chapter 26 ray optics
TRANSCRIPT
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Chapter 26 Geometric Opticsinstead of as waves, looking at light as rays along in the direction of propagation
Reection
a process in which light bounces off of a surface
the angle of incidence is equal to the angle of reection, law of reection
i=
r
angles are measured from the normal to the surface
if the surface is smooth such that all light rays are reected in the samedirection, this is called specular reection
if the surface is rough such that all light rays are not reected in the same direction, this is calleddiffused reection
Plane Mirror
a plane mirror is just a at mirror
mirror
object
observer
image
mirror
object
observer
image
the object distance is the distance between the mirror and object
the image distance is the distance between the mirror and image
for a plane mirror these two distances are the same
the magnication is +1 because the apparent size of the image is the same as the object and theobject and image have the same orientation
mirror
object image
objectdistance
imagedistance
objectsize
imagesize
objectorientation imageorientation
normal
surface
i
r
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Example 26-1
An observer at table level a distance d behind the object which is also a distance d from a planemirror looks at the top of the image. What is the height of the location where the light from the topof the object bounce off of the mirror (in terms of the height of the object)?
mirror
object
observer
imagedd
yh
for the labeled diagram below,
dd
yh
h-y
the shaded triangles say that
tan =h y
d=
y2d
2(h y) = y 2h 2y = y y =23
h
Example: mirror size
What is the minimum height of a mirror required for a person to see himself entirely?
h
observer
h/2
Due to the law of reection, 1/2 height of the person.
Example: corner reector
Two mirrors are placed at right angle with each other. What is theangle of the reected ray with respect to the x axis if the incidentrays is 30 from the x axis?
By looking at the triangles, the angle of the reected ray is also 30.
Spherical Mirror
a mirror made of a section of a sphere
if the reective side is curved in then it is a concave mirror
if the reective side is curved out then it is a convex mirror
the radius is always perpendicular to the mirror surface so it isalways the normal
306060
3030
concave mirror
center focusprincipal
axis
radius
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for a concave mirror, the focus is dened by where parallelincident rays converge after reection
for a convex mirror, the focus is dened by where parallelincident rays appear to originate after reection
the focus is located half way between the mirror and the centerof curvature of the mirror
for a concave mirror, the actual light rays converge so the imageis called real
for a convex mirror, the actual light rays diverge so the image is called virtual
center focus
incidentlight ray
r
i
center focus
incidentlight ray
r i
Spherical Aberration
for large mirrors, parallel rays focus perfectly for only parabolic mirrors
for spherical mirrors, the reected light rays don t exactly line up at the focus so the image isalways slightly fuzzy
Ray Tracing
method to nd the image due to an optical instrument graphically
where these three rays intersect (or appear to originate) is where the image is locatedthe 3 principal rays
1. " incident ray parallel to the principal axis reects through the focus2. " incident ray going through the focus reects back parallel to the principal axis3. " incident ray going through the center of curvature returns reects back through the center of
curvature
concave mirror: object outside the focus
f c
principal ray 1
o
principal ray 2
f co
principal ray 3
f co
convex mirror
center focusprincipal
axis
radius
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the image is real, upside-down, and smaller
image
f c
o
concave mirror: object inside the focus
f c
principal ray 1
o
principal ray 2
f c
o
principal ray 3
f c
o
the image is virtual, right-side-up, and larger
demo: concave mirror
convex mirror: object anywhere
principal ray 3principal ray 1 principal ray 2
the image is virtual, right-side-up, and smaller
animation: convex mirror
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Mirror Equation
mathematical method for nding the properties of animage due to a mirror
the focal length is dened as
f =
1
2R
+ for a concave and for a convex mirrorthe mirror equation says where the image is
1
f =
1
do
+1
di
the size of the image is given by the magnication
m =h
i
ho
= d
i
do
Mirror Sign Conventions
focal length: concave mirror is positive and convex mirror is negativeobject distance: in front of mirror is positive and behind the mirror is negativeimage distance: in front of mirror is positive and behind the mirror is negative
image magnication: same orientation as the object is positive and opposite orientation as theobject is negative
Example
The concave side of a spoon has a focal length of 5.00 cm. What are the image distances andmagnications of an object whose object distances are (a) 25.0 cm, (b) 9.00 cm, and (c) 2.00 cm?
A diagram of part (a) shows that the image distance is approximately +6 cm.
510
25
Using the mirror equation,
d i =
1
f 1
do
1
=
1
5 cm 1
25 cm
1
= 6.25 cm
The magnication is
m = d
i
do
= 6.2525
= 0.25 = 1
4
do>0
d i>0
ho>0
h i0
do>0
d i0
h i>0
f
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A diagram of part (b) shows that the image distance is about +10 cm.
Using the mirror equation,
di
=
1
f
1
do
1
=
1
5 cm
1
9 cm
1
= 11.3 cm
The magnication is
m = d id
o
= 11.39
= 0.25 = 1.25
A diagram of part (c) shows that the image distance is about -1.3cm.
Using the mirror equation,
di
=
1
f
1
do
1
=
1
5 cm
1
2 cm
1
= 3.33 cm
At least the sign is correct and it is in the ball park.
The magnication is
m = d
i
do
= 3.33
2= + 1.7
Example
A convex mirror has a radius of curvature of 20.0 cm. What is the image distance for an object6.33 cm from the mirror? What is the magnication of the image?
10 206.33 ~4.5
It appears that the image distance should be about -4.5 cm and the magnication is about +0.6.
The mirror equation says that
di
=
1
f
1
do
1
=
1
10 cm
1
6.33 cm
1
=
3.88 cm
The magnication is
m = d
i
do
= 3.88 cm6.33 cm
= + 0.61
510
9
510
2
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Example
A 1.75 m person standing 1.10 m from a sphere of 8.50 cm diameter casts an image of what sizeand location?
The object size is 1.75 m. The object distance is 1.10 m. The radius of curvature is 0.0425 m.The focal length is 0.02125 m. Since the mirror is convex, the focal length is actually -0.02125 m.Using the mirror equation,
1
di
=
1
f 1
do
d i=
1
f 1
do
1
=
1
0.02125 m 1
1.10 m
1
= 0.0208 m = 2.08 cm
This is 2.08 cm behind the mirror. The size of the image is
m = d
i
do
= 0.0208 m
1.10 m= + 0.0190 h
i= 0.0190h
o= 0.0332 m = 3.32 cm
Example
To look at the back side of a tooth with a greater than 1 magnication and an upright image, whatkind of mirror must be used?
If the mirror is placed 1.5 cm from the tooth and a magnication of +2.0 is desired, what should bethe focal length of the mirror?
To meet both requirements, a concave mirror is necessary.
The mirror equation says that
1
f =
1
do
+1
di
f =1
do
+1
di
1
=1
1.5 cm+
1
di
1
Since the magnication is
m = d i
d o= + 2.0, d i = 2d o
combining the two equations gives
f =
11.5
+1
2(1.5)
1
= 3 cm
Refraction
the bending of light path passing through the interface between two materials with differentspeeds of light
the speed of the medium is measured using the index of refraction, n, dened as
v =c
n
or n =c
v
for some substances: diamond 2.4, glass 1.4 to 1.7, ice 1.3, water 1.33, air 1.000
the angle of refracted light depends on the angle of incident measured from the normal
Snell s Law relates the angles and indexes of refraction
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n i sin i = n r sin r
alternatively
sin r
=
ni
nr
sin i
if light travels from a lower index medium to a higher one, the refracted angle is smaller than theincident angle
if light travels from a higher index medium to a lower one, the refracted angle is larger than theincident angle
no refraction occurs if the incident angle is 0 or if the indices were the same
Example
You shine a laser at an object that is under water. The beam startsout 1.8 m above the water and it hits a spot 2.4 m away on thesurface. The water is 5.5 m deep. How far away is the object fromyou?
From the diagram, the incident angle is
tan
i=
2.4 m
1.8 m
i= 0.927 rad = 53.1
The index of refraction of air is 1.00 and of water is 1.33. So therefracted angle is
(1.00)sin(0.927 rad)= (1.33)sin r r
= 0.645 rad = 37.0
This means the distance x is
tan
r=
x
5.5 m x = 4.14 m
and the distance of the object from you is 6.5 m. And it appearto come from a shallower location.
Total Internal Reection
when a ray comes from a higher index of refraction medium toone that is lower, the refracted angle is larger than the incidentangle
the refracted angle will reach 90 before the incident angle
the incident angle at which the refracted angle is 90 is calledthe critical angle
The light is reected back from the interface and is called totalinternal reection
nisin
i= n
rsin
r n
isin
c= n
rsin90 sin
c=
nr
ni
the reected light is total polarized parallel to the interfacesurface when the reected and refracted angles areperpendicular to each other
5.5 m
1.8 m
2.4 m
x
object
i
r
actual objectlocation
apparent objectlocation
incidentlight
partiallyreflected
light
refractedlight
incidentlight
totallyreflected
light
refractedlight
criticalangle
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Example
What are the critical angles for light traveling from glass (n = 1.50) to air (1.00) and from glass towater (n = 1.33)?
From glass to air,
n i sin
i= n r sin r (1.50)sin c
= (1.00)sin90 = 1.00 c= 41.8
From glass to water,
n i sin i=
n r sin r (1.50)sin c=
(1.33)sin90 =
1.33 c=
62.5
Demo: light pipe
critical angle path
fiber (n=1.5)
cladding (n
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the image is made of actual, converging light rays so it is real
object inside the focus
lens
principal axisfocus object
image
the image is made of light rays that don t converge to form an image so the image is virtual
Demo: converging lens animation
Diverging Lens
the 3 principal rays
1. " ray parallel to the principal axis appears to come from the focus on the same side2. " ray going toward the opposite focus comes out parallel to the principal axis3. " ray going through the center of the lens passes through unchanged
for an object beyond the focus, the image is virtual since the light rays do not converge but looksto come from some other location
lens
principal axisfocus
object image
for an object inside the focus, the image is still virtuallens
principal axisfocus
object
image
Demo: diverging lens animation
Thin Lens Equationmathematical way to describe a thin lens
1
f =
1
do
+1
di
with the magnication
m = d
i
do
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Thin Lens Sign Conventions
focal length: converging lens is positive and diverging lens is negative
object distance: positive (real) if the object is on the side where the light is coming toward the lens" negative (virtual) if the object is on the side where the light is leaving the lens
image distance: positive (real) if the image is on the side where the light is leaving the lens" negative (virtual) if the image is on the side where the light is coming toward the lens
image magnication: same orientation as object is positive and opposite object is negative
Example
A glass converging lens has a focal length of f. Would the focal length of the same lens be largeror smaller if the lens were to be immersed in water?
The amount of refraction is decreased when immersed in water so the focal length increases.
Example
A lens produces a real image twice as larger as the original object. The image is located 15 cmfrom the lens. What is the object distance and what is the focal length of the lens?
Since the image is real, it means the lens is a converging lens. It also means that the image isupside-down. The second statements say the following
m = d
i
do
do
= d
i
m=
15 cm
2= + 7.5 cm
The object distance is +7.5 cm so the focal length is
1
f =
1
do
+1
di
f =1
do
+1
di
1
=1
7.5 cm+
1
15 cm
1
= 5.0 cm
ExampleA object is placed 12 cm from a diverging lens of focal length -7.9 cm. What is the image distanceand what is the magnication of the image?
The thin lens equation says that the image distance is
1
f =
1
do
+1
di
d i =1
f
1
do
1
=1
7.9 cm 1
12 cm
1
= 4.8 cm
The magnication then is
m = d
i
do
= 4.8 cm12 cm
= + 0.40
Dispersion
refraction is frequency dependent
the index of refraction is different for different frequencies
this frequency dependence is called dispersion
higher frequency light (blue) bends more (n is larger) than lowerfrequency light (red) (n is smaller)
higher index
lower index
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Rainbow
rainbows are formed from sunlight is refracted then reected thenrefracted again from water droplets in the sky
blue light is refracted more so it comes back at a sharper anglefrom the sun light
observer
sun
rain drops
sun light
rainbow is seen when you are between the sun and the waterdroplets
in fact, if you are elevated, you could see the entire rainbowwhich is circularly symmetric
observer sun
rain drops
sun light (white)
if the sun light is strong enough and there is enough rain drops, you can see a secondary rainbowin which the color order is reversed because of an extra bounce within the rain drop
more light is lost so it is also weaker
observer
sun
rain drops
sun light
sun light
sun light
angle-accurate diagram
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