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Lecture 14 Images Chapter 34
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• Law of Reflection
• Dispersion
• Snell’s Law
• Brewsters Angle
Preliminary topics before
mirrors and lenses
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•Law of Reflection
•Dispersion
•Snell’s Law
•Brewsters Angle
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Geometrical Optics:Study of reflection and refraction
of light from surfaces
The ray approximation states that light travels in straight lines
until it is reflected or refracted and then travels in straight lines again.
The wavelength of light must be small compared to the size of
the objects or else diffractive effects occur.
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Law of Reflection
1A
B
!I= !
R
!i
!r
Mirror
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Drawing Normals
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Fermat’s Principle
Using Fermat’s Principle you can prove the
Reflection law. It states that the path taken
by light when traveling from one point to
another is the path that takes the shortest
time compared to nearby paths.
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Two light rays 1 and 2 taking different paths
between points A and B and reflecting off a
vertical mirror
1
2
A
B Plane Mirror
Use calculus - method
of minimization
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t = 1
C( h
1
2+ y
2+ h
2
2+ (w ! y)
2)
dt
dy=
2y
h1
2+ y
2+
!2(w ! y)
h2
2+ (w ! y)
2)= 0
y
h1
2+ y
2=
(w ! y)
h2
2+ (w ! y)
2)
sin" I = sin"R
!I= !
R
Write down time as a function of y
and set the derivative to 0.
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Law of Refraction: Snells Law
�
n1sin!
1= n
2sin!
2
n1
n2
How do we prove it?
1
v1
sin!1=1
v2
sin!2
Air 1.0
Glass 1.33
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t = (1
v1
h1
2+ y
2+
1
v2
h2
2+ (w ! y)
2)
dt
dy= 0
1
v1
sin" I = 1
v2
sin"R
n1sin!
I= n
2 sin!
R
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JAVA APPLET
Show Fermat’s principle simulator
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What allows you to see various colors when
white light passes through a prism
Dispersion
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How does a Rainbow work?
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Dispersion: Different wavelengths have different velocities and
therefore different indices of refraction. This leads to different
refractive angles for different wavelengths. Thus the light is dispersed.
The frequency dose not change when n changes.
v = f!
! changes when medium changes
f does not change when medium changes
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Why is light totally reflected inside a fiber
optics cable? Internal reflection
�
n1sin!
1= n
2sin!
2
�
(1.33)sin!1 = (1.00)sin90 =1.00
When !1 " sin#1 1
1.33" 48.75 deg
light won't get out of the material
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Fiber Cable
Same here
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Corner Reflector?
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Show Total Internal reflection
simulator
Halliday, Resnick, Walker: Fundamentals of Physics, 7th
Edition - Student Companion Site
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What causes a Mirage
1.06
1.09
1.08
1.07 1.07
1.08
1.09
sky eye
Hot road causes gradient in the index of refraction that increases
as you increase the distance from the road
Index of refraction
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Inverse Mirage Bend
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47. In the figure, a 2.00-m-long vertical pole extends from the bottom
of a swimming pool to a point 50.0 cm above the water. What is the
length of the shadow of the pole on the level bottom of the pool?
Consider a ray that grazes the top of the
pole, as shown in the diagram below. Here
!1 = 35o, l1 = 0.50 m, and l2 = 1.50 m.
!2
!1
l2
l1
L x
air
water
shadow
x is given by
x = l1tan!1 = (0.50m)tan35o = 0.35 m.
The length of the shadow is x + L.
L is given by
L=l2tan !2
Use Snells Law to find !2
Snells Law Example
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According to the law of refraction, n2sin!2 =
n1sin!1. We take n1 = 1 and n2 = 1.33
o
o
n55.25
33.1
35sinsin
sinsin
1
2
11
2=!!
"
#$$%
&=!!
"
#$$%
&=
'' ((
L is given by
.72.055.25tan)50.1(tan 22 mmlLo=== !
The length of the shadow is L+x.
L+x = 0.35m + 0.72 m = 1.07 m.
!2
!1
l2
l1
L x
air
water
shadow
Calculation of L
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Polarization by Reflection Brewsters Law
(1.)sin!B= nsin!
r
!B+!
r= 90 we get 100% polarized reflected wave
sin !B = nsin(90 "!B) = ncos!
B
!B= tan"1
n Brewsters Law
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Plane Mirrors Where is the image formed
Mirrors and Lenses
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Plane mirrors
Normal
Angle of
incidence
Angle of reflectioni = - p
Real sideVirtual side
Virtual image
eye
Object distance = - image distance
Image size = Object size
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Problem: Two plane mirrors make an angle of 90o. How
many images are there for an object placed between
them?
object
eye
1
2
3
mirror
mirror
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Problem: Two plan mirrors make an angle of 60o. Find all
images for a point object on the bisector.
object
2
4
5,6
3
1
mirror
mirror
eye
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Assuming no spin
Assuming an elastic collision
No cushion deformation
d
d
Using the Law of Reflection to
make a bank shot
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What happens if we bend the mirror?
i = - p magnification = 1
Concave mirror.
Image gets magnified.
Field of view is diminished
Convex mirror.
Image is reduced.
Field of view increased.
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Rules for drawing images for mirrors
• Initial parallel ray reflects through focal point.
•Ray that passes in initially through focal point reflects parallel
from mirror
•Ray reflects from C the radius of curvature of mirror reflects along
itself.
• Ray that reflects from mirror at little point c is reflected
symmetrically
1
p+1
i=1
f
m =!i
p
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Concept Simulator/Illustrations
Halliday, Resnick, Walker: Fundamentals of Physics, 7th
Edition - Student Companion Site
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Spherical refracting surfaces
n1
p+n2
i=n2! n
1
r
Using Snell’s Law and assuming small
Angles between the rays with the central
axis, we get the following formula:
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Apply this equation to Thin Lenses where the thickness is
small compared to object distance, image distance, and
radius of curvature. Neglect thickness.
Converging lens
Diverging lens
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Thin Lens Equation
�
1
f=1
p+1
i
Lensmaker Equation
�
1
f= (n !1)(
1
r1
!1
r2
)
What is the sign convention?
Lateral Magnification for a Lensm = !
i
p
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Sign Convention
p
Virtual side - V Real side - R
i
Light
Real object - distance p is pos on V side (Incident rays are diverging)
Radius of curvature is pos on R side.
Real image - distance is pos on R side.
Virtual object - distance is neg on R side. Incident rays are converging)
Radius of curvature is neg on the V side.
Virtual image- distance is neg on the V side.
r2r1
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Rules for drawing rays to locate images from
a lens
•A ray initially parallel to the central axis will pass through the focal point.
•A ray that initially passes through the focal point will emerge from the lens
parallel to the central axis.
• A ray that is directed towards the center of the lens will go straight
through the lens undeflected.
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Example of drawing images
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24(b). Given a lens with a focal length f = 5 cm and object distance p
= +10 cm, find the following: i and m. Is the image real or virtual?
Upright or inverted? Draw 3 rays.
pfi
111!=
�
m =! y
y= "
i
p
�
1
i=1
5!1
10= +
1
10
Image is real,
inverted.
. .F1
F2p
Virtual side Real side
�
m = !10
10= !1
�
i = +10 cm
Example
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24(e). Given a lens with the properties (lengths in cm) r1 = +30, r2 =
-30, p = +10, and n = 1.5, find the following: f, i and m. Is the image
real or virtual? Upright or inverted? Draw 3 rays.
( ) !!"
#$$%
&''=
21
111
1
rrn
f
( )30
1
30
1
30
115.1
1=!
"#
$%&
'''=
f
cmf 30=
pfi
111!=
15
1
10
1
30
11!=!=
i
cmi 15!=
�
m =! y
y= "
i
p
�
m = !!15
10= +1.5
Image is virtual,
upright.
Virtual sideReal side
r1. .F1 F2
pr2
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27. A converging lens with a focal length of +20 cm is located 10 cm to
the left of a diverging lens having a focal length of -15 cm. If an object is
located 40 cm to the left of the converging lens, locate and describe
completely the final image formed by the diverging lens. Treat each lens
Separately.
f1
f1
Lens 1 Lens 2
f2
f2
10
40
+20 -15
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f1
f1
Lens 1 Lens 2
f2
f2
10
40
+20 -15
Ignoring the diverging lens (lens 2), the image formed by the
converging lens (lens 1) is located at a distance
�
1
i1
=1
f1
!1
p1
=1
20cm!
1
40cm. i
1= 40cm
40
This image now serves as a virtual object for lens 2, with p2 = - (40 cm - 10 cm) = - 30 cm.
30
Since m = -i1/p1= - 40/40= - 1 , the image is inverted
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�
1
i2
=1
f2
!1
p2
=1
!15cm!
1
!30cm i
2= !30cm.
Thus, the image formed by lens 2 is located 30 cm to the left of lens 2. It is
virtual (since i2 < 0).
f1
f1
Lens 1 Lens 2
f2
f2
10
40
+20 -15
40
30
The magnification is m = (-i1/p1) x (-i2/p2) = (-40/40)x(30/-30) =+1, so the image
has the same size orientation as the object.
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Optical Instruments
Magnifying lens
Compound microscope
Refracting telescope