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.
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1
A Spherical Mirror: Central Rays
A few rays are
easy to figure out
where they go.
All rays satisfy
the “angle of incidence
= angle of reflection”
measured to the normal
to the surface
All rays through
the center strike
the mirror perpendicular
to the surface and
bounce back
along their
incoming path.
center
of sphere
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2
A Spherical Mirror: Central Ray
A few rays are
easy to figure out
where they go.
All rays satisfy
the “angle of incidence
= angle of reflection”
measured to the normal
to the surface
The ray hitting the
central line of the
diagram is particularly
simple.
center
of sphere
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3
A Spherical Mirror: Parallel Rays
A few rays are
easy to figure out
where they go.
All rays satisfy
the “angle of incidence
= angle of reflection”
measured to the normal
to the surface
All rays parallel to
and near an axis of
the sphere reflect through
a single point on the
axis (the focal point)
center
of sphere
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4
A Real Image Formed by a Concave Mirror
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5
Images in a Spherical Mirror: 1
Physical
center of sphere
focal point
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6
Tactics: Ray Tracing for a Spherical Mirror
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7
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8
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:
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PHYS 132
9
You see an upright, magnified image of your face when
you look into magnifying “cosmetic mirror.” The image
is located
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PHYS 132
it.
.
be
ca
us
e
su
rfa
ce
.
yo
ur
m
ir r
or
’s
On
ly
in
m
th
e
in
d
su
rfa
ce
.
m
irr
or
’s
th
e
On
Be
hi
nd
fr
on
to
ft
he
m
ir r
or
’s
su
rfa
ce
.
25% 25% 25% 25%
In
A. In front of the mirror’s
surface.
B. On the mirror’s surface.
C. Behind the mirror’s surface.
D. Only in your mind because
it’s a virtual image.
10
Example 23.17 Analyzing a Concave Mirror
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11
Example 23.17 Analyzing a Concave Mirror
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12
Example 23.17 Analyzing a Concave Mirror
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13
Example 23.17 Analyzing a Concave Mirror
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14
Image Formation with Spherical Mirrors
A city skyline is reflected in this polished sphere.
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15
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.
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16
A Real Image Formed by a Convex Mirror
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17
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.
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18
Converging Lenses
 A converging lens is
thicker in the center
than at the edges.
 The focal length f is the
distance from the lens
at which rays parallel to
the optical axis
converge.
 The focal length is a
property of the lens,
independent of how the
lens is used.
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19
Diverging Lenses
 A diverging lens is
thicker at the edges
than in the center.
 The focal length f is the
distance from the lens
at which rays parallel to
the optical axis appear
to diverge.
 The focal length is a
property of the lens,
independent of how the
lens is used.
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20
You can use the sun’s rays and a lens to
start a fire. To do so, you should use
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33%
di
v.
..
or
a
Ei
th
er
ac
A
di
ve
rg
in
g
en
s
co
nv
er
gi
ng
l
A
33%
le
ns
.
.
33%
on
ve
rg
in
g
A. A converging lens.
B. A diverging lens.
C. Either a converging
or a diverging lens
will work if you use
it correctly.
PHYS 132
21
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.
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22
Thin Lenses: Ray Tracing
 Three situations form
the basis for ray
tracing through a
thin converging
lens.
 Situation 2:
A ray through the
near focal point of a
thin lens becomes
parallel to the optic
axis after passing
through the lens.
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23
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.
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24
Thin Lenses: Ray Tracing
Rays from an object point P are refracted by the lens and converge to a real image at point P′.
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25
A lens produces a sharply
focused, inverted image on a
screen. What will you see on
the screen if the lens is
removed?
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PHYS 132
im
ag
e
An
in
ve
rte
d
bu
tb
lu
rr
yi
th
m
at
ag
is
e.
di
m
A
m
sh
er
ar
bu
p,
...
up
r
ig
A
ht
bl
im
ur
ry
ag
,u
e.
pr
ig
ht
im
ag
No
e.
im
ag
e
at
al
l.
20% 20% 20% 20% 20%
An
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.
26
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?
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PHYS 132
im
ag
e
An
in
ve
rte
d
bu
tb
lu
rr
yi
th
m
a
t
ag
On
i
s
e.
ly
di
th
m
m
e
er
to
p
bu
On
ha
t..
ly
lf
.
th
of
e
th
bo
e
im
tto
ag
m
e.
ha
lf
of
th
e
...
No
im
ag
e
at
al
l.
20% 20% 20% 20% 20%
An
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.
27
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?
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PHYS 132
im
ag
e
An
in
ve
rte
d
bu
tb
lu
rr
yi
th
m
a
t
ag
On
i
s
e.
ly
di
th
m
m
e
er
to
p
bu
On
ha
t..
ly
lf
.
th
of
e
th
bo
e
im
tto
ag
m
e.
ha
lf
of
th
e
...
No
im
ag
e
at
al
l.
20% 20% 20% 20% 20%
An
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.
28
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.
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29
Tactics: Ray Tracing for a Converging Lens
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30
Tactics: Ray Tracing for a Converging Lens
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31
A lens creates an image
as shown. In this
situation, the object
distance s is
PHYS 132
f.
33%
en
gt
h
le
ng
th
f.
f.
en
gt
h
th
e
Sh
or
te
rt
ha
n
Eq
ua
l
to
th
e
fo
ca
ll
fo
ca
ll
th
e
th
an
La
rg
er
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33%
fo
ca
l
A. Larger than the focal length f.
B. Equal to the focal length f.
C. Shorter than the focal length f.
33%
32
A lens creates an image
as shown. In this
situation, the image
distance s′ is
PHYS 132
f.
33%
en
gt
h
le
ng
th
f.
f.
en
gt
h
th
e
Sh
or
te
rt
ha
n
Eq
ua
l
to
th
e
fo
ca
ll
fo
ca
ll
th
e
th
an
La
rg
er
4/28/2014
33%
fo
ca
l
A. Larger than the focal length f.
B. Equal to the focal length f.
C. Shorter than the focal length f.
33%
33
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|.
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34
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.
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35
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.
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36
Example 23.9 Magnifying a Flower
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37
Example 23.9 Magnifying a Flower
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38
Example 23.9 Magnifying a Flower
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39
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.
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40
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.
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41
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.
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42
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?
4/28/2014
20%
E
20%
D
20%
C
20%
B
A
20%
PHYS 132
43
Tactics: Ray Tracing for a Diverging Lens
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44
Example 23.10 Demagnifying a Flower
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45
Example 23.10 Demagnifying a Flower
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46
Example 23.10 Demagnifying a Flower
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47
Wave Model
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48
The second model for light:
Electromagnetic wave
• Light is an oscillating
electromagnetic wave. (Long story)
• A “close-up” of a ray: a plane wave
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49
It’s hard to picture EM waves in 3D
• Let’s build some intuition by working through
a simpler example.
Waves on the surface of water
(treating the height of the surface only –
that moves up and down – transvers to the
wave motion: the actual bits of water move
in small circles)
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50
http://www.falstad.com/ripple/
Ripple tank analogy
Can two sources lead
to both “bright spots”
and “dark spots”?
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PHYS 132
51
Chapter 22 Preview
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52
100 micron slit
Spot actually gets wider…
Does this mean light has a “size”?
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53
Chapter 22 Preview
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54
What a difference a slit makes
The big deal here is that opening an additional slit
makes it darker in some places.
No way this happens in either
the ray or photon model.
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55
Diffraction of Light
 When red light
passes through
an opening that
is only 0.1 mm wide,
it does spread out.
 Diffraction of light
is observable if
the hole is
sufficiently small.
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56
Young’s Double-Slit Experiment
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57
Young’s Double-Slit Experiment
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58
Analyzing Double-Slit Interference
 The figure shows the
“big picture” of
the double-slit
experiment.
 The next slide
zooms in on the area
inside the circle.
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59
Analyzing Double-Slit Interference
 The figure shows a
magnified portion of the
double-slit experiment.
 The wave from the lower
slit travels an extra
distance.
 Bright fringes (constructive
interference) will occur at
angles θm such that ∆r = mλ,
where m = 0, 1, 2, 3, …
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60
Analyzing Double-Slit Interference
 The mth bright fringe emerging from the double slit
is at an angle:
where θm is in radians, and we have used the smallangle approximation.
 The y-position on the screen of the mth bright fringe on
a screen a distance L away is:
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61
A laboratory experiment produces a double-slit interference
pattern on a screen. The point on the screen marked with
a dot is how much farther from the left slit than from the
right slit?
A.
B.
C.
D.
E.
1.0 λ.
1.5 λ.
2.0 λ.
2.5 λ.
3.0 λ.
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62
A laboratory experiment produces a double-slit
interference pattern on a screen. If the screen is moved
farther away from the slits, the fringes will be
A. closer together.
B. in the same
positions.
C. farther apart.
D. fuzzy and out of
focus.
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m
e
fa
rt
he
ra
pa
fu
rt.
zz
ya
nd
ou
to
ff
oc
us
.
in
th
e
sa
clo
se
rt
og
et
he
r.
po
si t
io
ns
.
25% 25% 25% 25%
63
A laboratory experiment produces a double-slit
interference pattern on a screen. If green light is used,
with everything else the same, the bright fringes will be
4/28/2014
fa
rt
he
ra
pa
rt.
no
fr
in
ge
sb
ec
...
be
ill
w
r.
th
e
sa
m
e
rt
og
et
he
po
si t
io
ns
.
Th
er
e
∆y =
25% 25% 25% 25%
in
closer together.
in the same positions.
farther apart.
There will be no
fringes because the
conditions for
interference won’t be
satisfied.λ L
clo
se
A.
B.
C.
D.
and green light has a shorter wavelength.
d
PHYS 132
64
A laboratory experiment produces a double-slit
interference pattern on a screen. If the slits are moved
closer together, the bright fringes will be
4/28/2014
PHYS 132
fa
rt
he
ra
pa
be
rt.
no
fr
in
ge
sb
ec
...
ill
w
r.
sa
m
e
rt
og
et
he
po
si t
io
ns
.
Th
er
e
d
and d is smaller.
th
e
∆y =
25% 25% 25% 25%
in
closer together.
in the same positions.
farther apart.
There will be no
fringes because the
conditions for
interference won’t be
satisfied.
λL
clo
se
A.
B.
C.
D.
65
The figure shows what happens if you put
white light through the same slit-screen
system. Why are the different colors
separated on either side of the center?
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66
Intensity of the Double-Slit Interference Pattern
The intensity of the
double-slit interference
pattern at position y is:
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67
Intensity of the Double-Slit Interference Pattern
The actual intensity
from a double-slit
experiment slowly
decreases as |y|
increases.
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68
Single-Slit Diffraction
 Diffraction through
a tall, narrow slit is
known as single-slit
diffraction.
 A viewing screen is
placed distance L
behind the slit of
width a, and we will
assume that L >> a.
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69
Huygens’ Principle: Plane Waves
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70
Huygens’ Principle: Spherical Waves
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71
Analyzing Single-Slit Diffraction
 The figure shows a
wave front passing
through a narrow slit
of width a.
 According to Huygens’
principle, each point
on the wave front can
be thought of as the
source of a spherical
wavelet.
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72
Single-Slit Diffraction
 The light pattern from a
single slit consists of a
central maximum flanked
by a series of weaker
secondary maxima and
dark fringes.
 The dark fringes occur at
angles:
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73
The Width of a Single-Slit Diffraction Pattern
 The central maximum of this single-slit
diffraction pattern is much brighter than
the secondary maximum.
 The width of the central maximum on
a screen a distance L away is twice
the spacing between the dark fringes
on either side:
 The farther away from the screen (larger L), the wider the
pattern of light becomes.
 The narrower the opening (smaller a), the wider the pattern
of light becomes!
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74
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PHYS 132
25%
25%
D
25%
C
A
25%
B
A laboratory experiment
produces a double-slit
interference pattern on a
screen. If the left slit is blocked,
the screen will look like
75