Chapter 26 Ray Optics

8/6/2019 Chapter 26 Ray Optics

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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.


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.


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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Chapter 26 Ray Optics

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