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Wave Fronts and Rays
Mirrors are usually close at hand. It is difficult, for example, to put on makeup,
shave, or drive a car without them. We see images in mirrors because some of the light that
strikes them is reflected into our eyes. To discuss reflection, it is necessary to introduce the
concepts of a wave front and a ray of light, and we can do so by taking advantage of the
familiar topic of sound waves (see Chapter 16). Both sound and light are kinds of waves.
Sound is a pressure wave, whereas light is electromagnetic in nature. However, the ideas
of a wave front and a ray apply to both.
Consider a small spherical object whose surface is pulsating in simple harmonic
motion. A sound wave is emitted that moves spherically outward from the object at a
constant speed. To represent this wave, we draw surfaces through all points of the wave
that are in the same phase of motion. These surfaces of constant phase are called wavefronts. Figure 25.1 shows a hemispherical view of the wave fronts. In this view they
appear as concentric spherical shells about the vibrating object. If the wave fronts are
drawn through the condensations, or crests, of the sound wave, as they are in the picture,
the distance between adjacent wave fronts equals the wavelength �. The radial lines pointing
outward from the source and perpendicular to the wave fronts are called rays. The rays
point in the direction of the velocity of the wave.
Figure 25.2a shows small sections of two adjacent spherical wave fronts. At large
distances from the source, the wave fronts become less and less curved and approach the
shape of flat surfaces, as in part b of the drawing. Waves whose wave fronts are flat surfaces
(i.e., planes) are known as plane waves and are important in understanding the properties
of mirrors and lenses. Since rays are perpendicular to the wave fronts, the rays for a plane
wave are parallel to each other.
The concepts of wave fronts and rays can also be used to describe light waves. For
light waves, the ray concept is particularly convenient when showing the path taken by the
light. We will make frequent use of light rays, which can be regarded essentially as narrow
beams of light much like those that lasers produce.
When you look into a plane (flat) mirror, you see an image of yourself that has
three properties:
1. The image is upright.
2. The image is the same size as you are.
3. The image is located as far behind the mirror as you are in front of it.
As Figure 25.5a illustrates, the image of yourself in the mirror is also reversed right to left
and left to right. If you wave your right hand, it is the left hand of the image that waves
back. Similarly, letters and words held up to a mirror are reversed. Ambulances and other
emergency vehicles are often lettered in reverse, as in Figure 25.5b, so that the letters will
appear normal when seen in the rearview mirror of a car.
To illustrate why an image appears to originate from behind a plane mirror, Figure 25.6ashows a light ray leaving the top of an object. This ray reflects from the mirror (angle of
reflection equals angle of incidence) and enters the eye. To the eye, it appears that the ray
originates from behind the mirror, somewhere back along the dashed line. Actually, rays
going in all directions leave each point on the object, but only a small bundle of such rays
is intercepted by the eye. Part b of the figure shows a bundle of two rays leaving the top of
the object. All the rays that leave a given point on the object, no matter what angle � they
have when they strike the mirror, appear to originate from a corresponding point on the
image behind the mirror (see the dashed lines in part b). For each point on the object, there
is a single corresponding point on the image, and it is this fact that makes the image in a
plane mirror a sharp and undistorted one.
Although rays of light seem to come from the image, it is evident from Figure 25.6bthat they do not originate from behind the plane mirror where the image appears to be.
Because none of the light rays actually emanate from the image, it is called a virtual image. In this text the parts of the light rays that appear to come from a virtual image are
represented by dashed lines. Curved mirrors, on the other hand, can produce images from
which all the light rays actually do emanate. Such images are known as real images and
are discussed later.
With the aid of the law of reflection, it is possible to show that the image is located as
far behind a plane mirror as the object is in front of it. In Figure 25.7 the object distance
is do and the image distance is d i. A ray of light leaves the base of the object, strikes the
mirror at an angle of incidence �, and is reflected at the same angle. To the eye, this ray
appears to come from the base of the image. For the angles �1 and �2 in the drawing it
follows that � � �1 � 90� and � � �2 � 90�. But the angle � is equal to the angle of
reflection �, since the two are opposite angles formed by intersecting lines. Therefore,
�1 � �2. As a result, triangles ABC and DBC are identical (congruent) because they share
770 ■ Chapter 25 The Reflection of Light: Mirrors
not. The pivoting action can occur as fast as 1000 times per second, leading to a series of
light pulses for each pixel that the eye and the brain combine and interpret as a continu-
ously changing image. Present-generation digital micromirror projectors use up to several
million micromirrors to reproduce each of the three primary colors (red, green, and blue)
25.3 The Formation of Images by a Plane Mirror ■ 771
a common side BC and have equal angles (�1 � �2) at the top and equal angles (90�) at
the base. Thus, the magnitude of the object distance do equals the magnitude of the image
distance d i.
By starting with a light ray from the top of the object, rather than the bottom, we can
use the same line of reasoning to show also that the height of the image equals the height
of the object.
Conceptual Examples 1 and 2 discuss some interesting features of plane mirrors.
B
90°
Normal
90°
1β 2β
αθ
θ
dido
A C D
H
M
D
C
P
B
AF
E
H D
CE
90°90°
θ
θ
(b)
Mirror 2Mirror 1
Image 1 Image 2
Image 3
Person
(a)
Figure 25.7 This drawing illustrates the
geometry used with a plane mirror to show
that the image distance d i equals the object
distance do.
Figure 25.8 A woman stands in front of a
plane mirror and sees her full image.
Full-Length Versus Half-Length Mirrors
In Figure 25.8 a woman is standing in front of a plane mirror. Is the minimum mirror height
that is necessary for her to see her full image (a) equal to her height, or (b) equal to one-half
her height?
Reasoning The woman sees her image because light emanating from her body is reflected by
the mirror (labeled ABCD in Figure 25.8) and enters her eyes. Consider, for example, a ray of
light from her foot F. This ray strikes the mirror at B and enters her eyes at E. According to the
law of reflection, the angles of incidence and reflection are both �. This law will allow us to
deduce how the height of the mirror is related to her own height.
Answer (a) is incorrect. The mirror in Figure 25.8 is the same height as the woman. Any
light from her foot that strikes the mirror below B is reflected toward a point on her body that
is below her eyes. Since light striking the mirror below B does not enter her eyes, the part
of the mirror between B and A may be removed. Thus, the necessary minimum height of the
mirror is not equal to the woman’s height.
Answer (b) is correct. As discussed above, the section AB of the mirror is not necessary
in order for the woman to see her full image. The section BC of the mirror that produces the
image is one-half the woman’s height between F and E. This follows because the right triangles
FBM and EBM are identical. They are identical because they share a common side BM and have
two angles, � and 90�, that are the same. The blowup in Figure 25.8 illustrates a similar line of
reasoning, starting with a ray from the woman’s head at H. This ray is reflected from the
mirror at P and enters her eyes. The top mirror section PD may be removed without disturbing
this reflection. The necessary section CP is one-half the woman’s height between her head at Hand her eyes at E. We find, then, that only the sections BC and CP are needed for the woman
to see her full length. The height of section BC plus section CP is exactly one-half the woman’s
height. The conclusions here are valid regardless of how far the person stands from the mirror.
Thus, to view one’s full length in a mirror, only a half-length mirror is needed.
Related Homework: Problems 7, 39 ■
Conceptual Example 1
Multiple Reflections
A person is sitting in front of two mirrors that intersect at a 90� angle. As Figure 25.9a illustrates,
the person sees three images of herself. (The person herself is only partially visible at the bottom
of the photo.) These images arise because rays of light emanate from her body, reflect from the
mirrors, and enter her eyes. Consider the light that enters her eyes and appears to come from each
of the three images identified in Figure 25.9b. The following table shows three possibilities for the
number of reflections that the light undergoes before entering her eyes. Which one is correct?
Reasoning Images of the woman are formed when light emanating from her body enters her
eyes after being reflected by one, or both, mirrors. For each reflection, the angle of the light
reflected from a mirror is equal to the angle of the light incident on the mirror (law of reflection).
We will see that there are three ways that light can reach her eyes from the two mirrors.
Answers (a) and (b) are incorrect. Figure 25.9b represents a top view of the person in
front of the two mirrors and has been repeated in the margin for convenience. It is a straight-
forward matter to understand two of the images that she sees. These are the images that are normally
seen when one sits in front of a mirror. Sitting in front of mirror 1, she sees image 1, which is
located as far behind that mirror as she is in front of it. She also sees image 2 behind mirror 2,
at a distance that matches her distance in front of that mirror. Each of these images arises from
light emanating from her body and reflecting only once from a single mirror. Therefore, each
ray of light does not reflect two or three times before entering her eyes.
Answer (c) is correct. As discussed above, images 1 and 2 arise, respectively, from singlereflections from mirrors 1 and 2. The third image arises when light undergoes two reflections
in sequence, first from one mirror and then from the other. When such a double reflection
occurs, an additional image becomes possible. Figure 25.9b shows two rays of light that strike
mirror 1. Each one, according to the law of reflection, has an angle of reflection that equals the
angle of incidence. The rays then strike mirror 2, where they again are reflected according to
the law of reflection. When the outgoing rays are extended backward (see the dashed lines in
the drawing), they intersect and appear to originate from image 3. Thus, the third image arises
when an incident ray of light is reflected twice, once from each mirror, before entering her eyes.
Related Homework: Problem 1 ■
Check Your Understanding
(The answers are given at the end of the book.)
1. The drawing shows a light ray undergoing multiple
reflections from a mirrored corridor. The walls of the
corridor are either parallel or perpendicular to one
another. If the initial angle of incidence is 35�, what
is the angle of reflection when the ray makes its last
reflection?
2. A sign painted on a store window is reversed when
viewed from inside the store. If a person inside the
store views the reversed sign in a plane mirror,
does the sign appear as it would when viewed from
outside the store? (Try it by writing some letters
on a transparent sheet of paper and then holding the
back side of the paper up to a mirror.)
3. If a clock is held in front of a mirror, its image is reversed left to right. From the point of view
of a person looking into the mirror, does the image of the second hand rotate in the reverse
(counterclockwise) direction?
Spherical Mirrors
The most common type of curved mirror is a spherical mirror. As Figure 25.10
shows, a spherical mirror has the shape of a section from the surface of a hollow sphere.
If the inside surface of the mirror is polished, it is a concave mirror. If the outside surface
is polished, it is a convex mirror. The drawing shows both types of mirrors, with a light
ray reflecting from the polished surface. The law of reflection applies, just as it does
for a plane mirror. For either type of spherical mirror, the normal is drawn perpendicular
to the mirror at the point of incidence. For each type, the center of curvature is located at
point C, and the radius of curvature is R. The principal axis of the mirror is a straight line
drawn through C and the midpoint of the mirror.
Figure 25.11 shows a tree in front of a concave mirror. A point on this tree lies on the
principal axis of the mirror and is beyond the center of curvature C. Light rays emanate
from this point and reflect from the mirror, consistent with the law of reflection. If the rays
are near the principal axis, they cross it at a common point after reflection. This point is
called the image point. The rays continue to diverge from the image point as if there were
an object there. Since light rays actually come from the image point, the image is a real
image.
If the tree in Figure 25.11 is infinitely far from the mirror, the rays are parallel to each
other and to the principal axis as they approach the mirror. Figure 25.12 shows rays near
and parallel to the principal axis, as they reflect from the mirror and pass through an
image point. In this special case the image point is referred to as the focal point F of the
mirror. Therefore, an object infinitely far away on the principal axis gives rise to an image
at the focal point of the mirror. The distance between the focal point and the middle of the
mirror is the focal length f of the mirror.
We can show that the focal point F lies halfway between the center of curvature C and
the middle of a concave mirror. In Figure 25.13, a light ray parallel to the principal axis
strikes the mirror at point A. The line CA is the radius of the mirror and, therefore, is the
normal to the spherical surface at the point of incidence. The ray reflects from the mirror,
and the angle of reflection � equals the angle of incidence. Furthermore, the angle ACFis also � because the radial line CA is a transversal of two parallel lines. Since two of
its angles are equal, the colored triangle CAF is an isosceles triangle; thus, sides CF and
FA are equal. However, when the incoming ray lies close to the principal axis, the angle of
incidence � is small, and the distance FA does not differ appreciably from the distance FB.
Therefore, in the limit that � is small, CF � FA � FB, and so the focal point F lies halfway
between the center of curvature and the mirror. In other words, the focal length f is one-
half of the radius R:
Focal length of a concave mirror (25.1)
Rays that lie close to the principal axis are known as paraxial rays,* and Equation 25.1
is valid only for such rays. Rays that are far from the principal axis do not converge
to a single point after reflection from the mirror, as Figure 25.14 shows. The result is
a blurred image. The fact that a spherical mirror does not bring all rays parallel to the
principal axis to a single image point is known as spherical aberration. Spherical aberra-
tion can be minimized by using a mirror whose height is small compared to the radius of
curvature.
A sharp image point can be obtained with a large mirror, if the mirror is parabolic in
shape instead of spherical. The shape of a parabolic mirror is such that all light rays
parallel to the principal axis, regardless of their distance from the axis, are reflected
through a single image point. However, parabolic mirrors are costly to manufacture and are
used where the sharpest images are required, as in research-quality telescopes.
f � 1
2 R
25.4 Spherical Mirrors ■ 773
C Principalaxis
Concavemirror
Imagepoint
C F Principalaxis
f
C F B
A
R
θ
θθ
C F
Figure 25.11 A point on the tree lies on the
principal axis of the concave mirror. Rays
from this point that are near the principal axis
are reflected from the mirror and cross the
axis at the image point.
Figure 25.12 Light rays near and parallel to
the principal axis are reflected from a concave
mirror and converge at the focal point F. The
focal length f is the distance between F and
the mirror.
Figure 25.13 This drawing is used to show
that the focal point F of a concave mirror is
halfway between the center of curvature Cand the mirror at point B.
Figure 25.14 Rays that are farthest from
the principal axis have the greatest angle of
incidence and miss the focal point F after
reflection from the mirror.*Paraxial rays are close to the principal axis but not necessarily parallel to it.
25.5 The Formation of Images by Spherical Mirrors ■ 775
5. The photograph shows an experimental device at
Sandia National Laboratories in New Mexico. This
device is a mirror that focuses sunlight to heat
sodium to a boil, which then heats helium gas in an
engine. The engine does the work of driving a
generator to produce electricity. The sodium unit
and the engine are labeled in the photo. (a) What
kind of mirror, concave or convex, is being used?
(b) Where is the sodium unit located relative to the
mirror? Express your answer in terms of the focal
length of the mirror.
6. Refer to Figure 25.14 and the related discussion
about spherical aberration. To bring the top ray
closer to the focal point F after reflection, describe
how you would change the shape of the mirror.
Would you open it up to produce a more gently
curving shape or bring the top and bottom edges
closer to the principal axis?
Sodium unitand engine
(Courtesy Sandia National Laboratories)
Question 5
The Formation of Images by Spherical Mirrors
As we have seen, some of the light rays emitted from an object in front of a
mirror strike the mirror, reflect from it, and form an image. We can analyze the image
produced by either concave or convex mirrors by using a graphical method called raytracing. This method is based on the law of reflection and the notion that a spherical
mirror has a center of curvature C and a focal point F. Ray tracing enables us to find the
location of the image, as well as its size, by taking advantage of the following fact: paraxial
rays leave from a point on the object and intersect, or appear to intersect, at a corresponding
point on the image after reflection.
■ Concave Mirrors
Three specific paraxial rays are especially convenient to use in the ray-tracing method.
Figure 25.17 shows an object in front of a concave mirror, and these three rays leave from
a point on the top of the object. The rays are labeled 1, 2, and 3, and when tracing their
paths, we use the following reasoning strategy.
25.5
Reasoning Strategy Ray Tracing for a Concave Mirror
Ray 1. This ray is initially parallel to the principal axis and, therefore, passes through the focal
point F after reflection from the mirror.
Ray 2. This ray initially passes through the focal point F and is reflected parallel to the principal
axis. Ray 2 is analogous to ray 1 except that the reflected, rather than the incident, ray is
parallel to the principal axis.
Ray 3. This ray travels along a line that passes through the center of curvature C and follows a
radius of the spherical mirror; as a result, the ray strikes the mirror perpendicularly and
reflects back on itself.Figure 25.17 The rays labeled 1, 2, and 3
If rays 1, 2, and 3 are superimposed on a scale drawing, they converge at a point on the
top of the image, as can be seen in Figure 25.18a.* Although three rays have been used here
to locate the image, only two are really needed; the third ray is usually drawn to serve as a
check. In a similar fashion, rays from all other points on the object locate corresponding points
on the image, and the mirror forms a complete image of the object. If you were to place your
eye as shown in the drawing, you would see an image that is larger and inverted relative to
the object. The image is real because the light rays actually pass through the image.
If the locations of the object and image in Figure 25.18a are interchanged, the situation
in part b of the drawing results. The three rays in part b are the same as those in part a,
except that the directions are reversed. These drawings illustrate the principle of reversibility,which states that if the direction of a light ray is reversed, the light retraces its originalpath. This principle is quite general and is not restricted to reflection from mirrors. The image
is real, and it is smaller and inverted relative to the object.
When the object is placed between the focal point F and a concave mirror, as in
Figure 25.19a, three rays can again be drawn to find the image. Now, however, ray 2 does
not go through the focal point on its way to the mirror, since the object is closer to the
mirror than the focal point is. When projected backward, though, ray 2 appears to come
from the focal point. Therefore, after reflection, ray 2 is directed parallel to the principal
axis. In this case the three reflected rays diverge from each other and do not converge to a
common point. However, when projected behind the mirror, the three rays appear to come
from a common point; thus, a virtual image is formed. This virtual image is larger than the
object and upright.The physics of makeup and shaving mirrors. Makeup and shaving mirrors are concave mirrors.
When you place your face between the mirror and its focal point, you see an enlarged
virtual image of yourself, as Figure 25.19b shows.
776 ■ Chapter 25 The Reflection of Light: Mirrors
Realimage
(a)
Object
C F
3
1
2
Object(b)
Realimage
C F
*In the drawings that follow, we assume that the rays are paraxial, although the distance between the rays and
the principal axis is often exaggerated for clarity.
25.6 The Mirror Equation and the Magnification Equation ■ 779
The Mirror Equation and
the Magnification Equation
Ray diagrams drawn to scale are useful for determining the location and size of the
image formed by a mirror. However, for an accurate description of the image, a more
analytical technique is needed, so we will derive two equations, known as the mirror equationand the magnification equation. These equations are based on the law of reflection and
provide relationships between:
f � the focal length of the mirror
do � the object distance, which is the distance between the object and the mirror
d i � the image distance, which is the distance between the image and the mirror
m � the magnification of the mirror, which is the ratio of the height of the image to the
height of the object.
■ Concave Mirrors
We begin our derivation of the mirror equation by referring to Figure 25.22a, which shows
a ray leaving the top of the object and striking a concave mirror at the point where the
principal axis intersects the mirror. Since the principal axis is perpendicular to the mirror,
it is also the normal at this point of incidence. Therefore, the ray reflects at an equal angle
and passes through the image. The two colored triangles are similar triangles because they
have equal angles, so
where ho is the height of the object and hi is the height of the image. The minus sign appears
on the left in this equation because the image is inverted in Figure 25.22a. In part b another
ray leaves the top of the object, this one passing through the focal point F, reflecting parallel
to the principal axis, and then passing through the image. Provided the ray remains close to
the axis, the two colored areas can be considered to be similar triangles, with the result that
Setting the two equations above equal to each other yields do/d i � (do � f ) / f. Rearranging
this result gives the mirror equation:
Mirror equation
(25.3)
We have derived this equation for a real image formed in front of a concave mirror. In
this case, the image distance is a positive quantity, as are the object distance and the focal
length. However, we have seen in the last section that a concave mirror can also form a
virtual image, if the object is located between the focal point and the mirror. Equation 25.3
can also be applied to such a situation, provided that we adopt the convention that d i is
negative for an image behind the mirror, as it is for a virtual image.
In deriving the magnification equation, we remember that the magnification m of a
mirror is the ratio of the image height to the object height: m � h i/ho. If the image height
is less than the object height, the magnitude of m is less than one, and if the image is larger
than the object, the magnitude of m is greater than one. We have already shown that
ho/ (�h i) � do/d i , so it follows that
Magnificationequation (25.4)
As Examples 3 and 4 show, the value of m is negative if the image is inverted and positive
25.6 The Mirror Equation and the Magnification Equation ■ 781
■ Convex Mirrors
The mirror equation and the magnification equation can also be used with convex mirrors,
provided the focal length f is taken to be a negative number, as indicated explicitly in
Equation 25.2. One way to remember this is to recall that the focal point of a convex mirror
lies behind the mirror. Example 5 deals with a convex mirror.
A Virtual Image Formed by a Convex Mirror
A convex mirror is used to reflect light from an object placed 66 cm in front of the mirror. The
focal length of the mirror is f � �46 cm (note the minus sign). Find (a) the location of the
image and (b) the magnification.
Reasoning We have seen that a convex mirror always forms a virtual image, as in
Figure 25.21a, where the image is upright and smaller than the object. These characteristics
should also be indicated by the results of our analysis here.
Solution (a) With do � 66 cm and f � �46 cm, the mirror equation gives
The negative sign for di indicates that the image is behind the mirror and, therefore, is a virtual
image.
(b) According to the magnification equation, the magnification is
The image is smaller (m is less than one) and upright (m is positive) with respect to the object.
■
0.41m � �d i
do
� �(�27 cm)
66 cm�
di � �27 cm1
di
�1
f�
1
do
�1
�46 cm�
1
66 cm� �0.037 cm�1 or
Example 5
Convex mirrors, like plane (flat) mirrors, always produce virtual images behind the
mirror. However, the virtual image in a convex mirror is closer to the mirror than it would
be if the mirror were planar, as Example 6 illustrates.
A Convex Versus a Plane Mirror
An object is placed 9.00 cm in front of a mirror. The image is 3.00 cm closer to the mirror when
the mirror is convex than when it is planar (see Figure 25.23). Find the focal length of the
convex mirror.
Reasoning For a plane mirror, the image and the object are the same distance on either side
of the mirror. Thus, the image would be 9.00 cm behind a plane mirror. If the image in a
convex mirror is 3.00 cm closer than this, the image must be located 6.00 cm behind the convex
mirror. In other words, when the object distance is do � 9.00 cm, the image distance for the
convex mirror is d i � �6.00 cm (negative because the image is virtual). The mirror equation
can be used to find the focal length of the mirror.
Solution According to the mirror equation, the reciprocal of the focal length is
■
f � �18 cm1
f�
1
do
�1
d i
�1
9.00 cm�
1
�6.00 cm� �0.056 cm�1 or
Example 6
Contact lenses are worn to correct vision problems. Optometrists take advantage of
the mirror equation and the magnification equation in providing lenses that fit the patient’s
eyes properly, as the next example illustrates.
9.00 cm3.00 cm
9.00 cm 9.00 cm
Figure 25.23 The object distance (9.00 cm)
is the same for the plane mirror (top part of
drawing) as for the convex mirror (bottom
part of drawing). However, as discussed in
Example 6, the image formed by the convex
mirror is 3.00 cm closer to the mirror.
■ Problem-Solving Insight. When using the mirror equation, it is useful toconstruct a ray diagram to guide your thinking and to check your calculation.
25.3 The Formation of Images by a Plane Mirror A virtual image is one from which all the
rays of light do not actually come, but only appear to do so. A real image is one from which all the
rays of light actually do emanate.
A plane mirror forms an upright, virtual image that is located as far behind the mirror as the
object is in front of it. In addition, the heights of the image and the object are equal.
25.4 Spherical Mirrors A spherical mirror has the shape of a section from the surface of a
hollow sphere. If the inside surface of the mirror is polished, it is a concave mirror. If the outside
surface is polished, it is a convex mirror.
The principal axis of a mirror is a straight line drawn through the center of curvature and the
middle of the mirror’s surface. Rays that are close to the principal axis are known as paraxial rays.
Paraxial rays are not necessarily parallel to the principal axis. The radius of curvature R of a mirror
is the distance from the center of curvature to the mirror.
The focal point of a concave spherical mirror is a point on the principal axis, in front of the
mirror. Incident paraxial rays that are parallel to the principal axis converge to the focal point after
being reflected from the concave mirror.
The focal point of a convex spherical mirror is a point on the principal axis, behind the mirror.
For a convex mirror, incident paraxial rays that are parallel to the principal axis diverge after reflecting
from the mirror. These rays seem to originate from the focal point.
The fact that a spherical mirror does not bring all rays parallel to the principal axis to a single
image point after reflection is known as spherical aberration.
The focal length f indicates the distance along the principal axis between the focal point and
the mirror. The focal length and the radius of curvature R are related by Equations 25.1 and 25.2.
25.5 The Formation of Images by Spherical Mirrors The image produced by a mirror can
be located by a graphical method known as ray tracing.
For a concave mirror, the following paraxial rays are useful for ray tracing (see Figure 25.17):
Ray 1. This ray leaves the object traveling parallel to the principal axis. The ray reflects from the
mirror and passes through the focal point.
Ray 2. This ray leaves the object and passes through the focal point. The ray reflects from the
mirror and travels parallel to the principal axis.
Ray 3. This ray leaves the object and travels along a line that passes through the center of curvature.
The ray strikes the mirror perpendicularly and reflects back on itself.
For a convex mirror, the following paraxial rays are useful for ray tracing (see Figure 25.21a):
Ray 1. This ray leaves the object traveling parallel to the principal axis. After reflection from the
mirror, the ray appears to originate from the focal point of the mirror.
Ray 2. This ray leaves the object and heads toward the focal point. After reflection, the ray travels
parallel to the principal axis.
Ray 3. This ray leaves the object and travels toward the center of curvature. The ray strikes the
mirror perpendicularly and reflects back on itself.
25.6 The Mirror Equation and the Magnification Equation The mirror equation
(Equation 25.3) specifies the relation between the object distance do, the image distance d i, and the
focal length f of the mirror. The mirror equation can be used with either concave or convex mirrors.
The magnification m of a mirror is the ratio of the image height h i to the object height ho:
The magnification is also related to d i and do by the magnification equation (Equation 25.4). The
algebraic sign conventions for the variables appearing in these equations are summarized in the
Reasoning Strategy at the end of Section 25.6.
m � h i /ho.
(Concave mirror) (25.1)
(Convex mirror) (25.2)f � �1
2R
f � 1
2R
(25.3)1
do
�1
di
�1
f
Note to Instructors: The numbering of the questions shown here reflects the fact that they are only a representative subset of the total number that are available online. However, all of the questions are available for assignment via an online homework management program such as WileyPLUS or WebAssign.
Section 25.1 Wave Fronts and Rays
2. A ray is _______. (a) always parallel to other rays (b) parallel to
the velocity of the wave (c) perpendicular to the velocity of the wave
(d) parallel to the wave fronts
Section 25.2 The Reflection of Light
4. The drawing shows a top view of an object located to the right of a
mirror. A single ray of light is shown leaving the object. After reflection
from the mirror, through which location, A, B, C, or D, does the ray pass?
14. An object is placed at a known distance in front of a mirror whose
focal length is also known. You apply the mirror equation and find that the
image distance is a negative number. This result tells you that _______.
(a) the image is larger than the object (b) the image is smaller than the
object (c) the image is inverted relative to the object (d) the image is
real (e) the image is virtual
15. An object is situated at a known distance in front of a convex mirror
whose focal length is also known. A friend of yours does a calculation
that shows that the magnification is �2. After some thought, you
conclude correctly that _______. (a) your friend’s answer is correct
(b) the magnification should be �2 (c) the magnification should be
(d) the magnification should be �1
2
�1
2
A
C
D
B
Mirror Object
Ray
Question 4
Note to Instructors: Most of the homework problems in this chapter are available for assignment via an online homework management program such asWileyPLUS or WebAssign, and those marked with the icons and are presented in WileyPLUS using a guided tutorial format that provides enhanced interactivity. See the Preface for additional details.
ssm Solution is in the Student Solutions Manual. This icon represents a biomedical application.
mmh Problem-solving help is available online at www.wiley.com/college/cutnell.
Problems
Section 25.2 The Reflection of Light
Section 25.3 The Formation of Images by a Plane Mirror
1. Review Conceptual Example 2. Suppose that in Figure 25.9b the two
perpendicular plane mirrors are represented by the �x and �y axes
of an x, y coordinate system; mirror 1 is the �x axis, and mirror 2 is
the �y axis. An object is in front of these mirrors at a point whose
coordinates are x � �2.0 m and y � �1.0 m. Find the coordinates that
locate each of the three images.
2. On the �y axis a laser is located at
y � �3.0 cm. The coordinates of a
small target are x � �9.0 cm and
y � �6.0 cm. The �x axis represents
the edge-on view of a plane mirror. At
what point on the �x axis should the
laser be aimed in order for the laser
light to hit the target after reflection?
3. ssm You are trying to photograph
a bird sitting on a tree branch, but a
tall hedge is blocking your view.
However, as the drawing shows, a
plane mirror reflects light from the
bird into your camera. For what
distance must you set the focus of the
camera lens in order to snap a sharp
picture of the bird’s image?
4. Suppose that you are walking perpendicularly with a velocity of
�0.90 m/s toward a stationary plane mirror. What is the velocity of your
image relative to you? The direction in which you walk is the positive
direction.
5. ssm Two plane mirrors are separated by 120�, as the drawing
illustrates. If a ray strikes mirror M1 at a 65� angle of incidence, at what
angle � does it leave mirror M2?
6. The drawing shows a laser beam
shining on a plane mirror that is
perpendicular to the floor. The
beam’s angle of incidence is 33.0�.The beam emerges from the laser at a
point that is 1.10 m from the mirror
and 1.80 m above the floor. After
reflection, how far from the base of the
mirror does the beam strike the floor?
65°120°
θM2
M1
4.3 m
2.1 m
3.7 m
Problem 3
1.10 m
33.0°1.80 m
Floor
What does the image of these words look like? (a) DOG TOP(b) (c) TOP DOG (d) DOG TOP TOP DOG(e)
Section 25.4 Spherical Mirrors
8. Rays of light coming from the sun (a very distant object) are near and
parallel to the principal axis of a concave mirror. After reflecting from the
mirror, where will the rays cross each other at a single point? The rays
_________. (a) will not cross each other after reflecting from a concave
mirror (b) will cross at the point where the principal axis intersects the
mirror (c) will cross at the center of curvature (d) will cross at the
focal point (e) will cross at a point beyond the center of curvature
26. A convex mirror has a focal length of 27.0 cm. Find the
magnification produced by the mirror when the object distance is 9.0 cm
and 18.0 cm.
27. ssm When viewed in a spherical mirror, the image of a setting sun
is a virtual image. The image lies 12.0 cm behind the mirror. (a) Is the
mirror concave or convex? Why? (b) What is the radius of curvature of
the mirror?
28. A concave mirror has a focal length of 12 cm. This mirror
forms an image located 36 cm in front of the mirror. What is the magni-
fication of the mirror?
* 29. An object is located 14.0 cm in front of a convex mirror, the
image being 7.00 cm behind the mirror. A second object, twice as tall as
the first one, is placed in front of the mirror, but at a different location.
The image of this second object has the same height as the other image.
How far in front of the mirror is the second object located?
* 30. A dentist’s mirror is placed 2.0 cm from a tooth. The
enlarged image is located 5.6 cm behind the mirror. (a) What
kind of mirror (plane, concave, or convex) is being used? (b) Determine
the focal length of the mirror. (c) What is the magnification? (d) How
is the image oriented relative to the object?
* 31. A tall tree is growing across a river from you. You would like to
know the distance between yourself and the tree, as well as its height, but
are unable to make the measurements directly. However, by using a
mirror to form an image of the tree and then measuring the image
distance and the image height, you can calculate the distance to the tree
�
Additional Problems ■ 789
as well as its height. Suppose that this mirror produces an image of
the sun, and the image is located 0.9000 m from the mirror. The same
mirror is then used to produce an image of the tree. The image of the tree
is 0.9100 m from the mirror. (a) How far away is the tree? (b) The
image height of the tree has a magnitude of 0.12 m. How tall is the tree?
* 32. A spherical mirror is polished on both sides. When the concave side
is used as a mirror, the magnification is �2.0. What is the magnification
when the convex side is used as a mirror, the object remaining the same
distance from the mirror?
* 33. Consult Multiple-Concept Example 7 to see a model for solving
this type of problem. A concave makeup mirror is designed so the virtual
image it produces is twice the size of the object when the distance
between the object and the mirror is 14 cm. What is the radius of curvature
of the mirror?
** 34. A concave mirror has a focal length of 30.0 cm. The distance between
an object and its image is 45.0 cm. Find the object and image distances,
assuming that (a) the object lies beyond the center of curvature and
(b) the object lies between the focal point and the mirror.
** 35. ssm A spacecraft is in a circular orbit about the moon, 1.22 � 105 m
above its surface. The speed of the spacecraft is 1620 m/s, and the radius
of the moon is 1.74 � 106 m. If the moon were a smooth, reflective
sphere, (a) how far below the moon’s surface would the image of the
spacecraft appear, and (b) what would be the apparent speed of the
spacecraft’s image? (Hint: Both the spacecraft and its image have thesame angular speed about the center of the moon.)
Additional Problems36. An object is placed in front of a convex mirror. Draw the convex
mirror (radius of curvature � 15 cm) to scale, and place the object 25 cm
in front of it. Make the object height 4 cm. Using a ray diagram, locate the
image and measure its height. Now move the object closer to the mirror, so
the object distance is 5 cm. Again, locate its image using a ray diagram. As
the object moves closer to the mirror, (a) does the magnitude of the
image distance become larger or smaller, and (b) does the magnitude of
the image height become larger or smaller? (c) What is the ratio of the
image height when the object distance is 5 cm to its height when the
object distance is 25 cm? Give your answer to one significant figure.
37. ssm An object that is 25 cm in front of a convex mirror has an
image located 17 cm behind the mirror. How far behind the mirror is the
image located when the object is 19 cm in front of the mirror?
38. A concave mirror has a focal length of 42 cm. The image formed
by this mirror is 97 cm in front of the mirror. What is the object distance?
39. Review Conceptual Example 1 before attempting this problem. A
person whose eyes are 1.70 m above the floor stands in front of a plane
mirror. The top of her head is 0.12 m above her eyes. (a) What is
the height of the shortest mirror in which she can see her entire image?
(b) How far above the floor should the bottom edge of the mirror be
placed?
40. A drop of water on a countertop reflects light from a flower
held 3.0 cm directly above it. The flower’s diameter is 2.0 cm, and the
diameter of the flower’s image is 0.10 cm. What is the focal length of the
water drop, assuming that it may be treated as a convex spherical mirror?
41. ssm A small postage stamp is placed in front of a concave mirror (radius � R) so that the image distance equals the object distance. (a) Interms of R, what is the object distance? (b) What is the magnification ofthe mirror? (c) State whether the image is upright or inverted relative tothe object. Draw a ray diagram to guide your thinking.
* 42. Identical objects are located at the same distance from two
spherical mirrors, A and B. The magnifications produced by the mirrors
are mA � 4.0 and mB � 2.0. Find the ratio
fA/ fB of the focal lengths of the mirrors.
* 43. You walk at an angle of � � 50.0�toward a plane mirror, as in the drawing. Your
walking velocity has a magnitude of 0.90 m/s.
What is the velocity of your image relative to
you (magnitude and direction)?
* 44. A candle is placed 15.0 cm in front of a convex mirror. When the
convex mirror is replaced with a plane mirror, the image moves 7.0 cm
farther away from the mirror. Find the focal length of the convex mirror.
* 45. ssm mmh An object is placed in front of a convex mirror, and the
size of the image is one-fourth that of the object. What is the ratio do/ f of
the object distance to the focal length of the mirror?
** 46. A man holds a double-sided spherical mirror so that he is looking
directly into its convex surface, 45 cm from his face. The magnification
of the image of his face is �0.20. What will be the image distance when
he reverses the mirror (looking into its concave surface), maintaining the
same distance between the mirror and his face? Be sure to include the
algebraic sign (� or �) with your answer.
** 47. ssm A lamp is twice as far in front of a plane mirror as a person is.
Light from the lamp reaches the person via two paths, reflected and
direct. It strikes the mirror at a 30.0� angle of incidence and reflects from
it before reaching the person. The total time for the light to travel this
path includes the time to travel to the mirror and the time to travel from
the mirror to the person. The light also travels directly to the person
without reflecting. Find the ratio of the total travel time along the
reflected path to the travel time along the direct path.