a ray that leaves a point source (the object) on the central axis and that makes an angle $\alpha$ with that axis.
First, find the point of intersection of the ray with the mirror. If the coordinates of this intersection point are $x$ and $y$ and the origin is placed at the center of curvature, then $y=(x+p-r) \tan \alpha$ and $x^{2}+y^{2}=r^{2}$, where $p$ is the object distance and $r$ is the mirror's radius of curvature. Next, use $\tan \beta=y / x$ to find the angle $\beta$ at the point of intersection, and then use $\alpha+\gamma=2 \beta$ to find the value of $\gamma$. Finally, use the relation tan $\gamma=y /(x+i-r)$ to find the distance $i$ of the image.
(a) Suppose $r=12 \mathrm{~cm}$ and $p=20 \mathrm{~cm} .$ For each of the following values of $\alpha$, find the position of the image $-$ that is, the position of the point where the reflected ray crosses the central axis: $0.500,0.100,0.0100 \mathrm{rad}$. Compare the results with those obtained with the equation $1 / p+1 / i=2 / r .$ (b) Repeat the calculations for $p=4.00 \mathrm{~cm}$.

Zachary Warner

Numerade Educator

Problem 130

A small cup of green tea is positioned on the central axis of a spherical mirror. The lateral magnification of the cup is $+0.250$, and the distance between the mirror and its focal point is $2.00 \mathrm{~cm}$.
(a) What is the distance between the mirror and the image it produces? (b) Is the focal length positive or negative?
(c) Is the image real or virtual?

Averell Hause

Carnegie Mellon University

Problem 131

A $20-\mathrm{mm}$ -thick layer of water $(n=1.33)$ floats on a $40-\mathrm{mm}$ -thick layer of carbon tetrachloride $(n=1.46)$ in a tank. $\mathrm{A}$ coin lies at the bottom of the tank. At what depth below the top water surface do you perceive the coin? (Hint: Use the result and assumptions of Problem 112 and work with a ray diagram.)

Zachary Warner

Numerade Educator

Problem 132

A millipede sits $1.0 \mathrm{~m}$ in front of the nearest part of the surface of a shiny sphere of diameter $0.70 \mathrm{~m}$. (a) How far from the surface does the millipede's image appear? (b) If the millipede's height is $2.0$ $\mathrm{mm}$, what is the image height? (c) Is the image inverted?

Averell Hause

Carnegie Mellon University

Problem 133

Show that if the object $O$ in Fig. $34-19 c$ is moved from focal point $F_{1}$ toward the observer's eye, the image moves in from infinity and the angle $\theta^{\prime}$ (and thus the angular magnification $m_{\theta}$ ) increases. (b) If you continue this process, where is the image when $m_{g}$ has its maximum usable value? (You can then still increase $m_{b}$, but the image will no longer be clear.) (c) Show that the maximum usable value of $m_{\theta}$ is $1+(25 \mathrm{~cm}) / f$. (d) Show that in this situation the angular magnification is equal to the lateral magnification.

Zachary Warner

Numerade Educator

Problem 134

Isaac Newton, having convinced himself (erroneously as it
turned out) that chromatic aberration is an inherent property of refracting telescopes, invented the reflecting telescope, shown schematically in Fig. $34-59 .$ He presented his second model of this telescope, with a magnifying power of 38 , to the Royal Society (of London), which still has it. In Fig. $34-59$ incident light falls, closely parallel to the telescope axis, on the objective mirror $M$. After reflection from small mirror $M^{\prime}$ (the figure is not to scale), the rays form a real, inverted image in the focal plane (the plane perpendicular to the line of sight, at focal point $F$. This image is then viewed through an eyepiece. (a) Show that the angular
magnification $m_{\theta}$ for the device is given by Eq. $34-15:$
$$
m_{\theta}=-f_{\mathrm{ob}} / f_{\mathrm{cy}}
$$
where $f_{o b}$ is the focal length of the objective mirror and $f_{e y}$ is that of the eyepiece. (b) The 200 in. mirror in the reflecting telescope at Mt. Palomar in California has a focal length of $16.8 \mathrm{~m}$. Estimate the size of the image formed by this mirror when the object is a meter stick $2.0 \mathrm{~km}$ away. Assume parallel incident rays. (c) The mirror of a different reflecting astronomical telescope has an effective radius of curvature of $10 \mathrm{~m}$ ("effective" because such mirrors are ground to a parabolic rather than a spherical shape, to eliminate spherical aberration defects). To give an angular magnification of 200, what must be the focal length of the eyepiece?

Averell Hause

Carnegie Mellon University

Problem 135

A narrow beam of parallel light rays is incident on a glass sphere from the left, directed toward the center of the sphere. (The sphere is a lens but certainly not a thin lens.) Approximate the angle of incidence of the rays as $0^{\circ}$, and assume that the index of refraction of the glass is $n<2.0 .$ (a) In terms of $n$ and the sphere radius $r$, what is the distance between the image produced by the sphere and the right side of the sphere? (b) Is the image to the left or right of that side? (Hint: Apply Eq. $34-8$ to locate the image that is produced by refraction at the left side of the sphere; then use that image as the object for refraction at the right side of the sphere to locate the final image. In the second refraction, is the object distance $p$ positive or negative?)

Zachary Warner

Numerade Educator

Problem 136

A corner reflector, much used in optical, microwave, and other applications, consists of three plane mirrors fastened together to form the corner of a cube. Show that after three reflections, an incident ray is returned with its direction exactly reversed.

Averell Hause

Carnegie Mellon University

Problem 137

A cheese enchilada is $4.00 \mathrm{~cm}$ in front of a converging lens. The magnification of the enchilada is $-2.00 .$ What is the focal length of the lens?

Zachary Warner

Numerade Educator

Problem 138

A grasshopper hops to a point on the central axis of a spherical mirror. The absolute magnitude of the mirror's focal length is $40.0 \mathrm{~cm}$, and the lateral magnification of the image produced by the mirror is $+0.200 .$ (a) Is the mirror convex or concave? (b) How far from the mirror is the grasshopper?

Averell Hause

Carnegie Mellon University

Problem 139

A grasshopper hops to a point on the central axis of a spherical mirror. The absolute magnitude of the mirror's focal length is $40.0 \mathrm{~cm}$, and the lateral magnification of the image produced by the mirror is $+0.200$. (a) Is the mirror convex or concave? (b) How far from the mirror is the grasshopper?

Averell Hause

Carnegie Mellon University

Problem 140

Suppose the farthest distance a person can see without visual aid is $50 \mathrm{~cm} .$ (a) What is the focal length of the corrective lens that will allow the person to see very far away? (b) Is the lens converging or diverging? (c) The power $P$ of a lens (in diopters) is equal to $1 / f$, where $f$ is in meters. What is $P$ for the lens?

Averell Hause

Carnegie Mellon University

Problem 141

A simple magnifier of focal length $f$ is placed near the eye of someone whose near point $P_{n}$ is $25 \mathrm{~cm}$. An object is positioned so that its image in the magnifier appears at $P_{n^{-}}$ (a) What is the angular magnification of the magnifier? (b) What is the angular magnification if the object is moved so that its image appears at infinity? For $f=10 \mathrm{~cm}$, evaluate the angular magnifications of (c) the situation in (a) and (d) the situation in (b). (Viewing an image at $P_{n}$ requires effort by muscles in the eye, whereas viewing an image at infinity requires no such effort for many people.)

Eduard Sanchez

Numerade Educator