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Chapter 23: Geometric Optics
© 2013 Pearson Education, Inc.
Review Basic Geometry!
Ray Approximation
• The rays are straight
lines perpendicular to
the wave fronts
• With the ray
approximation, we
assume that a wave
moving through a
medium travels in a
straight line in the
direction of its rays
Light Rays: Ignore Diffraction and
Interference of waves!
Diffraction depends on SLIT WIDTH: the smaller the width,
relative to wavelength, the more bending and diffraction.
We will assume that λ<<d , where d is the diameter of the opening.
This approximation is good for the study of mirrors, lenses, prisms, etc.
The Ray Model of Light
 Let us define a light ray as a line in the direction along
which light energy is flowing.
 Any narrow beam of light, such as a laser beam, is
actually a bundle of many parallel light rays.
 You can think of a single light ray as the limiting case
of a laser beam whose diameter approaches zero.
© 2013 Pearson Education, Inc.
Slide 23-19
The Ray Model of Light
 Light travels through a
transparent material in
straight lines called light rays.
 The speed of light is v  c/n,
where n is the index of
refraction of the material.
 Light rays do not interact with
each other.
 Two rays can cross without
either being affected in any
way.
© 2013 Pearson Education, Inc.
Slide 23-20
The Ray Model of Light
 An object is a source of light rays.
 Rays originate from every point on the
object, and each point sends rays in all
directions.
 The eye “sees” an object
when diverging bundles
of rays from each point
on the object enter the
pupil and are focused to
an image on the retina.
© 2013 Pearson Education, Inc.
Slide 23-22
The Ray Model of Light
Light interacts with matter in four different ways:
At an interface between two materials, light can be
either reflected or refracted.
Within a material, light can be either scattered or
absorbed.
© 2013 Pearson Education, Inc.
Slide 23-21
Objects
 Objects can be
either self-luminous,
such as the sun,
flames, and
lightbulbs, or
reflective.
 Most objects are
reflective.
© 2013 Pearson Education, Inc.
Slide 23-23
Objects
 The diverging rays from a point source are emitted in
all directions.
 Each point on an object is a point source of light rays.
 A parallel bundle of rays could be a laser beam, or
light from a distant object.
© 2013 Pearson Education, Inc.
Slide 23-24
Ray Diagrams
 Rays originate from every point on an object and
travel outward in all directions, but a diagram trying to
show all these rays would be messy and confusing.
 To simplify the picture, we use a ray diagram showing
only a few rays.
© 2013 Pearson Education, Inc.
Slide 23-25
Reflection & Refraction
i   r
n1 sin 1  n2 sin  2
Law of Reflection
• The normal is a line
perpendicular to the
surface
i   r
– It is at the point where the
incident ray strikes the
surface
• The incident ray makes an
angle of θ1 with the
normal
• The reflected ray makes
an angle of θ1’ with the
normal
Specular
Reflection
• Specular reflection is
reflection from a
smooth surface
• The reflected rays are
parallel to each other
• All reflection in this
text is assumed to be
specular
Diffuse
Reflection
• Diffuse reflection is
reflection from a rough
surface
• The reflected rays travel in
a variety of directions
• A surface behaves as a
smooth surface as long as
the surface variations are
much smaller than the
wavelength of the light
Following the Reflected and
Refracted Rays
•Ray  is the
incident ray.
•Ray  is the
reflected ray.
•Ray  is refracted
into the lucite.
•Ray  is internally
reflected in the lucite.
•Ray  is refracted as
it enters the air from
the lucite.
Section 35.5
How many times will the
incident beam shown be reflected
by each of the parallel mirrors?
https://www.youtube.com/watch?v=BQNw8HSixGk
Mirror Mirror
QuickCheck 23.3
You are looking at the image of a
pencil in a mirror. What do you see
in the mirror if the top half of the
mirror is covered with a piece of
dark paper?
A. The full image of the
pencil.
B. The top half only of the
pencil.
C. The bottom half only of
the pencil.
D. No pencil, only the paper.
© 2013 Pearson Education, Inc.
Slide 23-41
QuickCheck 23.3
You are looking at the image of a
pencil in a mirror. What do you see
in the mirror if the top half of the
mirror is covered with a piece of
dark paper?
A. The full image of the
pencil.
B. The top half only of the
pencil.
C. The bottom half only of
the pencil.
D. No pencil, only the paper.
© 2013 Pearson Education, Inc.
Slide 23-42
Why are most materials Opaque?
(Opaque – Can’t see through)
They absorb light without re-emitting it. Vibrations given
by the light to their atoms and molecules are turned into
random kinetic energy – they become slightly warmer.
Opacity: Mirrors
Free electrons in opaque reflective surfaces
can vibrate, absorb & re-emit at any frequency.
Transparency
Selective Absorption
Glass resonates strongly with UV and
infrared, absorbing those frequencies
while transmitting visible frequencies.
Refraction:
Bending Light into Focus
Refraction: Bending of Light
Transmitted through Materials
Light Bends because it Slows Down.
Atoms are Optical Tuning Forks
Light slows down as it travels through
glass because it takes time to be
absorbed and re-emitted.
Light Slows Down in Materials
Light Bends Toward the Normal when going from a
medium of lower refractive index to one that has a
higher refractive index and visa versa.
lower n
higher n
Index of Refraction
c
n
v
n 1
Vacuum: 1
Water: 1.33
Glass: 1.46
Diamond: 2.4
The Index of Refraction
• Refraction: Light Bends in
Transmission
• The speed of light in any
material is less than its speed
in vacuum
• The index of refraction, n,
of a medium can be defined
as
• For a vacuum, n = 1
– We assume n = 1 for air
speed of light in a vacuum c
λ
n
 
also
speed of light in a m edium v λn
• For other media, n > 1
λ  λ in vacuum  • n is a dimensionless number
greater than unity, not
n


λn  λ in a medium 
necessarily an integer
Some Indices of Refraction
Variation of Index of Refraction with Wavelength
sp e e d o f lig h t in a v a cu u m c
λ
n
 
sp e e d o f lig h t in a m e d iu m v λ n
• This dependence of n on λ
is called dispersion
• The index of refraction for
a material generally
decreases with increasing
wavelength
• Violet light bends more
than red light when passing
into a refracting material
Frequency Doesn’t Change!
• As light travels from one
medium to another, its
frequency does not
change
– Both the wave speed and
the wavelength do change
– The wavefronts do not
pile up, nor are created or
destroyed at the
boundary, so ƒ must stay
the same
Snell’s Law of Refraction
Angles are always measured from the normal.
n1 sin 1  n2 sin  2
Snell’s Law
Light is refracted into a crown glass
slab. n1 = 1.00 and n2 = 1.52
If θ1 = 30.0o, θ2 = ?
θ2 = sin-1(n1 / n2) sin θ1 = 19.2o
The ray bends toward the normal,
as expected because n2 > n1
Snell’s Law of Refraction
In general:
n1 sin 1  n2 sin  2
n11  n22
n1
 2  1
n2
If n2  n1 , then 1  2
measured from the normal!
QuickCheck 23.4
A laser beam passing
from medium 1 to
medium 2 is refracted as
shown. Which is true?
A. n1 < n2.
B. n1 > n2.
C. There’s not enough
information to compare
n1 and n2.
© 2013 Pearson Education, Inc.
Slide 23-53
QuickCheck 23.4
A laser beam passing
from medium 1 to
medium 2 is refracted as
shown. Which is true?
A. n1 < n2.
B. n1 > n2.
C. There’s not enough
information to compare
n1 and n2.
© 2013 Pearson Education, Inc.
Slide 23-54
Prelab
Emerging Beam is Parallel to Incident Beam but
offset distance d, called the Lateral Shift and is the
subject of this week’s lab!
Fig. 35-15, p. 989
Prelab for Next Lab:
https://www.youtube.com/watch?v=1-0eQGeWLfY&t=2s
Real and Apparent Depth
Real and apparent depth. The
refraction of light at the surface of
water makes ponds and swimming
pools appear shallower than they
really are. A 1m deep pond would
only appear to be 0.75 m deep
when viewed from directly above.
When light emerges from glass or
water into air it speeds up again.
© 2013 Pearson Education, Inc.
Apparent Depth
© 2013 Pearson Education, Inc.
QuickCheck 23.6
A fish in an aquarium with flat sides looks out at a hungry
cat. To the fish, the distance to the cat appears to be
A. Less than the actual distance.
B. Equal to the actual distance.
C. More than the actual distance.
© 2013 Pearson Education, Inc.
Slide 23-71
QuickCheck 23.6
A fish in an aquarium with flat sides looks out at a hungry
cat. To the fish, the distance to the cat appears to be
A. Less than the actual distance.
B. Equal to the actual distance.
C. More than the actual distance.
© 2013 Pearson Education, Inc.
Slide 23-72
Beam & Refraction Directions
• Possible directions of the
beam are indicated by rays
numbered 1 through 5
• The refracted rays are bent
away from the normal since
n 1 > n2
• An application of internal
reflection
• Plastic or glass rods are used to
“pipe” light from one place to
another
• Applications include:
– medical use of fiber optic
cables for diagnosis and
correction of medical problems
– Telecommunications
• A flexible light pipe is called an
optical fiber
• A bundle of parallel fibers
(shown) can be used to construct
an optical transmission line
Fiber Optics
Construction of an Optical Fiber
• The transparent core is
surrounded by
cladding
– The cladding has a lower n
than the core
– This allows the light in the
core to experience total
internal reflection
• The combination is
surrounded by the
jacket
Total Internal Reflection
 2  90
n1 sin 1  n2 sin  2
n2
sin  C 
n1
The Critical Angle
Critical Angle
• There is a particular angle
of incidence that will
result in an angle of
refraction of 90°
– This angle of incidence is
called the critical angle, θC
n2
sin θ C 
(for n1  n 2 )
n1
Critical Angle Sample Problem
A ray of light, emitted by a laser located beneath the surface of an
unknown liquid with air above it, undergoes total internal
refection as shown. What is the index of refraction for the liquid?
What is its likely identification?
QuickCheck 23.5
A laser beam undergoes two
refractions plus total internal
reflection at the interface
between medium 2 and
medium 3. Which is true?
A. n1 < n3.
B. n1 > n3.
C. There’s not enough
information to
compare n1 and n3.
© 2013 Pearson Education, Inc.
Slide 23-62
QuickCheck 23.5
A laser beam undergoes two
refractions plus total internal
reflection at the interface
between medium 2 and
medium 3. Which is true?
A. n1 < n3.
B. n1 > n3.
C. There’s not enough
information to
compare n1 and n3.
© 2013 Pearson Education, Inc.
Slide 23-63
If you pass white light through a prism,
it separates into its component colors.
long wavelengths
short wavelengths
R.O.Y. G. B.I.V
The index of refraction depends on
WAVELENGTH.
long wavelengths
short wavelengths
R.O.Y. G. B.I.V
The speed and wavelength change but
the FREQUENCY does NOT.
Fr
Frequency depends on the oscillating source!
long wavelengths
short wavelengths
R.O.Y. G. B.I.V
Why does Violet Light bend more
than Red Light?
Violet light slows down more because the atoms in the material
are tuned to higher frequencies. As the violet light travels
through glass it takes more time to be absorbed and re-emitted.
Variation of Index of Refraction with Wavelength
sp e e d o f lig h t in a v a cu u m c
λ
n
 
sp e e d o f lig h t in a m e d iu m v λ n
• This dependence of n on λ
is called dispersion
• The index of refraction for
a material generally
decreases with increasing
wavelength
• Violet light bends more
than red light when passing
into a refracting material
Refraction in a Prism
•Since all the colors have
different angles of deviation,
white light will spread out
into a spectrum.
– Violet deviates the
most.
– Red deviates the least.
– The remaining colors
are in between.
Section 35.7
Compare: Dispersion via
Diffraction and Refraction
Why does Violet Light bend more
than Red Light?
Violet light slows down more because the atoms in the material
are tuned to higher frequencies. As the violet light travels
through glass it takes more time to be absorbed and re-emitted.
Angle of Deviation
• Since all the colors
have different angles
of deviation, white
light will spread out
into a spectrum
– Violet deviates the most
– Red deviates the least
– The remaining colors are
in between
Dispersion Sample Problem
The index of refraction for
violet light in silica flint glass
is 1.66, and that for red light is
1.62. What is the angular
dispersion of visible light
passing through a prism of
apex angle 60.0° if the angle of
incidence is 50.0°? red (660
nm) violet (410 nm)
Use Snell’s Law twice and some
geometry! Angles are always
measured from the normal.
n1 sin 1  n2 sin  2
How are Rainbows Formed?
Dispersion: Raindrops Act like Prisms
• A ray of light strikes a drop
of water in the atmosphere
• It undergoes both reflection
and refraction
– First refraction at the
front of the drop
• Violet light will
deviate the most
• Red light will deviate
the least
The Rainbow
• At the back surface the light is
reflected
• It is refracted again as it
returns to the front surface and
moves into the air
• The rays leave the drop at
various angles
– The angle between the white
light and the most intense violet
ray is 40°
– The angle between the white
light and the most intense red ray
is 42°
Observing the Rainbow
• If a raindrop high in the sky is observed, the red ray is seen
• A drop lower in the sky would direct violet light to the
observer
• The other colors of the spectra lie in between the red and
the violet
The droplets form a circular arc, with each droplet within the arc
dispersing light and reflecting it back towards the observer with
the greatest concentration of outgoing rays found at these 40-42
degree angles of deviation. Every droplet within the arc is
refracting and dispersing the entire visible light spectrum
(ROYGBIV).
Rainbow facts
• an observer is in a position to see only a single color from any
one droplet of water.
• your rainbow is slightly different from the rainbow seen by
others
• your rainbow moves with you
• disk within the bow is brighter because of overlapping of
multiple refractions (which don’t occur outside the disk)
A line drawn from your eye to the top of the rainbow forms a 42degree angle with the imaginary line from the sun through your
eye. (If there is a secondary rainbow, it forms an angle of 51degrees). Because these angles determine the position of the
rainbow in the sky, it will sink as the sun rises and rise as the sun
sinks. At some points, the entire rainbow, not just the bottom half,
will be below the horizon where you can't see it. That's why you'll
never see a summer rainbow at midday.
Double Rainbow
• The secondary rainbow is
fainter than the primary
• The secondary rainbow
arises from light that
makes two reflections
from the interior surface
before exiting the raindrop
• Higher-order rainbows are
possible, but their
intensity is low
•
•
Halos are caused by the light of the sun or moon passing through a very thin layer of
cirruform (ice-crystal) clouds in the upper atmosphere. The ice crystals refract the light
of the moon, similar to the way water droplets in the lower atmosphere can refract
sunlight to produce a rainbow. Just like a rainbow, strong halos can have bands of color
in them, due to slightly different refractive properties of the ice crystals for different
colors. Essentially, halos ARE rainbows caused by primary refraction in ice crystals.
Some interesting facts about halos: Halos always occur exactly 22 degrees away from
the sun or moon. Occasionally, intense halos can be double halos, just as intense
rainbows can be doubled. Intense halos can also produce "moondogs" or "sundogs,"
very bright regions on the halo evenly spaced at 90 degree intervals around the halo.
Colored Filters and Colored Objects
 Green glass is green because it absorbs any light
that is “not green.”
 If a green filter and a red filter are overlapped, no
light gets through.
 The green
filter transmits
only green
light, which is
then absorbed
by the red
filter because
it is “not red.”
© 2013 Pearson Education, Inc.
Slide 23-83
Colored Filters and Colored Objects
 The figure below shows the absorption curve of
chlorophyll, which is essential for photosynthesis in
green plants.
 The chemical reactions of photosynthesis absorb red
light and blue/violet light from sunlight and puts it to use.
 When you look at
the green leaves
on a tree, you’re
seeing the light that
was reflected
because it wasn’t
needed for
photosynthesis.
© 2013 Pearson Education, Inc.
Slide 23-84
Light Scattering: Blue Skies and Red Sunsets
Rayleigh scattering
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Light Scattering: Blue Skies and Red Sunsets
 Light can scatter
from small particles
that are suspended
in a medium.
 Rayleigh scattering
from atoms and
molecules depends
inversely on the
fourth power of the
wavelength:
Iscattered 4
© 2013 Pearson Education, Inc.
Slide 23-85
Light Scattering: Blue Skies and Red Sunsets
Sunsets are red because all the blue light has scattered as the sunlight passes through the atmosphere.
© 2013 Pearson Education, Inc.
Slide 23-86
Galileo
In the early 17th century, many scientists believed that there was no
such thing as the "speed of light"; they thought light could travel any
distance in no time at all. Galileo disagreed, and he came up with an
experiment to measure light's velocity: he and his assistant each took
a shuttered lantern, and they stood on hilltops one mile apart. Galileo
flashed his lantern, and the assistant was supposed to open the shutter
to his own lantern as soon as he saw Galileo's light. Galileo would
then time how long it took before he saw the light from the other
hilltop. The problem was that the speed of light is simply too fast to
be measured this way; light takes such a short time (about 0.000005
seconds, in fact) to travel one mile that there's no way the interval
could have been measured using the tools Galileo had.
The Speed of Light?
•
•
•
•
186,000 miles per second
300,000 kilometers per second
3 x 108 m/s
first successfully determined by
Danish astronomer Ole Roemer in
1675: 2.3 x 108 m/s
• First Terrestrial Measurement by
Fizeau in 1849: 2.9979 x 108 m/s
• In 1926, Michelson used a rotating
prism to measure the time it took
light to make a round trip from
Mount Wilson to Mount San
Antonio in California, a distance
of about 22 miles (36 km). The
precise measurements yielded a
speed of 186,285 miles per second
(299,796 kilometres per second).
Lenses and Mirrors
Image Formation
Why aren’t images of objects
produced on the wall without a
lens or hole?
Why aren’t images of objects
produced on the wall without a
lens or hole?
Images

Image formation is a consequence of
light traveling in straight lines

The first camera—the pinhole camera—
illustrates this fact.
Apertures
 A camera obscura is a
darkened room with a
single, small hole,
called an aperture.
 The geometry of the
rays causes the image
to be upside down.
 The object and image
heights are related
by:
© 2013 Pearson Education, Inc.
Slide 23-26
Apertures
We can apply the ray model to more complex apertures,
such as the L-shaped aperture below.
© 2013 Pearson Education, Inc.
Slide 23-27
QuickCheck 23.1
The dark screen has a small
hole, 2 mm in diameter. The
lightbulb is the only source of
light. What do you see on the
viewing screen?
© 2013 Pearson Education, Inc.
Slide 23-28
QuickCheck 23.1
The dark screen has a small
hole, 2 mm in diameter. The
lightbulb is the only source of
light. What do you see on the
viewing screen?
© 2013 Pearson Education, Inc.
Slide 23-29
QuickCheck 23.2
Two point sources of light
illuminate a narrow vertical
aperture in a dark screen.
What do you see on the
viewing screen?
© 2013 Pearson Education, Inc.
Slide 23-30
QuickCheck 23.2
Two point sources of light
illuminate a narrow vertical
aperture in a dark screen.
What do you see on the
viewing screen?
© 2013 Pearson Education, Inc.
Slide 23-31
Lenses
A lens nicely bends the straight-line paths
of light.
Real and Virtual Images
Real images can be
displayed on screens
Virtual Images can not be
displayed onto screens.
Lenses
Lenses
 two common types

converging (convex) lens



thicker at the center than edges
converges light
diverging (concave) lens


thinner at the center than edges
diverges light
Lens Refraction
Converging & Diverging
Converging Lens
Diverging Lens
Concave vs Convex
Focal Length: Where Parallel
rays come to a focus
Lenses
 The photos below show parallel light rays entering two
different lenses.
 The left lens, called a converging lens, causes the
rays to refract toward the optical axis.
 The right lens, called a diverging lens, refracts
parallel rays away from the optical axis.
Slide 23-87
Focal Length:
Converging Lens
Focal Length:
Diverging Lens
QuickCheck 23.8
You can use the sun’s rays and a lens to start a fire. To do
so, you should use
A. A converging lens.
B. A diverging lens.
C. Either a converging or a diverging lens will work if
you use it correctly.
Slide 23-90
QuickCheck 23.8
You can use the sun’s rays and a lens to start a fire. To do
so, you should use
A. A converging lens.
B. A diverging lens.
C. Either a converging or a diverging lens will work if
you use it correctly.
Slide 23-91
Thin Lenses: Ray Tracing
 Three situations form
the basis for ray
tracing through a thin
converging lens.
 Situation 1:
A ray initially parallel
to the optic axis will
go through the far
focal point after
passing through the
lens.
Slide 23-92
Thin Lenses: Ray Tracing
 Three situations form
the basis for ray
tracing through a thin
diverging lens.
 Situation 2:
A ray directed along a
line toward the far
focal point becomes
parallel to the optic
axis after passing
through the lens.
Slide 23-116
Thin Lenses: Ray Tracing
 Three situations form
the basis for ray
tracing through a thin
converging lens.
 Situation 3:
A ray through the
center of a thin lens is
neither bent nor
displaced but travels
in a straight line.
Slide 23-94
Thin Lenses: Ray Tracing
Rays from an object point P are refracted by the lens and converge to a real image at point P.
Slide 23-95
Lenses
In an actual lens, rays refract twice, at spherical
surfaces having radii of curvature R1 and R2.
Slide 23-130
Lenses
Key features of lenses

principal axis


focal point


line joining the centers of curvature of the two lens
surfaces
point at which all the light rays come together
focal length

distance between the center of the lens and either
focal point
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:
Denoted by q
The lateral magnification of the
mirror or lens is the ratio of the
image height to the object height

Denoted by M
The focal point: f

The radius of curvature: R = 2f
M
Image height h'

Object height h
1 1 2 1
  
p q R f
FRONT is on the same side as the object and BACK is the other side!
The Principal axis goes through the focal point and the center of
curvature of the lens or mirror!!
Signs for Thin Lenses
1 1 2 1
  
p q R f
h'
q
M

h
p
https://phet.colorado.edu/en/simulation/legacy/geometricoptics
The Thin Lens Equation
The object distance s is related to the image distance
s by:
where f is the focal length of the lens, which can be
found from:
where R1 is the radius of curvature of the first surface,
and R2 is the radius of curvature of the second surface,
and the material surrounding the lens has n = 1.
© 2013 Pearson Education, Inc.
Slide 23-131
Thin Lenses: Refraction Theory
If an object is located at distance s from a spherical
refracting surface, an image will be formed at distance
s given by:
Slide 23-125
Tactics: Ray Tracing for a Converging Lens
© 2013 Pearson Education, Inc.
Slide 23-103
© 2013 Pearson Education, Inc.
Ray Diagram for Converging
Lens, p > f



The image is real (q>0)
The image is inverted (M<0)
The image is on the back side of the lens (q>0)
Ray Diagram for Converging
Lens, p < f




The image is virtual (q < 0)
The image is upright (M>0)
The image is larger than the object (M>1)
The image is on the front side of the lens (q<0)
Magnifying Glass: Virtual Images
 You can “see” a virtual
image by looking
through the lens.
 This is exactly what you
do with a magnifying
glass, microscope or
binoculars.
Slide 23-111
QuickCheck 23.15
A lens creates an image as
shown. In this situation,
A.
B.
C.
D.
s < f.
f < s < 2f.
s > 2f.
There’s not enough information
to compare s to f.
© 2013 Pearson Education, Inc.
Slide 23-132
QuickCheck 23.15
A lens creates an image as
shown. In this situation,
A.
B.
C.
D.
s < f.
f < s < 2f.
s > 2f.
There’s not enough information
to compare s to f.
The image is real, which requires s > f.
The image is taller than the object, and
s’ > s requires s < 2f.
© 2013 Pearson Education, Inc.
Slide 23-133
Tactics: Ray Tracing for a Diverging Lens
© 2013 Pearson Education, Inc.
Slide 23-120
Ray Diagram for Diverging Lens

For a diverging lens, the image is always virtual
and upright (M>0)


This is regardless of where the object is placed
The image is on the front side of the lens (q<0)
Example 23.10 Demagnifying a Flower
© 2013 Pearson Education, Inc.
Slide 23-122
Example 23.10 Demagnifying a Flower
© 2013 Pearson Education, Inc.
Slide 23-123
Example 23.17 Analyzing a Concave Mirror
© 2013 Pearson Education, Inc.
Slide 23-150
Example 23.17 Analyzing a Concave Mirror
© 2013 Pearson Education, Inc.
Slide 23-151
Example 23.17 Analyzing a Concave Mirror
© 2013 Pearson Education, Inc.
Slide 23-152
QuickCheck 23.14
Light rays are converging to
point 1. The lens is inserted into
the rays with its focal point at
point 1. Which picture shows
the rays leaving the lens?
© 2013 Pearson Education, Inc.
Slide 23-118
QuickCheck 23.14
Light rays are converging to
point 1. The lens is inserted into
the rays with its focal point at
point 1. Which picture shows
the rays leaving the lens?
© 2013 Pearson Education, Inc.
Slide 23-119
Flat Mirrors Make Virtual Images
Virtual Image: An image that cannot be projected onto a surface.
A virtual image only appears like light rays came from the
location of the image, they are not really there.
Flat mirrors make Virtual Images.
Images Formed by
Flat Mirrors






Light rays leave the source and
are reflected from the mirror
Images are always located by
extending diverging rays back to
a point at which they intersect
One ray starts at point P, travels
to Q and reflects back on itself
Another ray follows the path PR
and reflects according to the law
of reflection
h’ = h for all images
Flat mirrors make virtual images
Reversals in a Flat Mirror

A flat mirror produces
an image that has an
apparent left-right
reversal

For example, if you
raise your right hand
the image you see
raises its left hand
Properties of the Image Formed
by a Flat Mirror – Summary

The image is as far behind the mirror as the
object is in front


|p| = |q|
The image is unmagnified

The image height is the same as the object height



The image is virtual
The image is upright


h’ = h and M = 1
It has the same orientation as the object
There is a front-back reversal in the image
Mirror Reflection
Convex & Concave
“Object” on the left, image on the right.
Convex Mirror
Convave Mirror
Focal Length& Radius of
Curvature



When the object is very far
away, then p → ∞ and the
incoming rays are essentially
parallel
In this special case, the image
point is called the focal point
The distance from the mirror to
the focal point is called the
focal length


The focal length is ½ the radius of
curvature
R = 2f
Lateral Magnification
Image height h'
M

Object height h
Magnification does not always mean bigger, the
size can either increase or decrease.
M>1: Increase
Positive: Upright
M<1: Decrease
Negative: Inverted
Ray Diagrams:Concave Mirrors




Ray 1 is drawn from the top of the object parallel to the principal axis and is
reflected through the focal point, F
Ray 2 is drawn from the top of the object through the focal point and is reflected
parallel to the principal axis
Ray 3 is drawn through the center of curvature, C, and is reflected back on itself
The intersection of any two of the rays at a point locates the image. The third ray
serves as a check of the construction
Concave Mirror, p > R




The center of curvature is between the object
and the concave mirror surface (f >0)
The image is real (q>0)
The image is inverted (M<0)
The image is smaller than the object (absM<1)
1 1 2 1
  
p q R f
A Real Image Formed by a Concave Mirror
Slide 23-140
Concave Mirror, p < f




The object is between the mirror surface and the
focal point (p>0)
The image is virtual (q<0)
The image is upright (M>0)
The image is larger than the object (M>1)
Ray Diagrams:Convex Mirrors



Ray 1 is drawn from the top of the object parallel to the principal
axis and is reflected away from the focal point, F
Ray 2 is drawn from the top of the object toward the focal point
and is reflected parallel to the principal axis
Ray 3 is drawn through the center of curvature, C, on the back
side of the mirror and is reflected back on itself
Convex Mirror





The object is in front of a convex mirror (p>0)
The focal point distance q is negative (q <0)
The image is always virtual and upright (M>0)
As the object distance decreases, the virtual image size increases
The image is smaller than the object (0<M<1)
Image Formation with Spherical Mirrors
A city skyline is reflected in this polished sphere.
Slide 23-143
Sign Conventions: Mirrors
1 1 2 1
  
p q R f
The Mirror Equation
For a spherical mirror with negligible thickness, the
object and image distances are related by:
where the focal
length f is related
to the mirror’s
radius of
curvature by:
© 2013 Pearson Education, Inc.
Slide 23-146
Compare Signs for Mirrors
and Thin Lenses
Thin Lenses
QuickCheck 23.9
A lens produces a sharply focused,
inverted image on a screen. What
will you see on the screen if the lens
is removed?
A. An inverted but blurry
image.
B. An image that is dimmer but
otherwise unchanged.
C. A sharp, upright image.
D. A blurry, upright image.
E. No image at all.
Slide 23-96
QuickCheck 23.9
A lens produces a sharply focused,
inverted image on a screen. What
will you see on the screen if the lens
is removed?
A. An inverted but blurry
image.
B. An image that is dimmer but
otherwise unchanged.
C. A sharp, upright image.
D. A blurry, upright image.
E. No image at all.
Slide 23-97
QuickCheck 23.10
A lens produces a sharply focused,
inverted image on a screen. What
will you see on the screen if a piece
of dark paper is lowered to cover the
top half of the lens?
A. An inverted but blurry image.
B. An image that is dimmer but otherwise
unchanged.
C. Only the top half of the image.
D. Only the bottom half of the image.
E. No image at all.
Slide 23-98
QuickCheck 23.10
A lens produces a sharply focused,
inverted image on a screen. What
will you see on the screen if a piece
of dark paper is lowered to cover the
top half of the lens?
A. An inverted but blurry image.
B. An image that is dimmer but
otherwise unchanged.
C. Only the top half of the image.
D. Only the bottom half of the image.
E. No image at all.
Slide 23-99
QuickCheck 23.11
A lens produces a sharply focused,
inverted image on a screen. What will
you see on the screen if the lens is
covered by a dark mask having only a
small hole in the center?
A.
B.
C.
D.
E.
An inverted but blurry image.
An image that is dimmer but
otherwise unchanged.
Only the middle piece of the image.
A circular diffraction pattern.
No image at all.
Slide 23-100
QuickCheck 23.11
A lens produces a sharply focused,
inverted image on a screen. What will
you see on the screen if the lens is
covered by a dark mask having only a
small hole in the center?
A.
B.
C.
D.
E.
An inverted but blurry image.
An image that is dimmer but
otherwise unchanged.
Only the middle piece of the image.
A circular diffraction pattern.
No image at all.
Slide 23-101
Image Formation
 The figure is a close-up
view of the rays very
near the image plane.
 To focus an image, you
must either move the
screen to coincide with
the image plane or
move the lens or object
to make the image
plane coincide with the
screen.
Slide 23-102
Example 23.12 A Goldfish in a Bowl
Slide 23-126
Example 23.12 A Goldfish in a Bowl
s  8.3 cm
Slide 23-128
Tactics: Ray Tracing for a Converging Lens
Slide 23-103
Tactics: Ray Tracing for a Converging Lens
Slide 23-104
QuickCheck 23.12
A lens creates an image as
shown. In this situation, the
object distance s is
A. Larger than the focal length f.
B. Equal to the focal length f.
C. Smaller than focal length f.
Slide 23-105
QuickCheck 23.12
A lens creates an image as
shown. In this situation, the
object distance s is
A. Larger than the focal length f.
B. Equal to the focal length f.
C. Smaller than focal length f.
Slide 23-106
QuickCheck 23.13
A lens creates an image as
shown. In this situation, the
image distance s is
A. Larger than the focal length f.
B. Equal to the focal length f.
C. Smaller than focal length f.
Slide 23-107
QuickCheck 23.13
A lens creates an image as
shown. In this situation, the
image distance s is
A. Larger than the focal length f.
B. Equal to the focal length f.
C. Smaller than focal length f.
Slide 23-108
Lateral Magnification
 The image can be either larger or smaller than the
object, depending on the location and focal length of
the lens.
 The lateral magnification m is defined as:
 A positive value of m indicates that the image is
upright relative to the object.
 A negative value of m indicates that the image is
inverted relative to the object.
 The absolute value of m gives the size ratio of the
image and object: h/h  |m|.
Slide 23-109
Virtual Images
 Consider a converging
lens for which the object
is inside the focal point,
at distance s < f.
 You can see all three
rays appear to diverge
from point P.
 Point P is an upright,
virtual image of the
object point P.
Slide 23-110
Virtual Images
 You can “see” a virtual
image by looking
through the lens.
 This is exactly what you
do with a magnifying
glass, microscope or
binoculars.
Slide 23-111
Example 23.9 Magnifying a Flower
Slide 23-112
Example 23.9 Magnifying a Flower
Slide 23-113
Example 23.9 Magnifying a Flower
Slide 23-114
Thin Lenses: Ray Tracing
 Three situations form the
basis for ray tracing
through a thin diverging
lens.
 Situation 1:
A ray initially parallel to
the optic axis will appear
to diverge from the near
focal point after passing
through the lens.
Slide 23-115
Thin Lenses: Ray Tracing
 Three situations form
the basis for ray
tracing through a thin
diverging lens.
 Situation 2:
A ray directed along a
line toward the far
focal point becomes
parallel to the optic
axis after passing
through the lens.
Slide 23-116
Thin Lenses: Ray Tracing
 Three situations form
the basis for ray
tracing through a thin
diverging lens.
 Situation 3:
A ray through the
center of a thin lens is
neither bent nor
displaced but travels in
a straight line.
Slide 23-117
QuickCheck 23.14
Light rays are converging to point
1. The lens is inserted into the rays
with its focal point at point 1.
Which picture shows the rays
leaving the lens?
Slide 23-118
QuickCheck 23.14
Light rays are converging to point
1. The lens is inserted into the rays
with its focal point at point 1.
Which picture shows the rays
leaving the lens?
Slide 23-119
Tactics: Ray Tracing for a Diverging Lens
Slide 23-120
Example 23.10 Demagnifying a Flower
Slide 23-121
Example 23.10 Demagnifying a Flower
Slide 23-122
Example 23.10 Demagnifying a Flower
Slide 23-123
Thin Lenses: Refraction Theory
 Consider a spherical boundary between two
transparent media with indices of refraction n1 and n2.
 The sphere has radius of curvature R and is centered
at point C.
Slide 23-124
Thin Lenses: Refraction Theory
If an object is located at distance s from a spherical
refracting surface, an image will be formed at distance
s given by:
Slide 23-125
Example 23.12 A Goldfish in a Bowl
Slide 23-126
Example 23.12 A Goldfish in a Bowl
Slide 23-127
Example 23.12 A Goldfish in a Bowl
Slide 23-128
Example 23.12 A Goldfish in a Bowl
s  8.3 cm
Slide 129
Lenses
In an actual lens, rays refract twice, at spherical
surfaces having radii of curvature R1 and R2.
Slide 23-130
The Thin Lens Equation
The object distance s is related to the image distance
s by:
where f is the focal length of the lens, which can be
found from:
where R1 is the radius of curvature of the first surface,
and R2 is the radius of curvature of the second surface,
and the material surrounding the lens has n = 1.
Slide 23-131
QuickCheck 23.15
A lens creates an image as
shown. In this situation,
A.
B.
C.
D.
s < f.
f < s < 2f.
s > 2f.
There’s not enough information
to compare s to f.
Slide 23-132
QuickCheck 23.15
A lens creates an image as
shown. In this situation,
A.
B.
C.
D.
s < f.
f < s < 2f.
s > 2f.
There’s not enough information
to compare s to f.
The image is real, which requires s > f.
The image is taller than the object, and
s’ > s requires s < 2f.
Slide 23-133
Example 23.13 Focal Length of a Meniscus Lens
Slide 23-134
Example 23.13 Focal Length of a Meniscus Lens
Slide 23-135
Example 23.15 A Magnifying Lens
Slide 23-136
Example 23.15 A Magnifying Lens
Slide 23-137
Example 23.15 A Magnifying Lens
Slide 23-138
Image Formation with Concave Spherical Mirrors
 The figure shows a
concave mirror, a
mirror in which the
edges curve toward
the light source.
 Rays parallel to the
optical axis reflect and
pass through the focal
point of the mirror.
Slide 23-139
A Real Image Formed by a Concave Mirror
Slide 23-140
Image Formation with Convex Spherical Mirrors
 The figure shows parallel
light rays approaching a
mirror in which the edges
curve away from the light
source.
 This is called a convex
mirror.
 The reflected rays appear
to come from a point
behind the mirror.
Slide 23-141
A Real Image Formed by a Convex Mirror
Slide 23-142
Image Formation with Spherical Mirrors
A city skyline is reflected in this polished sphere.
Slide 23-143
Tactics: Ray Tracing for a Spherical Mirror
Slide 23-144
Tactics: Ray Tracing for a Spherical Mirror
Slide 23-145
The Mirror Equation
For a spherical mirror with negligible thickness, the
object and image distances are related by:
where the focal
length f is related
to the mirror’s
radius of
curvature by:
Slide 23-146
QuickCheck 23.16
You see an upright, magnified image of your face when
you look into magnifying “cosmetic mirror.” The image
is located
A.
B.
C.
D.
In front of the mirror’s surface.
On the mirror’s surface.
Behind the mirror’s surface.
Only in your mind because it’s a virtual image.
Slide 23-147
QuickCheck 23.16
You see an upright, magnified image of your face when
you look into magnifying “cosmetic mirror.” The image
is located
A.
B.
C.
D.
In front of the mirror’s surface.
On the mirror’s surface.
Behind the mirror’s surface.
Only in your mind because it’s a virtual image.
Slide 23-148
Example 23.17 Analyzing a Concave Mirror
Slide 23-149
Example 23.17 Analyzing a Concave Mirror
Slide 23-150
Example 23.17 Analyzing a Concave Mirror
Slide 23-151
Example 23.17 Analyzing a Concave Mirror
Slide 23-152
Chapter 23 Summary Slides
Slide 23-153
General Principles
Slide 23-154
General Principles
Slide 23-155
Important Concepts
Slide 23-156
Important Concepts
Slide 23-157