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CHAPTER 23
Light: Geometric Optics
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Units
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The Ray Model of Light
Reflection; Image Formed by a Plane Mirror
Formation of Images by Spherical Mirrors
Index of Refraction
Refraction: Snell’s Law
Total Internal Reflection; Fiber Optics
Thin Lenses; Ray Tracing
The Thin Lens Equation; Magnification
Combinations of Lenses
Lensmaker’s Equation
The Ray Model of Light
Light very often travels in straight lines. We represent
light using rays, which are straight lines emanating from
an object. This is an idealization, but is very useful for
geometric optics.
The Reflection of Light
If a stone is dropped into a pond, circular waves emanate
from the point where it landed. Rays, perpendicular to the
wave fronts, give the direction in which the waves
propagate.
As one moves farther from a point wave
source, the wave fronts become more nearly
flat.
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Reflection; Image Formation by a Plane Mirror
Law of reflection: the angle of reflection (that the ray
makes with the normal to a surface) equals the angle of
incidence.
The Reflection of Light
Reflection from a smooth surface is called specular reflection; if the surface is rough, it is diffuse
reflection.
Reflection; Image Formation by a Plane Mirror
With diffuse reflection, your eye sees reflected light at all angles. With specular reflection (from a
mirror), your eye must be in the correct position.
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Notation for Mirrors and Lenses
The object distance is the distance from the object to the mirror or lens
– Denoted by p
The image distance is the distance from the image to the mirror or lens
– Images are formed at the point where rays actually intersect or appear to originate
– Denoted by q
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The lateral magnification of the mirror or lens is the ratio of the image height to the object
height
– Denoted by M
Types of Images for Mirrors and Lenses
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A real image is one in which light actually passes through the image point
– Real images can be displayed on screens
A virtual image is one in which the light does not pass through the image point
– The light appears to diverge from that point
– Virtual images cannot be displayed on screens
More about Images
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To find where an image is formed, it is always necessary to follow at least two rays of light
as they reflect from the mirror
Forming Images with a Plane Mirror
Light reflected from the flower and vase hits the mirror. Obeying the law of reflection, it enters the
eye. The eye interprets the ray as having had a straight-line path, and sees the image behind the
mirror.
Properties of Mirror Images Produced by Plane Mirrors:
• A mirror image is upright, but appears reversed right to left.
• A mirror image appears to be the same distance behind the mirror that the object is in front of the
mirror.
• A mirror image is the same size as the object.
Reflection; Image Formation by a Plane Mirror
What you see when you look into a plane (flat) mirror is an
image, which appears to be behind the mirror.
This is called a virtual image, as the light does not go
through it. The distance of the image from the mirror is
equal to the distance of the object from the mirror.
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Flat Mirror
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Simplest possible mirror
Properties of the image can be determined by
geometry
One ray starts at P, follows path PQ and reflects
back on itself
A second ray follows path PR and reflects
according to the Law of Reflection
Properties of the Image Formed by a Flat Mirror
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The image is as far behind the mirror as the object is in front
– q=p
The image is unmagnified
– The image height is the same as the object height
• h’ = h and M = 1
The image is virtual
The image is upright
– It has the same orientation as the object
There is an apparent left-right reversal in the image
Application – Day and Night Settings on Auto Mirrors
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With the daytime setting, the bright
beam of reflected light is directed
into the driver’s eyes
With the nighttime setting, the dim
beam of reflected light is directed
into the driver’s eyes, while the bright
beam goes elsewhere
Spherical Mirrors
A spherical mirror has the shape of a section of a sphere. If the
outside is mirrored, it is convex; if the inside is mirrored, it is
concave.
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Formation of Images by Spherical Mirrors
Spherical mirrors are shaped like
sections of a sphere, and may be
reflective on either the inside
(concave) or outside (convex).
Spherical Mirrors
Spherical mirrors have a central
axis (a radius of the sphere)
and a center of curvature (the
center of the sphere).
Formation of Images by Spherical Mirrors
Rays coming from a faraway object are
effectively parallel.
Parallel rays striking a spherical mirror do not all converge at
exactly the same place if the curvature of the mirror is large; this
is called spherical aberration.
Concave Mirror, Notation
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The mirror has a radius of curvature of R
Its center of curvature is the point C
Point V is the center of the spherical segment
A line drawn from C to V is called the principle axis of the
mirror
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Spherical Aberration
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Rays are generally assumed to make small angles with
the mirror
When the rays make large angles, they may converge to
points other than the image point
This results in a blurred image
This effect is called spherical aberration
When the Hubble Space Telescope was first launched, its optics
were marred by spherical aberration. This was fixed with
corrective optics.
Formation of Images by Spherical Mirrors
If the curvature is small, the focus is much more
precise; the focal point is where the rays converge.
Ray Tracing and the Mirror Equation
We use three principal rays in finding the image
produced by a concave mirror.
• The parallel ray (P ray) reflects through the
focal point.
• The focal ray (F ray) reflects parallel to the
axis.
• The center-of-curvature ray (C ray) reflects
back along its incoming path.
These three rays are illustrated here.
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Example 1: Image in a concave mirror.
A 1.50-cm-high diamond ring is placed 20.0 cm from a concave mirror with radius of curvature 30.0
cm. Determine
a) the position of the image
1 1 1
1
1
  

 0.0167cm1
di f do 15.0cm 20.0cm
di  1/ 0.0167cm1  60.0cm
d1 is positive, the image is 60.0 cm in front of the mirror.
b) The magnification is
m
di
60.0cm

 3.00
do
20.0cm
The image height is 3.0 times the object height, and is
hi  mho  (3.00)(1.5cm)  4.5cm
The minus sign states the image is inverted.
Ray Tracing and the Mirror Equation
This image shows how these three
rays are used to find the image formed
by a convex mirror. The image is
located where the projections of the
three rays cross. The size of the image
can also be determined.
The process is similar for a
concave mirror, although there
are different results depending
on where the object is placed.
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We derive the mirror
equation using the ray
diagrams:
Using the similar triangles and the fact that f = ½ R, we get the mirror equation:
Here, do is the distance from the mirror to the object, di is the distance from the mirror to the
image, and f is the focal length.
Formation of Images by Spherical Mirrors
Geometrically, we can derive an
equation that relates the object
distance, image distance, and focal
length of the mirror:
We can also find the magnification (ratio of image height to object height).
The negative sign indicates that the image is inverted. This object is between the center of
curvature and the focal point, and its image is larger, inverted, and real.
Example 2: Object closer to concave mirror. A
1.00-cm-high object is placed 10.0cm from a
concave mirror whose radius of curvature is
30.0cm.
a) Draw a ray diagram to locate (approximately)
the position of the image.
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b) Determine the position of the image and the magnification analytically.
Since f  r / 2  15.0cm the object is between the mirror and the focal point.
1 1 1
1
1
23
1
  



di f do 15.0cm 10cm 30.0cm
30.0cm
Therefore, di  30.0cm The minus sign means the image is behind the mirror.
m  di / do  (30.0cm) /(10.0cm)  3.00
The + sign means that the image is upright.
Example 3: Convex rearview mirror.
An external rearview car mirror is convex with a radius of curvature of 16.0 m. Determine the
location of the image and its magnification for an object 10.0 m from the mirror. The center of
curvature of a convex mirror is behind the mirror, as is its focal point, so set r = -16m. So the focal
length is f  r / 2  8.0m . The object is in front of the mirror, f  r / 2  8.0m .
1 1 1
1
1
10.0  8.0
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  



di f do 8.0m 10.0m
80.0m
80.0m
Thus
di  80.0m /18  4.4m
di
(4.4m)

 0.44
do
(10.0m)
Sign conventions: The image distance is negative, -44m, so the image is behind the mirror. The
magnification is m = +0.44, so the image is upright and less than half as tall as the object.
m
Ray Tracing and the Mirror Equation
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Here are the sign conventions for concave and convex mirrors:
Sign Conventions for Refracting Surfaces
Ray Diagrams
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A ray diagram can be used to determine the position and size of an image
They are graphical constructions which tell the overall nature of the image
They can also be used to check the parameters calculated from the mirror and
magnification equations
Drawing a Ray Diagram
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To make the ray diagram, you need to know
– The position of the object
– The position of the center of curvature
Three rays are drawn
– They all start from the same position on the object
The intersection of any two of the rays at a point locates the image
– The third ray serves as a check of the construction
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The Rays in a Ray Diagram
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Ray 1 is drawn parallel to the principle axis and is reflected back through the focal point, F
Ray 2 is drawn through the focal point and is reflected parallel to the principle axis
Ray 3 is drawn through the center of curvature and is reflected back on itself
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The rays actually go in all directions from the object
The three rays were chosen for their ease of construction
The image point obtained by the ray diagram must agree with the value of q calculated
from the mirror equation
Ray Diagram for Concave Mirror, p > R
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The object is outside the center of curvature of the mirror
The image is real
The image is inverted
The image is smaller than the object
Ray Diagram for a Concave Mirror, p < f
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The object is between the mirror and the focal point
The image is virtual
The image is upright
The image is larger than the object
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Ray Diagram for a Convex Mirror
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The object is in front of a convex mirror
The image is virtual
The image is upright
The image is smaller than the object
Formation of Images by Spherical Mirrors
Notes on Images
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With a concave mirror, the image may be either real or virtual
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When the object is outside the focal point, the image is real
When the object is at the focal point, the image is infinitely far away
When the object is between the mirror and the focal point, the image is virtual
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With a convex mirror, the image is always virtual and upright
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As the object distance increases, the virtual image gets smaller
Index of Refraction
In general, light slows somewhat when traveling through a
medium. The index of refraction of the medium is the ratio of the
speed of light in vacuum to the speed of light in the medium:
Example 4: Light’s speed in diamond.
Calculate the speed of light in diamond.
v
c
1

 0.413c
n 2.42
3.00 x108 m / s
v
 1.24 x108 m / s
2.42
Willebrord Snell was an early seventeenth century Dutch
mathematician who is best known for determining that transparent
materials have different indices of refraction depending upon their
composition.
Refraction: Snell’s Law
Light changes direction when
crossing a boundary from one
medium to another. This is
called refraction, and the angle
the outgoing ray makes with the
normal is called the angle of
refraction.
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We can now write the angle of refraction in terms of the index of refraction:
If light enters a medium of lower index of refraction, it will be bent away from the normal. If the
angle of incidence is large enough, the angle of refraction is 90°; at larger incident angles the light
will be totally reflected.
Basic properties of refraction:
Example 5: Refraction through flat glass.
Light traveling in air strikes a flat piece of uniformly thick glass at an
incident of 60o , as shown. If the index of refraction of the glass is 1.50
a) What is the angle of refraction in the glass?
1.00
sin  A 
sin 60o  0.577  35.2o
1.50
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b) What is the angle at which the ray emerges from the glass?
Since the faces of the glass are parallel, the incident angle at the second surface is just  A , so
sin  A  0.577 . At this second interface, n1  1.50 and n2  1.00 . Thus the ray re-enters the air
at an angle  B ( 2 )
sin  B 
1.50
sin  A  0.866  60o
1.00
Total Internal Reflection; Fiber Optics
If the angle of incidence is larger
than this, no transmission occurs.
This is called total internal
reflection.
The Refraction of Light
Refraction can make objects
immersed in water appear broken,
and can create mirages.
Atmospheric Refraction
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There are many interesting results of refraction
in the atmosphere
– Sunsets
– Mirages
Light rays from the sun are bent as they pass
into the atmosphere
It is a gradual bend because the light passes
through layers of the atmosphere
– Each layer has a slightly different index of refraction
The Sun is seen to be above the horizon even after it has fallen below it
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Atmospheric Refraction and Mirages
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A mirage can be observed when the air above the
ground is warmer than the air at higher elevations
The rays in path B are directed toward the ground
and then bent by refraction
The observer sees both an upright and an inverted
image
The Refraction of Light
This is called total internal reflection, and the incident angle at which the angle of refraction is 90°
is called the critical angle, C. Total internal reflection is used in some binoculars and in optical
fibers.
There is a special angle called Brewster’s angle; light
reflected at this angle is totally polarized.
Reflected light is completely polarized when the reflected
and refracted beams are at right angles to one another.
The direction of polarization is parallel to the reflecting
surface.
Brewster’s angle can be calculated using the appropriate
geometry:
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Thin Lenses; Ray Tracing
Thin lenses are those whose thickness is small compared to
their radius of curvature. They may be either converging (a) or
diverging (b).
Ray Tracing for Lenses
Lenses are used to focus light and form images. There are a variety of possible types; we will
consider only the symmetric ones, the double concave and the double convex.
If we think of a convex lens as consisting of prisms, we can see how light going through it
converges at a focal point (assuming the lens is properly shaped).
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A concave lens can also be modeled by prisms:
The three principal rays for lenses are similar to those for mirrors:
• The P ray—or parallel ray—approaches the lens parallel to its axis.
• The F ray is drawn toward (concave) or through (convex) the focal point.
• The midpoint ray (M ray) goes through the middle of the lens. Assuming the lens is thin
enough, it will not be deflected. This is the thin-lens approximation.
These diagrams show the principal rays for both types of lenses:
As with mirrors, we use these principal rays to locate the image:
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The convex lens forms different image types depending on where the object is located with respect
to the focal point:
The Thin-Lens Equation
We derive the thin-lens equation in the same way we did the
mirror equation, using these diagrams:
This gives us the thin-lens approximation, as well as the
magnification:
Sign Conventions for Thin Lenses
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The Thin-Lens Equation
Example 6: Image formed by converging lens.
a) What is the position of the image of a 7.6-cm-high flower placed 1.00m from a +50mm focal
length camera lens?
b) What is the size, of the image of the 7.6-cm-high flower placed 1.00m from a +50.0mm focal
length camera lens?
1 1 1
1
1
20.0  1.0 19.0
  



do f do 5.00cm 100cm
100cm
100cm
di 
100cm
 5.26cm Or 52.6mm behind the lens.
19.0
m
di
5.26cm

 0.0526
do
100cm
hi  mho  (0.0526)(7.6cm)  0.40cm
Thin Lenses; Ray Tracing
The power of a lens is the inverse of its focal length.
Lens power is measured in diopters, D.
-1
1D=1m
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Thin Lenses; Ray Tracing
Ray tracing for thin lenses is similar to that for mirrors. We have three key rays:
1. This ray comes in parallel to the axis and exits through the focal point.
2. This ray comes in through the focal point and exits parallel to the axis.
3. This ray goes through the center of the lens and is undeflected.
For a diverging lens, we can use the same three rays; the image is upright and virtual.
The Thin Lens Equation; Magnification
The thin lens equation is the same as the mirror equation:
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The Thin Lens Equation; Magnification
The magnification formula is also the same as that for a mirror:
The power of a lens is positive if it is converging and negative if it is diverging.
Combinations of Lenses
In lens combinations, the image formed by
the first lens becomes the object for the
second lens (this is where object distances
may be negative).
Lensmaker’s Equation
This useful equation relates the radii of curvature of the two
lens surfaces, and the index of refraction, to the focal length.
Dispersion and the Rainbow
The index of refraction varies slightly with the frequency of light; in
general, the higher the frequency, the higher the index of
refraction.
This means that refracted light is “spread out” in a rainbow of
colors; this phenomenon is known as dispersion.
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Dispersion and the Rainbow
Rainbows are created by the dispersion of light as it refracts
in a rain drop.
As the drop falls, all the colors of the rainbow arrive
at the eye.
Sometimes a faint secondary arc can be seen.
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CHAPTER 23
GEOMETRIC OPTICS
CONCEPTS
1. The diagram to the right represents an object 0.030 m high placed at point X,
0.60 m from the center of the lens. An image is formed at point Y, 0.30 m from the
center of the lens. The image formed is real and inverted.
2. A plane mirror produces an image of an object. Compared to the object, the image
appears reversed and the same size.
3. A convex (converging) lens can form images that are either real or virtual.
4. When the calculated image distance for an image formed using a curved mirror has a negative value, the image
must be virtual.
5. The convex spherical mirror found on the passenger side of many cars contains the
warning: “Objects are closer than they appear.” The best phrase that describes the
image of an object viewed in this mirror is – virtual and smaller than the object.
6. The diagram to the right represents an object in front of a concave mirror. The image
of the object formed by the mirror is virtual and larger than the object.
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7. The diagram to the right shows a convex (converging) lens with focal length F. The object
should be placed between F and the lens to produce a virtual image.
8. The diagram below that best represents the image of the letter “L” formed by the
plane mirror MM is B.
9. An object is placed in front of a plane mirror as shown in the diagram to the right.
The diagram below that best represents the image that is formed is C.
10. As the distance between a man and a plane mirror increases, the size of the image of the man produced by the
mirror remains the same.
11. The diagram to the right shows light ray R parallel to the principal axis of a spherical
concave (converging) mirror. Point F is the focal point of the mirror and C is the center
of curvature. After reflecting, the light ray will pass through point F.
12. A searchlight consists of a high-density light source at the focal point of a concave
(converging) mirror. The light reflected from the mirror will form a nearly parallel beam.
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13. The graph below that best represents the relationship between image distance (di) and object distance (do) for a
plane mirror is C.
14. A candle is located beyond the center of curvature, C, of a concave spherical mirror
having principal focus F, as shown in the diagram to the right. The candle’s image is
located between C and F.
15. The diagram on the right shows the principal axis of a concave spherical mirror. The focal
point is F, and C is the center of the curvature of the mirror. The focal length of the mirror is
0.10 m. If an object is placed at point B, its image is virtual and erect.
The diagram on the right shows a concave (converging) spherical mirror having
principal focus F and center of curvature C. Point A lies on the principal axis.
(Concepts 16 and 17)
16. When an object is placed at point A, its image is observed to the left of the mirror.
17. If an object is located at point A, its image is virtual and erect.
18. The phenomenon of light that accounts for the formation of images by a lens is refraction.
19. The phenomenon is represented by the diagram to the right is refraction.
20. The piece of glass that could be used to focus parallel rays of sunlight to a small spot of light is B.
21. An optical device that may form an enlarged image is a converging lens.
22. A student uses a magnifying glass to examine the crystals in a mineral specimen. The
magnifying glass contains a convex (converging) lens.
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23. The diagram below that best represents a lens being used to produce a real, enlarged image of object O is B.
24. The diagram below that shows the path of light rays as they pass from an object 2F through a converging lens to
the image formed at 2F’ is B.
25. When light rays pass through the film in a movie projector, an image of the film is produced on a screen. In order to
produce the image on the screen, the type of lens the projector uses and how far the lens must be placed from the film
is converging lens, at a distance greater than the focal length.
26. The focal length of a lens is not dependent on the distance of an object from the lens.
27. The diagram to the right shows a lens with an object located at position A. As the
object is moved from position A to position B, the image formed by lens will be
inverted and increasing in size.
28. The diagram to the right represents a convex (converging) lens with focal point F.
29. If an object is placed at 2F, the image will be real, inverted, and the same size as
the object.
30. A student’s solution to an optics problem had a negative value for the focal length. The optical device in the
problem was most likely a diverging (concave) lens.
31. The image formed by a diverging lens is virtual.
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32. The ray diagram below that is incorrect is B.
33. The diagram to the right shows the refraction of the blue and red components
of a white light beam. The phenomenon illustrated is chromatic aberration.
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