Optics: Mirrors and Reflection

• Reflection• Diffuse reflection•Plane mirrors• Spherical aberration• Concave and convex mirrors• Focal length & radius of curvature• Mirror / lens equation

Optics: Mirrors and Reflection

Objectives:

I will discover how the incident and reflected rays are related.

I will create ray diagrams for each of the cases for mirrors.

I will evaluate the experimental data for the mirror laboratories in terms of the mirror equations.

ReflectionLight is reflected off objects we see, since most objects don’t produce visible light. Most light is absorbed by objects. The wavelengths of the reflected light determine the colors we see. When white light hits an apple red wavelengths are reflected, most of the others are absorbed.

ReflectionA ray of light heading towards an object is called an incident ray. If it reflects off the object, it is called a reflected ray.

A perpendicular line to the surface is a normal. The angle between the incident ray and normal is the angle of incidence, ےi, and the angle between the reflected ray and the normal is the angle of reflection, ےr. The law of reflection states that the angle of incidence is always equal to the angle of reflection.

Law of Reflection

i r

i = r

Normal line (perpendicular to surface)

incident rays

refle

cted

rays

Diffuse ReflectionDiffuse reflection is when light bounces off a non-smooth surface. Each ray of light still obeys the law of reflection but is scattered.

Real vs. Virtual ImagesReal images are formed by mirrors when light rays actually converge and pass through the image. They are located in front of the mirror. A real image can be projected onto a piece of paper or a screen.

Virtual images occur where light rays only appear to have originated. For example, sometimes rays appear to be coming from a point behind the mirror. Virtual images can’t be projected since the light rays do not really converge there.

Plane Mirror

Rays from an object at point P strike the mirror and are reflected. After reflection, the rays continue to spread. When these rays are traced back behind the mirror, they intersect at point P’, where the virtual image of point P appears.

Object

Virtual Image

P P’

The image, I, formed by a plane mirror of an object, O, appears to be a distance di , behind the mirror, equal to the object distance do.

Plane Mirror

Two rays from object P strike the mirror at points B and M. Each ray is reflected such that i = r.

Triangles BPM and BP’M are congruent, which implies that do= di and h = h’. Thus, the image is the same distance behind the mirror as the object is in front of it, and the image is the same size as the object.

Plane Mirror

Object Image

P B

M

P’do di

h h’

Mirror

With plane mirrors, the image is reversed left to right. When you raise your left hand in front of a mirror, your image raises its right hand. Why aren’t top and bottom reversed?

object image

Plane Mirror

Concave and Convex MirrorsConcave and convex mirrors are curved mirrors.

light rayslight rays

Concave mirrors reflect light from their inner surface.

Convex mirrors reflect light from their outer surface.

Concave Mirrors• Concave mirrors are spherical and have a principal axis that goes through the center, C, of the imagined sphere and ends at the point at the center of the mirror, A. The principal axis is perpendicular to the surface of the mirror at A.

• CA is the radius of the sphere,or the radius of curvature of the mirror, R .

• Halfway between C and A is the focal point of the mirror, F. This is the point where rays parallel to the principal axis will converge when reflected off the mirror.

• The length of FA is the focal length, f.

• The focal length is half of the radius of the sphere (proven on next slide).

Concave Mirrors

Focusing Light with Concave Mirrors

Light rays parallel to the principal axis will be reflected through the focal point.

In reverse, light rays passing through the focus will be reflected parallel to the principal axis.

••CF

Spherical Mirror

Only parallel rays close to the principal axis of a spherical mirror will converge at the focal point. Rays farther away will converge at a point closer to the mirror. The image formed by a large spherical mirror will be a disk, not a point. This is known as spherical aberration.

Spherical Aberration

• •C

F

Parabolic Mirror

Parabolic mirrors don’t have spherical aberration. They are used to focus rays from stars in a telescope. They can also be used in flashlights and headlights since a light source placed at their focal point will reflect light in parallel beams.

Spherical Aberration

Spherical vs. Parabolic Mirrors

PA

RA

BO

LICvs.

SPH

ER

ICA

L

PA

RA

BO

LICvs.

SPH

ER

ICA

L

Parallel rays converge at the focal point of a spherical mirror only if they are close to the principal axis. The image formed in a large spherical mirror is a disk, not a point (spherical aberration).

Parabolic mirrors focuses all parallel rays at the focal point. They are used in telescopes, flashlights and car headlights.

Six Cases With Concave Mirrors

• •C

F

Parabolic Mirror

Case 1: All rays that arrive parallel to the principle axis converge at the focal point. (Object at infinity, image is at the focal point)

Concave Mirrors: Object beyond C

• •C F

object

image

The image formed when an object is placed beyond C is located between C and F. It is a real, inverted image that is smaller in size than the object.

Concave Mirrors: Object beyond C

• •C F

object

image

The image formed when an object is placed at C is located at C. It is a real, inverted image that is the same size as the object.

Concave Mirrors: Object between C and F

• •C F

object

image

The image formed when an object is placed between C and F is located beyond C. It is a real, inverted image that is larger in size than the object.

Concave Mirrors: Object in front of F

• •C F

objectimage

The image formed when an object is placed in front of F is located behind the mirror. It is a virtual, upright image that is larger in size than the object. It is virtual since it is formed only where light rays seem to be diverging from.

Concave Mirrors: Object at F

• •C F

object

No image is formed when an object is placed F since the rays come off parallel. The image is “infinitely large and located at infinity.”

Convex Mirrors

• A convex mirror has the same basic properties as a concave mirror but its focus and center are located behind the mirror.

• This means a convex mirror has a negative focal length.

• Light rays reflected from convex mirrors diverge, so only virtual images will be formed.

light rays

Convex Mirrors

light rays

• Rays parallel to the principal axis will reflect as if coming from the focus behind the mirror.

• Rays approaching the mirror on a path toward F will reflect parallel to the principal axis.

Convex Mirror Diagram

• •CF

objectimage

The image formed by a convex mirror no matter where the object is placed will be virtual, upright, and smaller than the object. As the object is moved closer to the mirror, the image will approach the size of the object.

Mirror/Lens Equation DerivationΔABO ≈ ΔA’B’O AAAAB/BO = A’B’/B’O Ratio of sides of similar triangles

•C

Aobject

image

di

OF

Since AB is the object height, hi, and A’B’ is the image height, ho, and BO is the object distance from the mirror, do, and B’O is the image distance from the mirror, di, then through substitution:

ho/do = hi/di

and by rearrangement

hi/ho = di/do = magnificationdo

•

A’

BB’

Magnification

m = magnification

hi = image height (negative means inverted)

ho = object height

m = hi

ho

By definition,

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

Mirror/Lens Equation Derivation

•C

Aobject

image

di

OF

Since AB is the object height, hi, and A’B’ is equal to DO the image height, ho, and Bf is the object distance minus the focal length, do- f and OF is the focal length, f, then through substitution:

hi/ho = f/do - f

do

•

A’

BB’

D

ΔABF ≈ ΔDOF AAA

DO/AB=OF/BF Ratio of sides of similar triangles

Mirror/Lens Equation Derivation

•C

Aobject

image

di

OF

do

•

A’

BB’

D

hi/ho = f/do – f

Since hi/ho = di/do

Then di/do = f/do – f

By cross multiplying

dido – dif = dof

By dividing everything by a common didof and canceling out like terms the mirror equation is derived.

1/f = 1/do + 1/di

Mirror 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

Magnification

m = magnification

hi = image height (negative means inverted)

ho = object height

m = hi

ho

By definition,

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

Mirror Equation Sample Problem

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

di =

hi =

• •C F

15.88 feet

-2.06 feet

Mirror Equation Sample Problem 2

• •CF

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

ho =

19.25 feet

2.75 feet

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

Concave Mirror Ray Diagrams

Simulation for the image location from a concave mirror.

Law of Reflection

Simulation of reflection from a plane mirror