• Law of Reflection• Law of Refraction or Snell’s Law• Chromatic Dispersion• Brewsters Angle
Chapter 33 ContinuedProperties of Light
Dispersion: Different wavelengths have different velocities andtherefore different indices of refraction. This leads to differentrefractive angles for different wavelengths. Thus the light is dispersed.The frequency does not change when n changes.
v = f!
! changes when medium changes
f does not change when medium changes
Snells LawRed
Snells Lawblue
Fiber Cable
Total Internal Reflection SimulatorHalliday, Resnick, Walker: Fundamentals of Physics,7th Edition - Student Companion Site
Why is light totally reflected inside a fiberoptics cable? Internal reflection
�
n1sin!
1= n
2sin!
2
(1.509)sin!1 = (1.00)sin90 = 1.00
!1 " sin#1 1
1.509" 41.505 deg
Total Internal Reflection
What causes a Mirage
1.06
1.091.08
1.07 1.071.08
1.09
sky eye
Hot road causes gradient in the index of refraction that increasesas you increase the distance from the road
Index of refraction
Inverse Mirage Bend
Chromatic Dispersion
How does a Rainbow get made?
Primary rainbowSecondary rainbowSupernumeraries
Polarization by Reflection: Brewsters Law
47. In the figure, a 2.00-m-long vertical pole extends from the bottomof a swimming pool to a point 50.0 cm above the water. What is thelength of the shadow of the pole on the level bottom of the pool?
Consider a ray that grazes the top of thepole, as shown in the diagram below. Hereθ1 = 35o, l1 = 0.50 m, and l2 = 1.50 m.
θ2
θ1
l2
l1
L x
airwater
shadow
x is given by
x = l1tanθ1 = (0.50m)tan35o = 0.35 m.
The length of the shadow is x + L.
L is given by
L=l2tan θ2
Use Snells Law to find θ2
Snells Law Example
According to the law of refraction, n2sinθ2 =n1sinθ1. 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.
θ2
θ1
l2
l1
L x
airwater
shadow
Calculation of L
Lecture 14 Images Chapter 34
•Geometrical Optics•Fermats Principle
-Law of reflection-Law of Refraction
•Plane Mirrors and Spherical Mirrors•Spherical refracting Surfaces•Thin Lenses•Optical Instruments
-Magnifying Glass, Microscope, Refracting telescope• Polling Questions
Geometrical Optics:Study of reflection and refractionof light from surfaces
The ray approximation states that light travels in straight linesuntil it is reflected or refracted and then travels in straight lines again.The wavelength of light must be small compared to the size ofthe objects or else diffractive effects occur.
Law of Reflection
1A
B
!I= !
R
!i
!r
Mirror
Fermat’s PrincipleUsing Fermat’s Principle you can prove theReflection law. It states that the path taken by light when traveling from one point toanother is the path that takes the shortesttime compared to nearby paths.
JAVA APPLET
Show Fermat’s principle simulator
Two light rays 1 and 2 taking different pathsbetween points A and B and reflecting off a
vertical mirror
1
2
A
B Plane Mirror
Use calculus - method of minimization
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 yand set the derivative to 0.
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
Plane Mirrors Where is the image formed
Mirrors and Lenses
Plane mirrors
Normal
Angle ofincidence
Angle of reflectioni = - p
Real sideVirtual side
Virtual image
eye
Object distance = - image distanceImage size = Object size
Problem: Two plane mirrors make an angle of 90o. Howmany images are there for an object placed betweenthem?
object
eye
1
2
3
mirror
mirror
Assuming no spinAssuming an elastic collisionNo cushion deformation
d
d
Using the Law of Reflection tomake a bank shot
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.
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
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:
Thin Lenses
n1
p+n2
i=n2! n
1
r
Apply this equation to Thin Lenses where the thickness issmall compared to object distance, image distance, and
radius of curvature. Neglect thickness.
Converging lens
Diverging lens
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 Lens m = !i
p
m = !image height
object height
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
Rules for drawing rays to locate images froma lens
1) A ray initially parallel to the central axis will pass through the focal point.
2) A ray that initially passes through the focal point will emerge from the lens parallel to the central axis.
3) A ray that is directed towards the center of the lens will go straightthrough the lens undeflected.
Real image ray diagram for aconverging lens
Object Distance=20 cmand lens focal length is +10 cm
Find: Image distanceMagnificationHeight of imageIs image erect or invertedIs it real or virtualDraw the 3 rays
Object Distance =5 cm and focal length is + 10 cm
Find: Image distanceMagnificationHeight of imageIs image erect or invertedIs it real or virtualDraw the 3 rays
p=Object Distance =10 cm
Virtual image ray diagram forconverging lens
Virtual image ray diagram for adiverging lens
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 the 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
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 imagereal 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 side Real side
r1. .F1 F2pr2
27. A converging lens with a focal length of +20 cm is located 10 cm tothe left of a diverging lens having a focal length of -15 cm. If an object islocated 40 cm to the left of the converging lens, locate and describecompletely the final image formed by the diverging lens. Treat each lensSeparately.
f1
f1
Lens 1 Lens 2
f2
f2
10
40
+20 -15
f1
f1
Lens 1 Lens 2
f2
f2
10
40
+20 -15
Ignoring the diverging lens (lens 2), the image formed by theconverging 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
�
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 isvirtual (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 imagehas the same size orientation as the object.
Optical Instruments
Magnifying lensCompound microscopeRefracting telescope
In Figure 34-35, A beam of parallel light rays from a laser is incident on asolid transparent sphere of index of refraction n.
Fig. 34-35(a) If a point image is produced at the back of the sphere, what is the
index of refraction of the sphere?
(b) What index of refraction, if any, will produce a point image at thecenter of the sphere? Enter 'none' if necessary.
Chapter 34 Problem 32
Chapter 34 Problem 91 Figure 34-43a shows the basic structure of a human eye. Lightrefracts into the eye through the cornea and is then further redirectedby a lens whose shape (and thus ability to focus the light) iscontrolled by muscles. We can treat the cornea and eye lens as asingle effective thin lens (Figure 34-43b). A "normal" eye can focusparallel light rays from a distant object O to a point on the retina atthe back of the eye, where processing of the visual informationbegins. As an object is brought close to the eye, however, themuscles must change the shape of the lens so that rays form aninverted real image on the retina (Figure 34-43c).
(a) Suppose that for the parallel rays of Figure 34-43a and Figure34-43b, the focal length f of the effective thin lens of the eye is2.52 cm. For an object at distance p = 48.0 cm, what focal lengthf' of the effective lens is required for the object to be seen clearly?
(b) Must the eye muscles increase or decrease the radii ofcurvature of the eye lens to give focal length f'?