Physics For Scientists And Engineers 6E - part 292

shows  such  a  telescope.  Note  that  in  the  reflecting  telescope  the  light  never  passes
through  glass  (except  through  the  small  eyepiece).  As  a  result,  problems  associated
with  chromatic  aberration  are  virtually  eliminated.  The  reflecting  telescope  can  be
made even shorter by orienting the flat mirror so that it reflects the light back toward
the  objective  mirror  and  the  light  enters  an  eyepiece  in  a  hole  in  the  middle  of  the
mirror.

The largest reflecting telescopes in the world are at the Keck Observatory on Mauna

Kea, Hawaii. The site includes two telescopes with diameters of 10 m, each containing
36 hexagonally shaped, computer-controlled mirrors that work together to form a large
reflecting surface. In contrast, the largest refracting telescope in the world, at the Yerkes
Observatory in Williams Bay, Wisconsin, has a diameter of only 1 m.

Summary

1165

The

lateral magnification M of the image due to a mirror or lens is defined as the

ratio of the image height h! to the object height h and is equal to the negative of the
ratio of the image distance q to the object distance p:

(36.1, 36.2)

In the paraxial ray approximation, the object distance p and image distance q for a

spherical mirror of radius R are related by the

mirror equation:

(36.4, 36.6)

where f " R/2 is the

focal length of the mirror.

An image can be formed by refraction from a spherical surface of radius R . The

object and image distances for refraction from such a surface are related by

(36.8)

where the light is incident in the medium for which the index of refraction is n

and is

refracted in the medium for which the index of refraction is n

The inverse of the

focal length f of a thin lens surrounded by air is given by the

lens makers’ equation:

(36.15)

Converging lenses have positive focal lengths, and diverging lenses have negative
focal lengths.

For a thin lens, and in the paraxial ray approximation, the object and image

distances are related by the

thin lens equation:

(36.16)

The ratio of the focal length of a camera lens to the diameter of the lens is called

the

f-number of the lens:

(36.18)

The intensity of light incident on the film in the camera varies according to:

(36.19)

I ,

(

f/D)

(

f -number)

f -number

1
p

1
f

(n $ 1)

1
p

1
f

M "

" $

S U M M A R Y

Take a practice test for

this chapter by clicking on
the Practice Test link at
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1166

C H A P T E R 3 6 • Image Formation

The maximum magnification of a single lens of focal length f used as a simple

magnifier is

(36.22)

The overall magnification of the image formed by a compound microscope is:

(36.24)

where f

and f

are the focal lengths of the objective and eyepiece lenses, respectively,

and L is the distance between the lenses.

The angular magnification of a refracting telescope can be expressed as

(36.25)

where f

and f

are the focal lengths of the objective and eyepiece lenses, respectively.

The angular magnification of a reflecting telescope is given by the same expression
where f

is the focal length of the objective mirror.

m " $

M " $

L
f

25 cm

max

1 &

25 cm

1. What is wrong with the caption of the cartoon shown in

Figure Q36.1?

5. Why does a clear stream, such as a creek, always appear to

be shallower than it actually is? By how much is its depth
apparently reduced?

6. Consider the image formed by a thin converging

lens. Under what conditions is the image (a) inverted,
(b) upright, (c) real, (d) virtual, (e) larger than the object,
and (f) smaller than the object?

7. Repeat Question 6 for a thin diverging lens.
8. Use the lens makers’ equation to verify the sign of the

focal length of each of the lenses in Figure 36.27.

9. If a solid cylinder of glass or clear plastic is placed above

the words LEAD OXIDE and viewed from above as shown

Q U E S T I O N S

Figure Q36.1 “Most mirrors reverse left and right. This one

reverses top and bottom.”

2. Consider a concave spherical mirror with a real object. Is

the image always inverted? Is the image always real? Give
conditions for your answers.

3. Repeat the preceding question for a convex spherical mirror.
4. Do the equations 1/p & 1/q " 1/f or M " $ q/p apply to

the image formed by a flat mirror? Explain your answer.

Figure Q36.9

Richard Megna/Fundamental Photographs, NYC

2003 Sidney Harris

Questions

1167

in Figure Q36.9, the LEAD appears inverted but the
OXIDE does not. Explain.

10. In Figure 36.28a, assume that the blue object arrow is

replaced by one that is much taller than the lens. How
many rays from the object will strike the lens? How many
principal rays can be drawn in a ray diagram?

11. A zip-lock plastic sandwich bag filled with water can act as

a crude converging lens in air. If the bag is filled with air
and placed under water, is the effective lens converging or
diverging?

12. Explain why a mirror cannot give rise to chromatic

aberration.
Why do some automobile mirrors have printed on them
the statement “Objects in mirror are closer than they
appear”? (See Fig. P36.19.)

14. Can a converging lens be made to diverge light if it is

placed into a liquid? What If? How about a converging
mirror?
Explain why a fish in a spherical goldfish bowl appears
larger than it really is.

16. Why do some emergency vehicles have the symbol

written on the front?

17. A lens forms an image of an object on a screen. What

happens to the image if you cover the top half of the lens
with paper?
Lenses used in eyeglasses, whether converging or diverg-
ing, are always designed so that the middle of the lens
curves away from the eye, like the center lenses of Figure
36.27a and b. Why?

19. Which glasses in Figure Q36.19 correct nearsightedness

and which correct farsightedness?

18.

AMBUL

15.

13.

22. In a Jules Verne novel, a piece of ice is shaped to form a mag-

nifying lens to focus sunlight to start a fire. Is this possible?

23. The f -number of a camera is the focal length of the lens

divided by its aperture (or diameter). How can the f -number
of the lens be changed? How does changing this number
affect the required exposure time?

24. A solar furnace can be constructed by using a concave

mirror to reflect and focus sunlight into a furnace enclo-
sure. What factors in the design of the reflecting mirror
would guarantee very high temperatures?

25. One method for determining the position of an image,

either real or virtual, is by means of parallax. If a finger or
other  object  is  placed  at  the  position  of  the  image,  as
shown  in  Figure  Q36.25,  and  the  finger  and  image  are
viewed  simultaneously  (the  image  is  viewed  through  the
lens  if  it  is  virtual),  the  finger  and  image  have  the  same
parallax; that is, if they are viewed from different positions,
the image will appear to move along with the finger. Use
this method to locate the image formed by a lens. Explain
why the method works.

Figure Q36.19

Finger

Image

Figure Q36.25

George Semple

20. A child tries on either his hyperopic grandfather’s or his

myopic brother’s glasses and complains that “everything
looks blurry.” Why do the eyes of a person wearing glasses
not look blurry? (See Figure Q36.19.)

21. Consider a spherical concave mirror, with the object

located to the left of the mirror beyond the focal point.
Using ray diagrams, show that the image moves to the left
as the object approaches the focal point.

26. Figure Q36.26 shows a lithograph by M. C. Escher titled

Hand with Reflection Sphere (Self-Portrait in Spherical Mirror).
Escher had this to say about the work: “The picture

Figure Q36.26

1168

C H A P T E R 3 6 • Image Formation

shows a spherical mirror, resting on a left hand. But as a
print  is  the  reverse  of  the  original  drawing  on  stone,  it
was  my  right  hand  that  you  see  depicted.  (Being  left-
handed,  I  needed  my  left  hand  to  make  the  drawing.)
Such  a  globe  reflection  collects  almost  one’s  whole
surroundings  in  one  disk-shaped  image.  The  whole
room,  four  walls,  the  floor,  and  the  ceiling,  everything,
albeit distorted, is compressed into that one small circle.
Your own head, or more exactly the point between your
eyes,  is  the  absolute  center.  No  matter  how  you  turn  or
twist  yourself,  you  can’t  get  out  of  that  central  point.
You are  immovably  the  focus,  the  unshakable  core,  of
your  world.”  Comment  on  the  accuracy  of  Escher’s
description.

27. You can make a corner reflector by placing three flat

mirrors in the corner of a room where the ceiling meets
the walls. Show that no matter where you are in the
room, you can see yourself reflected in the mirrors—
upside down.

28. A converging lens of short focal length can take light di-

verging from a small source and refract it into a beam of
parallel  rays.  A  Fresnel  lens,  as  shown  in  Figure  36.29,  is
used  for  this  purpose  in  a  lighthouse.  A  concave  mirror
can take light diverging from a small source and reflect it
into a beam of parallel rays. Is it possible to make a Fresnel
mirror?  Is  this  an  original  idea,  or  has  it  already  been
done? Suggestion: Look at the walls and ceiling of an audi-
torium.

Section 36.1 Images Formed by Flat Mirrors

1. Does your bathroom mirror show you older or younger

than you actually are? Compute an order-of-magnitude
estimate for the age difference, based on data that you
specify.

2. In a church choir loft, two parallel walls are 5.30 m apart.

The  singers  stand  against  the  north  wall.  The  organist
faces  the  south  wall,  sitting  0.800 m  away  from  it.  To
enable  her  to  see  the  choir,  a  flat  mirror  0.600 m  wide  is
mounted on the south wall, straight in front of her. What
width of the north wall can she see? Suggestion: Draw a top-
view diagram to justify your answer.
Determine the minimum height of a vertical flat mirror in
which  a  person  5!10. in  height  can  see  his  or  her  full
image. (A ray diagram would be helpful.)

Two  flat  mirrors  have  their  reflecting  surfaces  facing
each other, with the edge of one mirror in contact with
an  edge  of  the  other,  so  that  the  angle  between  the
mirrors  is  %. When  an  object  is  placed  between  the
mirrors,  a  number  of  images  are  formed.  In  general,  if
the angle % is such that n% " 360°, where n is an integer,
the number of images formed is n $ 1. Graphically, find
all  the  image  positions  for  the  case  n " 6  when  a  point
object  is  between  the  mirrors  (but  not  on  the  angle
bisector).

5. A person walks into a room with two flat mirrors on oppo-

site  walls,  which  produce  multiple  images.  When  the
person  is  located  5.00 ft  from  the  mirror  on  the  left  wall
and  10.0 ft  from  the  mirror  on  the  right  wall,  find  the
distance from the person to the first three images seen in
the mirror on the left.

A  periscope  (Figure  P36.6)  is  useful  for  viewing  objects
that  cannot  be  seen  directly.  It  finds  use  in  submarines
and  in  watching  golf  matches  or  parades  from  behind  a
crowd  of  people.  Suppose  that  the  object  is  a  distance

from the upper mirror and that the two flat mirrors

are separated by a distance h. (a) What is the distance of
the  final  image  from  the  lower  mirror?  (b)  Is  the  final
image  real  or  virtual?  (c)  Is  it  upright  or  inverted?
(d) What  is  its  magnification?  (e)  Does  it  appear  to  be
left–right reversed?

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= paired numerical and symbolic problems

P R O B L E M S

Figure P36.6

Section 36.2 Images Formed by Spherical Mirrors

7. A concave spherical mirror has a radius of curvature of

20.0 cm.  Find  the  location  of  the  image  for  object
distances of (a) 40.0 cm, (b) 20.0 cm, and (c) 10.0 cm. For
each  case,  state  whether  the  image  is  real  or  virtual  and
upright  or  inverted.  Find  the  magnification  in  each  case.

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