OpticsOpticsLCHS

Notation for Mirrors and Lenses

• The object distance is the distance from the object to the mirror or lens

• The image distance is the distance from the image to the mirror or lens

• The lateral magnification of the mirror or lens is the ratio of the image height to the object height

Types of Images

• A real image is formed when light rays pass through and diverge from the image point

• A virtual image is formed when light rays do not pass through the image point but only appear to diverge from that point

Flat MirrrorsFlat Mirrrors

Law of ReflectionLaw of Reflection

“The angle of incidence equals the angle of reflection.”

This is true for both flat mirrors and curved mirrors.

MIRROR

A Angle of Angle of IncidenceIncidence

BAngle of Angle of ReflectionReflection

Normal LineNormal Line

==

Diffuse ReflectionDiffuse Reflection

Locating the Image for Locating the Image for Plane MirrorsPlane Mirrors

1.1. Draw the image the same distance behind Draw the image the same distance behind the mirror as the object is in front.the mirror as the object is in front.

2.2. Draw a connector line from each object to Draw a connector line from each object to each image.each image.

3.3. If the connector line passes through the If the connector line passes through the mirror, the image will be seen. mirror, the image will be seen.

A C D EBA EDB CMirror ImagesMirror Images

These lines are These lines are pointed to the only pointed to the only images that will be images that will be seen from each of seen from each of

the original the original locations (A-E) locations (A-E)

NOTENOTE: No : No images will be images will be

seen from Eseen from E

Lateral MagnificationLateral MagnificationLateral magnification, M, is defined as

– This is the general magnification for any type of mirror

– It is also valid for images formed by lenses

– Magnification does not always mean bigger, the size can either increase or decrease

h'h

heightObjectheightageIm

M

Lateral Magnification of a Flat Lateral Magnification of a Flat MirrorMirror

• The lateral magnification of a flat mirror is 1

• This means that h' = h for all images

Reversals in a Flat MirrorReversals in a Flat Mirror

• A flat mirror produces an image that has an apparent left-right reversal– For example, if you raise

your right hand the image you see raises its left hand

• The reversal is not actually a left-right reversal• The reversal is actually a front-back reversal

– It is caused by the light rays going forward toward the mirror and then reflecting back from it

SummarySummary• The image is as far behind the mirror as the object

is in front– dd = |do|

• The image is unmagnified– The image height is the same as the object height

• h' = h and M = 1

• The image is virtual• The image is upright

– It has the same orientation as the object

• There is a front-back reversal in the image

The angle of incidence equals the The angle of incidence equals the angle of what?angle of what?

a) Dispersiona) Dispersion

b) Refractionb) Refraction

c) Reflectionc) Reflection

Specular reflections are images Specular reflections are images seen after __ surface(s).seen after __ surface(s).

a) rougha) rough

b) smoothb) smooth

c) noc) no

Can you see a reflection of Can you see a reflection of yourself in a diffuse reflection?yourself in a diffuse reflection?

a) yesa) yes

b) nob) no

Where do you draw the connector Where do you draw the connector lines?lines?

a) from the lens to objecta) from the lens to objectb) from image to lensb) from image to lensc) from object to each imagec) from object to each image

What happens if the connector line What happens if the connector line passes through the mirror?passes through the mirror?

a) image is invisiblea) image is invisibleb) image is seenb) image is seen

Light doesn’t pass through __ Light doesn’t pass through __ images.images.

a) virtuala) virtualb) realb) realc) largec) larged) smalld) small

SphericalSphericalMirrorsMirrors

Spherical MirrorsSpherical Mirrors• A spherical mirror has the shape of a segment of

a sphere

• The mirror focuses incoming parallel rays to a point

• A concave spherical mirror has the light reflected from the inner, or concave, side of the curve

• A convex spherical mirror has the light reflected from the outer, or convex, side of the curve

Concave and Convex MirrorsConcave and Convex Mirrors

Concave and convex mirrors are curved mirrors similar to portions of a sphere.

light rays light rays

Concave mirrors reflect light from their inner surface, like

the inside of a spoon.

Convex mirrors reflect light from their outer surface, like

the outside of a spoon.

Concave MirrorsConcave Mirrors

Light from Infinite DistanceLight from Infinite Distance

C F

Focuses at the focal point

Two Rules for Concave MirrorsTwo Rules for Concave Mirrors

• Any incident ray passing through the focal point on the way to the mirror will travel parallel to the principal axis upon reflection

• Any incident ray traveling parallel to the principal axis on the way to the mirror will pass through the focal point upon reflection

C F

C F

C F

C F

Virtual

Image

C F

Real ImageReal Image

C F

Virtual Virtual ImageImage

Convex MirrorsConvex Mirrors• A convex mirror is sometimes called a A convex mirror is sometimes called a

diverging diverging mirrormirror– The light reflects from the outer, convex sideThe light reflects from the outer, convex side

• The rays from any point on the object The rays from any point on the object diverge after reflection as though they were diverge after reflection as though they were coming from some point behind the mirror coming from some point behind the mirror

• The image is virtual because the reflected The image is virtual because the reflected rays only appear to originate at the image rays only appear to originate at the image pointpoint

F

Will an image Will an image ever focus at a ever focus at a

single point with a single point with a convex mirror?convex mirror?

Therefore, the images you see

are virtual!

Image Formed by a Convex MirrorImage Formed by a Convex Mirror

In general, the image formed by a convex mirror is upright, virtual, and smaller than the object

Notes on ImagesNotes on Images• With a With a concaveconcave mirror, the image may be either mirror, the image may be either

real or virtual. When the object isreal or virtual. When the object is – outsideoutside the focal point, the image is real the focal point, the image is real– atat the focal point, the image is infinitely far away the focal point, the image is infinitely far away– insideinside the focal point, the image is virtual the focal point, the image is virtual

• With a With a convexconvex mirror, the image is always mirror, the image is always virtual and uprightvirtual and upright– As the object distance decreases, the virtual image As the object distance decreases, the virtual image

increases in sizeincreases in size

Mirror Sign ConventionMirror Sign Convention

+ for real image

- for virtual image

+ for concave mirrors

- for convex mirrors

1f =

1do

1di

+

f = focal length

di = image distance

do = object distance

di

f

MagnificationMagnification

m = magnification

hi = image height (negative means inverted)

ho = object height

AND m = hi / ho = -di / do

m = hi

ho

By definition,

Magnification is simply the ratio of image height to object height. A positive magnification means an upright image.

• •CF

Casey decides to join in Casey decides to join in the fun and she finds a the fun and she finds a convex mirror to stand convex mirror to stand in front of. She sees her in front of. She sees her image reflected 7 feet image reflected 7 feet behind the mirror which behind the mirror which has a focal length of 11 has a focal length of 11 feet. Her image is 1 feet. Her image is 1 foot tall. Where is she foot tall. Where is she standing and how tall is standing and how tall is she?she?

do =

ho =

19.25 feet

2.75 feet

Mirror Equation Sample Problem

Suppose AllStar, who is 3 and Suppose AllStar, who is 3 and a half feet tall, stands 27 feet a half feet tall, stands 27 feet in front of a concave mirror in front of a concave mirror with a radius of curvature of with a radius of curvature of 20 feet. Where will his image 20 feet. Where will his image be reflectedbe reflected

di =

hi =

• •C F

15.88 feet

-2.06 feet

What will its size be?

Determine the image distance for a 5.00-cm tall object placed 10.0 cm from a concave mirror having a focal length of 15.0 cm.

Use 1 / f = 1 / do + 1 / di where f = 15 cm and do = 10.0 cm

di = -30.0 cm

Determine the image height for a 5.00-cm tall object placed 10.0 cm from a concave mirror having a focal length of 15.0 cm. (di = -30.0 cm)

Then use hi / ho = -di / do where ho = 5 cm, do = 45 cm, and di = -30.0 cm

hi = +15.0 cm

A 4.00-cm tall light bulb is placed a distance of A 4.00-cm tall light bulb is placed a distance of 45.7 cm from a concave mirror having a focal 45.7 cm from a concave mirror having a focal

length of 15.2 cm. Determine the image distance.length of 15.2 cm. Determine the image distance.

1/f = 1/do + 1/d1/f = 1/do + 1/dii

1/(15.2 cm) = 1/(45.7 cm) + 1/d1/(15.2 cm) = 1/(45.7 cm) + 1/d ii

0.0658 cm0.0658 cm-1-1 = 0.0219 cm = 0.0219 cm-1 -1 + 1/d+ 1/dii

0.0439 cm0.0439 cm-1-1 = 1/d = 1/dii

22.8 cm 22.8 cm = d= dii

A 4.00-cm tall light bulb is placed a distance of A 4.00-cm tall light bulb is placed a distance of 45.7 cm from a concave mirror having a focal 45.7 cm from a concave mirror having a focal length of 15.2 cm. Determine the image size.length of 15.2 cm. Determine the image size.

hi/ho = - di/do

hi /(4.0 cm) = - (22.8 cm)/(45.7 cm)

hi = - (4.0 cm) • (22.8 cm)/(45.7 cm)

hi = -1.99 cm-1.99 cm

Refraction Refraction

NormalNormalLineLine

More Dense

Less Dense

NormalNormalLineLine

More Dense

Less Dense

WATERWATER

AIRAIR

Normal Normal Line #1Line #1

SlowSlow

FastFast

Light BeamLight Beam

FastFastAIRAIR

Normal Normal Line #2Line #2

http://cougar.slvhs.slv.k12.ca.us/~pboomer/physicslectures/secondsemester/light/refraction/refraction.html

Snell’s Law

Snell’s law states that a ray of light bends in such a way that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant. Mathematically,

ni sin i = nr sinr

Here ni is the index of refraction in the original medium and nr is the index in the medium the light enters. i and r are the angles of incidence and refraction, respectively.

i

r

ni

nr

Willebrord Snell

Index of Refraction, nThe index of refraction of a substance is the ratio of the speed in light in a vacuum to the speed of light in that substance:

n = Index of Refraction

c = Speed of light in vacuum

v = Speed of light in medium

n =c v

Note that a large index of refraction corresponds to a relatively slow light speed in that medium.

Medium

Vacuum

Air (STP)

Water (20º C)

Ethanol

Glass

Diamond

n

1

1.00029

1.33

1.36

~1.5

2.42

Index of Refraction EquationsIndex of Refraction Equations• n = c/v = n = c/v = speed of light in a vacuumspeed of light in a vacuum speed of light in mediumspeed of light in medium

Index of Refraction EquationsIndex of Refraction Equations• n = c/v = speed of light in a vacuum speed of light in medium

• n = sin i/sin rn = sin i/sin r

Index of Refraction EquationsIndex of Refraction Equations• n = c/v = n = c/v = speed of light in a vacuumspeed of light in a vacuum speed of light in mediumspeed of light in medium

• n = sin i/sin rn = sin i/sin r

• sin sin AA / sin / sin BB = n = nBB / n / nAA

Index of Refraction Index of Refraction ProblemProblem

A diamond (n = 2.42) is in water (n = 1.33) and a ray of light shines on it making an angle of incidence of 55o. What is the angle of refraction inside the diamond?

sin A / sin B = nB / nA

sin 55o / sin B = 2.42/1.33

B = 27o

Total Internal Reflection...Total Internal Reflection...

…is the total reflection of light traveling in a medium when it strikes a surface of a less dense medium above

the critical angle

Critical Angle Animation

c = sin-1nr

ni

WATER

AIR

Light Light SourceSource

Critical AngleCritical Angle

Total Total Internal Internal

ReflectionReflection

RefractionRefraction

49

A diver basks in the reflection of the Northern A diver basks in the reflection of the Northern Lights underwater by George KarbusLights underwater by George Karbus

Index of Refraction ProblemIndex of Refraction ProblemWhat is the speed of light in water, What is the speed of light in water, which has an index of refraction of 1.33?which has an index of refraction of 1.33?

n = c/v n = c/v v = c/n v = c/n

v = (2.998 x 10v = (2.998 x 1088 m/s) / 1.33 m/s) / 1.33

V = 2.25 x 10V = 2.25 x 1088 m/s m/s

Index of Refraction ProblemIndex of Refraction ProblemA ray of light enters a piece of crown A ray of light enters a piece of crown glass at an angle of 57glass at an angle of 57oo and is refracted and is refracted to 31to 31oo inside the glass. What is the inside the glass. What is the index of refraction?index of refraction?

n = sin i/sin r n = sin i/sin r

= sin 57= sin 57oo / sin 31 / sin 31oo

= 1.63= 1.63

Air – Water InterfaceAir – Water Interface

sin θ = n2/n1

Air nair = 1 and Water n2 = 1.33

sin θ = 1.00/1.33 = 0.750

sin θ = 0.750

θ = sin-1 0.750

θ = 49o

Critical Angle Sample ProblemCritical Angle Sample ProblemCalculate the critical angle for the diamond (n = 2.42) -air (n = 1) boundary.

c = sin-1 (nr / ni)

= sin-1 (1 / 2.42)

= 24.4Any light shone on this

boundary beyond this angle will be reflected back into the

diamond.

c

air

diamond

LensesLenses

Focal Length and Focal Point of a Focal Length and Focal Point of a Thin LensThin Lens

A converging lens has a positive focal lengtho Therefore, it is sometimes called a positive

lensA diverging lens has a negative focal length

o It is sometimes called a negative lens

C F CF

Converging or Converging or Convex LensConvex Lens

F F

Converging or Converging or Convex LensConvex Lens

C F CF

Converging or Converging or Convex LensConvex Lens

C F CF

Converging Converging or Convex or Convex

LensLens

C F CF

Converging Converging or Convex or Convex

LensLens

C F CF

Converging Converging or Convex or Convex

LensLens

C F CF

Converging Converging or Convex or Convex

LensLens

C F CF

Converging Converging or Convex or Convex

LensLens

C F CF

Converging Converging or Convex or Convex

LensLens

C F CF

Converging Converging or Convex or Convex

LensLens

Lens Sign Convention

di

+ for real image

- for virtual image

f+ for convex lenses

- for concave lenses

1f =

1do

1di

+

f = focal length

di = image distance

do = object distance

Lens / Mirror Sign Convention

The general rule for lenses and mirrors is this:

di

+ for real image

- for virtual image

and if the lens or mirror has the ability to converge light, f is positive. Otherwise, f must be treated as negative for the mirror/lens equation to work correctly.

Lens Equation: dLens Equation: doo> C> Cf = 2 cm, C = 4 cm, hf = 2 cm, C = 4 cm, hoo = 2 cm, = 2 cm, ddoo = 5cm = 5cm, d, dii = ? = ?

1/f = 1/d1/f = 1/doo + 1/d + 1/dii

1/2 = 1/1/2 = 1/55 + 1/d + 1/dii

1/d1/di i = 1/2 - 1/= 1/2 - 1/55 = 0.5 – 0.2 = 0.3 = 0.5 – 0.2 = 0.3

ddi i = 3.33 cm= 3.33 cm

M = hM = hii/h/hoo = -d = -dii/d/do o (-h (-ho o x dx di i )/ d)/ doo = h = hii

hhii = (-2 x 3.3)/5 = (-2 x 3.3)/5

hhii = -1.3 cm = -1.3 cm

Lens Equation: dLens Equation: doo < f < ff = 2 cm, C = 4 cm, hf = 2 cm, C = 4 cm, hoo = 2 cm, = 2 cm, ddoo = 0.5 cm = 0.5 cm, d, dii = ? = ?

1/f = 1/d1/f = 1/doo + 1/d + 1/dii

1/2 = 1/1 + 1/d1/2 = 1/1 + 1/dii

1/d1/di i = 1/2 - 1/= 1/2 - 1/0.50.5 = 0.5 – 2.0 = -1.5 = 0.5 – 2.0 = -1.5

ddi i = -.67 cm= -.67 cm

M = hM = hii/h/hoo = -d = -dii/d/do o (-h (-ho o x dx di i )/ d)/ doo = h = hii

hhii = (-2 x -.67)/0.5 = (-2 x -.67)/0.5

hhii = +8/3 = +3.67 = +8/3 = +3.67

M = - dM = - di i / d/ doo = +1.33 = +1.33

Lens Sample Problem

•• • •F F 2F2F

Tooter, who stands 4 feet tall (counting his snorkel), finds himself 24 feet in front of a convex lens and he sees his image reflected 35 feet behind the lens. What is the focal length of the lens and how tall is his image?

f =

hi =

14.24 feet

-5.83 feet

C F CF

Diverging or Diverging or Concave LensConcave Lens

Rays traveling parallel to the principal axis of a concave lens will refract as if coming from the focus.

Rays traveling directly through the center of a concave lens will leave the lens traveling in the exact same direction, just as with a convex lens.

Concave Lenses

•• • •F F 2F

2F

•• • •F F 2F

2F

•• • •F F 2F

2F

Rays traveling toward the focus will refract parallel to the principal axis.

Concave Lens Diagram

•• • •F F 2F2F

object

image

No matter where the object is placed, the image will be on the same side as the object. The image is virtual, upright, and smaller than the object with a concave lens.

Image SummaryImage Summary• For a converging lens, when the object For a converging lens, when the object

distance is greater than the focal length (p >ƒ)distance is greater than the focal length (p >ƒ)– The image is real and invertedThe image is real and inverted

• For a converging lens, when the object is For a converging lens, when the object is between the focal point and the lens, (p<ƒ)between the focal point and the lens, (p<ƒ)– The image is virtual and uprightThe image is virtual and upright

• For a diverging lens, the image is always For a diverging lens, the image is always virtual and uprightvirtual and upright– This is regardless of where the object is placedThis is regardless of where the object is placed

For a converging lens, the object real For a converging lens, the object real and inverted when the object and inverted when the object

distance is __.distance is __.

a) Greater than the focal lengtha) Greater than the focal length

b) Less than the focal lengthb) Less than the focal length

c) Equal to the focal lengthc) Equal to the focal length

For a diverging lens, the image is For a diverging lens, the image is always what?always what?

a) virtuala) virtual

b) uprightb) upright

c) realc) real

d) a and bd) a and b

A converging lens has a __ focal A converging lens has a __ focal length and a diverging lens has a length and a diverging lens has a

__ focal length.__ focal length.

a) +, -a) +, -

c) +,+c) +,+

b) -, +b) -, +

d) -,-d) -,-

Fiber Optics

Fiber optic lines are strands of glass or transparent fibers that allows the transmission of light and digital information over long distances. They are used for the telephone system, the cable TV system, the internet, medical imaging, and mechanical engineering inspection.

Optical fibers have many advantages over copper wires. They are less expensive, thinner, lightweight, and more flexible. They aren’t flammable since they use light signals instead of electric signals. Light signals from one fiber do not interfere with signals in nearby fibers, which means clearer TV reception or phone conversations.

A fiber optic wire

spool of optical fiber

Continued…

Fiber Optics Cont.Fiber optics are often long strands of very pure glass. They are very thin, about the size of a human hair. Hundreds to thousands of them are arranged in bundles (optical cables) that can transmit light great distances. There are three main parts to an optical fiber:

• Core- the thin glass center where light travels.

• Cladding- optical material (with a lower index of refraction than the core) that surrounds the core that reflects light back into the core.

• Buffer Coating- plastic coating on the outside of an optical fiber to protect it from damage. Continued…

Fiber Optics (cont.)Light travels through the core of a fiber optic by continually reflecting off of the cladding. Due to total internal reflection, the cladding does not absorb any of the light, allowing the light to travel over great distances. Some of the light signal will degrade over time due to impurities in the glass.

There are two types of optical fibers:

• Single-mode fibers- transmit one signal per fiber (used in cable TV and telephones).

• Multi-mode fibers- transmit multiple signals per fiber (used in computer networks).

Mirage Pictures

Inferior Mirages

A person sees a puddle ahead on the hot highway because the road heats the air above it, while the air farther above the road stays cool. Instead of just two layers, hot and cool, there are really

many layers, each slightly hotter than the layer above it. The cooler air has a slightly higher index of refraction than the warm air beneath it. Rays of light coming toward the road gradually refract further from the normal, more parallel to the road. (Imagine the wheels and axle: on a light ray coming from the sky, the left wheel is always in slightly warmer air than the right wheel, so the left wheel continually moves faster, bending the axle more and more toward the observer.) When a ray is bent enough, it surpasses the critical angle and reflects. The ray continues to refract as it heads toward the observer. The “puddle” is really just an inverted image of the sky above. This is an example of an inferior mirage, since the cool are is above the hot air.

Observer

Apparent position of sun

Earth

Actual position of sun

Atmosphere

Lingering daylight after the sun is below the horizon is another effect of refraction. Light travels at a slightly slower speed in Earth’s atmosphere than in space. As a result, sunlight is refracted by the atmosphere. In the morning, this refraction causes sunlight to reach us before the sun is actually above the horizon. In the evening, the

Sunlight after Sunset

sunlight is bent above the horizon after the sun has actually set. So daylight is extended in the morning and evening because of the refraction of light. Note: the picture greatly exaggerates this effect as well as the thickness of the atmosphere.

way out of the droplet, the light is once more refracted and dispersed. Although each droplet produces a complete spectrum, an observer will only see a certain wavelength of light from each droplet. (The wavelength depends on the relative positions of the sun, droplet, and observer.) Because there are millions of droplets in the sky, a complete spectrum is seen. The droplets reflecting red light make an angle of 42o with respect to the direction of the sun’s rays; the droplets reflecting violet light make an angle of 40o.

Rainbows A rainbow is a spectrum formed when sunlight is dispersed by water droplets in the atmosphere. Sunlight incident on a water droplet is refracted. Because of dispersion, each color is refracted at a slightly different angle. At the back surface of the droplet, the light undergoes total internal reflection. On the

Primary Rainbow

Secondary RainbowThe secondary rainbow is a rainbow of radius 51, occasionally visible outside the primary rainbow. It is produced when the light entering a cloud droplet is reflected twice internally and then exits the droplet. The color spectrum is reversed in respect to the primary rainbow, with red appearing on its inner edge.

Primary

Secondary

Alexander’s dark region

Dispersion is the ___ of white light Dispersion is the ___ of white light into pure colors.into pure colors.

a) combinationa) combination

c) absorptionc) absorption

b) separationb) separation

d) decompositiond) decomposition

What color can bend the most?What color can bend the most?

a) violeta) violet

c) redc) red

b) cyanb) cyan

d) magentad) magenta

Which of these can raindrops not Which of these can raindrops not do to sunlight?do to sunlight?

a) Refracta) Refract

c) Reflectc) Reflect

b) Absorbb) Absorb

d) Dispersed) Disperse

CreditsSnork pics: http://www.geocities.com/EnchantedForest/Cottage/7352/indosnor.htmlSnorks icons: http://www.iconarchive.com/icon/cartoon/snorks_by_pino/Snork seahorse pic: http://members.aol.com/discopanth/private/snork.jpgMirror, Lens, and Eye pics:http://www.physicsclassroom.com/ Refracting Telescope pic: http://csep10.phys.utk.edu/astr162/lect/light/refracting.html Reflecting Telescope pic: http://csep10.phys.utk.edu/astr162/lect/light/reflecting.html Fiber Optics: http://www.howstuffworks.com/fiber-optic.htm

Willebrord Snell and Christiaan Huygens pics: http://micro.magnet.fsu.edu/optics/timeline/people/snell.html Chromatic Aberrations: http://www.dpreview.com/learn/Glossary/Optical/Chromatic_Aberrations_01.htm Mirage Diagrams: http://www.islandnet.com/~see/weather/elements/mirage1.htm Sir David Brewster pic: http://www.brewstersociety.com/brewster_bio.html Mirage pics: http://www.polarimage.fi/ http://www.greatestplaces.org/mirage/desert1.html http://www.ac-grenoble.fr/college.ugine/physique/les%20mirages.htmlDiffuse reflection: http://www.glenbrook.k12.il.us/gbssci/phys/Class/refln/u13l1d.htmlDiffraction: http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/grating.html