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Chapter 36

Image Formation

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Notation for Mirrors and

LensesThe object distance is the distance from theobject to the mirror or lens

Denoted by pThe image distance is the distance from theimage to the mirror or lens

Denoted by q

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

Denoted by M

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ImagesImages are always located by extendingdiverging rays back to a point at whichthey intersectImages are located either at a pointfrom which the rays of light actually diverge or at a point from which theyappear to diverge

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Types of ImagesA real image is formed when light rayspass through and diverge from theimage point

Real images can be displayed on screens

A virtual image is formed when light

rays do not pass through the imagepoint but only appear to diverge fromthat point

Virtual images cannot be displayed on screens

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Images Formed by Flat

MirrorsSimplest possiblemirror

Light rays leave thesource and arereflected from themirror

Point I is called theimage of the objectat point OThe image is virtual

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Images Formed by Flat

Mirrors, 2A flat mirror always produces a virtual imageGeometry can be used to determine the

properties of the imageThere are an infinite number of choices of direction in which light rays could leave eachpoint on the objectTwo rays are needed to determine where animage is formed

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Images Formed by Flat

Mirrors, 3One ray starts at pointP , travels to Q and

reflects back on itself Another ray followsthe path PR andreflects according to

the law of reflectionThe triangles PQR and PQR arecongruent

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Active Figure 36.2

(SLIDESHOW MODE ONLY)

http://../Active_Figures/Active%20Figures%20Media/AF_3602.htmlhttp://../Active_Figures/Active%20Figures%20Media/AF_3602.html

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Images Formed by Flat

Mirrors, 4To observe the image, the observer wouldtrace back the two reflected rays to P

Point P is the point where the rays appear tohave originatedThe image formed by an object placed infront of a flat mirror is as far behind the mirror as the object is in front of the mirror

| p | = | q |

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Lateral MagnificationLateral magnification, M , is defined as

This is the general magnification for any type of mirror It is also valid for images formed by lensesMagnification does not always mean bigger, thesize can either increase or decrease

M can be less than or greater than 1

Image height

Object height

' hM h

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Reversals in a Flat Mirror A flat mirror produces an image

that has an apparent left-right reversalFor example, if youraise your right hand

the image you seeraises its left hand

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Reversals, cont.The reversal is not actually a left-rightreversalThe reversal is actually a front-back reversal

It is caused by the light rays going forward

toward the mirror and then reflecting backfrom it

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Properties of the Image Formed

by a Flat Mirror SummaryThe image is as far behind the mirror as theobject is in front

| p | = | q |The image is unmagnifiedThe image height is the same as the object height

h = h and M = 1

The image is virtualThe image is upright

It has the same orientation as the object

There is a front-back reversal in the image

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Application Day and Night

Settings on Auto Mirrors

With the daytime setting, the bright beam of reflectedlight is directed into the drivers eyesWith the nighttime setting, the dim beam of reflectedlight is directed into the drivers eyes, while the brightbeam goes elsewhere

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Spherical MirrorsA spherical mirror has the shape of asection of a sphere

The mirror focuses incoming parallel rays to apointA concave spherical mirror has the silveredsurface of the mirror on the inner, or concave,

side of the curveA convex spherical mirror has the silveredsurface of the mirror on the outer, or convex,side of the curve

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Paraxial RaysWe use only rays that diverge from theobject and make a small angle with theprincipal axisSuch rays are called paraxial raysAll paraxial rays reflect through theimage point

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Image Formed by a Concave

Mirror Geometry can beused to determinethe magnification of the image

h is negative whenthe image is invertedwith respect to theobject

' h qM

h p

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Image Formed by a Concave

Mirror Geometry also shows the relationshipbetween the image and object distances

This is called the mirror equationIf p is much greater than R , then the imagepoint is half-way between the center of curvature and the center point of the mirror

p , then 1/ p 0 and q R /2

1 1 2

p q R

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Focal LengthWhen the object is veryfar away, then p and the incoming rays

are essentially parallelIn this special case, theimage point is called thefocal pointThe distance from themirror to the focal pointis called the focallength

The focal length is theradius of curvature

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Focal Point, cont.The colored beamsare traveling parallelto the principal axisThe mirror reflectsall three beams tothe focal point

The focal point iswhere all the beamsintersect

It is the white point

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Focal Point and Focal Length,

cont.The focal point is dependent solely onthe curvature of the mirror, not on the

location of the objectIt also does not depend on the materialfrom which the mirror is made

= R / 2The mirror equation can be expressedas 1 1 1

p q

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Focal Length Shown by

Parallel Rays

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Convex MirrorsA convex mirror is sometimes called adiverging mirror

The light reflects from the outer, convex sideThe rays from any point on the object divergeafter reflection as though they were comingfrom some point behind the mirror

The image is virtual because the reflectedrays only appear to originate at the imagepoint

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Image Formed by a Convex

Mirror

In general, the image formed by a convexmirror is upright, virtual, and smaller than theobject

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Sign ConventionsThese signconventions apply to

both concave andconvex mirrorsThe equations usedfor the concavemirror also apply tothe convex mirror

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Sign Conventions, Summary

Table

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Ray DiagramsA ray diagram can be used todetermine the position and size of animageThey are graphical constructions whichreveal the nature of the image

They can also be used to check theparameters calculated from the mirror and magnification equations

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Notes About the RaysThe rays actually go in all directionsfrom the objectThe three rays were chosen for their ease of constructionThe image point obtained by the raydiagram must agree with the value of q calculated from the mirror equation

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Ray Diagram for a Concave

Mirror, p > R

The center of curvature is between the object and theconcave mirror surface

The image is realThe image is inverted

The image is smaller than the object (reduced)

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Ray Diagram for a Concave

Mirror, p < f

The object is between the mirror surface and thefocal pointThe image is virtualThe image is upright

The image is larger than the object (enlarged)

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The Rays in a Ray Diagram

Convex MirrorsRay 1 is drawn from the top of theobject parallel to the principal axis and

is reflected away from the focal point, F Ray 2 is drawn from the top of theobject toward the focal point and isreflected parallel to the principal axisRay 3 is drawn through the center of curvature, C , on the back side of themirror and is reflected back on itself

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Ray Diagram for a Convex

Mirror

The object is in front of a convex mirror The image is virtualThe image is uprightThe image is smaller than the object (reduced)

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Active Figure 36.15

(SLIDESHOW MODE ONLY)

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Notes on ImagesWith a concave mirror, the image may beeither real or virtual

When the object is outside the focal point, theimage is realWhen the object is at the focal point, the image isinfinitely far awayWhen the object is between the mirror and the

focal point, the image is virtualWith a convex mirror, the image is alwaysvirtual and upright

As the object distance decreases, the virtual

image increases in size

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Images Formed by RefractionConsider twotransparent mediahaving indices of refraction n 1 and n 2The boundary betweenthe two media is a

spherical surface of radius R Rays originate from theobject at point O in the

medium with n = n 1

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Images Formed by Refraction,

2We will consider the paraxial raysleaving O

All such rays are refracted at thespherical surface and focus at theimage point, I

The relationship between object andimage distances can be given by

1 2 2 1n n n n

p q R

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Images Formed by Refraction,

3The side of the surface in which thelight rays originate is defined as the

front sideThe other side is called the back sideReal images are formed by refraction in

the back of the surfaceBecause of this, the sign conventions for q and R for refracting surfaces are oppositethose for reflecting surfaces

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Sign Conventions for

Refracting Surfaces

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Flat Refracting SurfacesIf a refractingsurface is flat, thenR is infiniteThen q = -( n 2 / n 1) p

The image formedby a flat refractingsurface is on thesame side of thesurface as the object

A virtual image isformed

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Active Figure 36.20

(SLIDESHOW MODE ONLY)

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LensesLenses are commonly used to formimages by refraction

Lenses are used in optical instrumentsCamerasTelescopes

Microscopes

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Images from LensesLight passing through a lensexperiences refraction at two surfaces

The image formed by one refractingsurface serves as the object for thesecond surface

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Locating the Image Formed by

a LensThe lens has an index of refraction n and two sphericalsurfaces with radii of R 1 and R 2

R 1 is the radius of curvature of the lens surface that the light of the object reaches firstR 2 is the radius of curvature of

the other surfaceThe object is placed at point O at a distance of p 1 in front of thefirst surface

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Locating the Image Formed by

a Lens, Image From Surface 1There is an image formed by surface 1Since the lens is surrounded by the air,

n 1 = 1 and

If the image due to surface 1 is virtual,q 1 is negative, and it is positive if theimage is real

1 2 2 1

1 1 1

1 1n n n n n n p q R p q R

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Image Formed by a Thin LensA thin lens is one whose thickness is smallcompared to the radii of curvature

For a thin lens, the thickness, t , of the lenscan be neglectedIn this case, p 2 = - q 1 for either type of image

Then the subscripts on p 1 and q 2 can beomitted

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Lens Makers EquationThe focal length of a thin lens is theimage distance that corresponds to an

infinite object distanceThis is the same as for a mirror

The lens makers equation is

1 2

1 1 1 1 1( 1)

n

p q R R

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Thin Lens EquationThe relationship among the focallength, the object distance and the

image distance is the same as for amirror 1 1 1

p q

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Notes on Focal Length and

Focal Point of a Thin LensBecause light can travel in either direction through a lens, each lens has

two focal pointsOne focal point is for light passing in onedirection through the lens and one is for light traveling in the opposite direction

However, there is only one focal lengthEach focal point is located the samedistance from the lens

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Focal Length of a Converging

Lens

The parallel rays pass through the lens and

converge at the focal pointThe parallel rays can come from the left or right of the lens

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Focal Length of a Diverging

Lens

The parallel rays diverge after passing

through the diverging lensThe focal point is the point where the raysappear to have originated

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Determining Signs for Thin

LensesThe front side of thethin lens is the side

of the incident lightThe back side of thelens is where thelight is refracted into

This is also valid for a refracting surface

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Magnification of ImagesThrough a Thin Lens

The lateral magnification of the image is

When M is positive, the image is upright andon the same side of the lens as the object

When M is negative, the image is invertedand on the side of the lens opposite theobject

' h qM

h p

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Thin Lens ShapesThese are examplesof converging lenses

They have positivefocal lengthsThey are thickest inthe middle

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More Thin Lens ShapesThese are examplesof diverging lenses

They have negativefocal lengthsThey are thickest atthe edges

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Ray Diagrams for Thin Lenses Converging

Ray diagrams are convenient for locating theimages formed by thin lenses or systems of lenses

For a converging lens, the following three raysare drawn:

Ray 1 is drawn parallel to the principal axis and thenpasses through the focal point on the back side of thelensRay 2 is drawn through the center of the lens andcontinues in a straight lineRay 3 is drawn through the focal point on the front of the lens (or as if coming from the focal point if p < )

and emerges from the lens parallel to the principal axis

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Ray Diagram for ConvergingLens, p > f

The image is realThe image is invertedThe image is on the back side of the lens

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Ray Diagram for ConvergingLens, p < f

The image is virtualThe image is uprightThe image is larger than the objectThe image is on the front side of the lens

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Ray Diagrams for Thin Lenses Diverging

For a diverging lens, the following three rays aredrawn:

Ray 1 is drawn parallel to the principal axis andemerges directed away from the focal point on the frontside of the lensRay 2 is drawn through the center of the lens andcontinues in a straight line

Ray 3 is drawn in the direction toward the focal pointon the back side of the lens and emerges from the lensparallel to the principal axis

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Ray Diagram for DivergingLens

The image is virtualThe image is uprightThe image is smaller The image is on the front side of the lens

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Active Figure 36.28

(SLIDESHOW MODE ONLY)

http://../Active_Figures/Active%20Figures%20Media/AF_3628.htmlhttp://../Active_Figures/Active%20Figures%20Media/AF_3628.html

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Image SummaryFor a converging lens, when the objectdistance is greater than the focal length( p > )

The image is real and invertedFor a converging lens, when the object isbetween the focal point and the lens, ( p < )

The image is virtual and uprightFor a diverging lens, the image is alwaysvirtual and upright

This is regardless of where the object is placed

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Fresnal LensRefraction occursonly at the surfacesof the lensA Fresnal lens isdesigned to takeadvantage of thisfactIt produces apowerful lenswithout greatthickness

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Fresnal Lens, cont.Only the surface curvature is important in therefracting qualities of the lens

The material in the middle of the Fresnal lensis removedBecause the edges of the curved segmentscause some distortion, Fresnal lenses areusually used only in situations where imagequality is less important than reduction of weight

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Combinations of Thin LensesThe image formed by the first lens islocated as though the second lens werenot presentThen a ray diagram is drawn for thesecond lensThe image of the first lens is treated asthe object of the second lensThe image formed by the second lens isthe final image of the system

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Combination of Thin Lenses, 2If the image formed by the first lens lies onthe back side of the second lens, then theimage is treated as a virtual object for thesecond lens

p will be negativeThe same procedure can be extended to asystem of three or more lensesThe overall magnification is the product of themagnification of the separate lenses

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Two Lenses in Contact

Consider a case of two lenses incontact with each other

The lenses have focal lengths of 1 and 2 For the first lens,

Since the lenses are in contact, p 2 = -q 1

1 1

1 1 1

p q

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Two Lenses in Contact, cont.For the second lens,

For the combination of the two lenses

Two thin lenses in contact with each other areequivalent to a single thin lens having a focallength given by the above equation

2 2 2 1

1 1 1 1 1

p q q q

21 1

1

1

+=

C bi i f Thi L

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Combination of Thin Lenses,example

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Lens AberrationsAssumptions have been:

Rays make small angles with the principal axisThe lenses are thin

The rays from a point object do not focus at asingle point

The result is a blurred image

The departures of actual images from theideal predicted by our model are calledaberrations

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Spherical AberrationThis results from the focalpoints of light rays far fromthe principal axis being

different from the focal pointsof rays passing near the axisFor a camera, a smallaperture allows a greater percentage of the rays to be

paraxialFor a mirror, parabolic shapescan be used to correct for spherical aberration

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Chromatic AberrationDifferent wavelengths of lightrefracted by a lens focus atdifferent points

Violet rays are refracted morethan red raysThe focal length for red light isgreater than the focal lengthfor violet light

Chromatic aberration can beminimized by the use of acombination of converging anddiverging lenses made of different materials

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The CameraThe photographiccamera is a simpleoptical instrumentComponents

Light-tight chamber Converging lens

Produces a realimage

Film behind the lensReceives the image

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Camera Operation, IntensityLight intensity is a measure of the rateat which energy is received by the film

per unit area of the imageThe intensity of the light reaching the film isproportional to the area of the lens

The brightness of the image formed onthe film depends on the light intensity

Depends on both the focal length and thediameter of the lens

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Camera, f -numbersThe -number of a camera lens is the ratio of the focal length of the lens to its diameter

-number = / DThe -number is often given as a description of the lens speed

A lens with a low f -number is a fast lens

The intensity of light incident on the film isrelated to the -number: I 1/(-number) 2

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Camera, f -numbers, cont.Increasing the setting from one -number tothe next higher value decreases the area of the aperture by a factor of 2The lowest -number setting on a cameracorresponds to the aperture wide open andthe use of the maximum possible lens areaSimple cameras usually have a fixed focallength and a fixed aperture size, with an -number of about 11

Most cameras with variable -numbers adjustthem automatically

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Camera, Depth of Field

A high value for the -number allows for a large depth of field

This means that objects at a wide range of distances from the lens form reasonablysharp images on the filmThe camera would not have to be focusedfor various objects

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Digital CameraDigital cameras are similar in operationThe image does not form on photographic

filmThe image does form on a charge-coupled device (CCD)

This digitizes the image and turns it into a binarycodeThe digital information can then be stored on amemory chip for later retrieval

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The EyeThe normal eye focuseslight and produces a sharpimage

Essential parts of the eye:Cornea light passesthrough this transparentstructureAqueous Humor clear liquid behind the cornea

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The Eye Parts, cont.The pupil

A variable aperture

An opening in the irisThe crystalline lensMost of the refraction takes place at theouter surface of the eye

Where the cornea is covered with a film of tears

The Eye Close up of the

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The Eye Close-up of theCornea

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The Eye Parts, finalThe iris is the colored portion of the eye

It is a muscular diaphragm that controlspupil sizeThe iris regulates the amount of lightentering the eye

It dilates the pupil in low light conditions

It contracts the pupil in high-light conditionsThe f -number of the eye is from about 2.8to 16

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The Eye Operation

The cornea-lens system focuses lightonto the back surface of the eye

This back surface is called the retinaThe retina contains sensitive receptorscalled rods and cones

These structures send impulses via theoptic nerve to the brainThis is where the image is perceived

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The Eye Operation, cont.

AccommodationThe eye focuses on an object by varying the

shape of the pliable crystalline lens throughthis processAn important component is the ciliary muscle which is situated in a circle around the rim of

the lensThin filaments, called zonules , run from thismuscle to the edge of the lens

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The Eye FocusingThe eye can focus on a distant object

The ciliary muscle is relaxed

The zonules tightenThis causes the lens to flatten, increasingits focal lengthFor an object at infinity, the focal length of

the eye is equal to the fixed distancebetween lens and retinaThis is about 1.7 cm

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The Eye Focusing, cont.

The eye can focus on near objectsThe ciliary muscle tenses

This relaxes the zonulesThe lens bulges a bit and the focal lengthdecreases

The image is focused on the retina

The Eye Near and Far

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The Eye Near and Far Points

The near point is the closest distance for which the lens can accommodate to focuslight on the retina

Typically at age 10, this is about 18 cmThe average value is about 25 cmIt increases with age

Up to 500 cm or greater at age 60

The far point of the eye represents the largestdistance for which the lens of the relaxed eyecan focus light on the retina

Normal vision has a far point of infinity

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The Eye Seeing ColorsOnly three types of color-sensitive cellsare present in theretina

They are called red,green and bluecones

What color is seendepends on whichcones arestimulated

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Conditions of the EyeEyes may suffer a mismatch between thefocusing power of the lens-cornea systemand the length of the eyeEyes may be:

FarsightedLight rays reach the retina before they converge to forman image

NearsightedPerson can focus on nearby objects but not those far away

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Farsightedness

Also called hyperopia

The near point of the farsighted person is muchfarther away than that of the normal eyeThe image focuses behind the retinaCan usually see far away objects clearly, but notnearby objects

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Correcting Farsightedness

A converging lens placed in front of the eye can

correct the conditionThe lens refracts the incoming rays more toward theprincipal axis before entering the eye

This allows the rays to converge and focus on the retina

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Correcting Nearsightedness

A diverging lens can be used to correct the

conditionThe lens refracts the rays away from theprincipal axis before they enter the eye

This allows the rays to focus on the retina

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Presbyopia and AstigmatismPresbyopia (literally, old-age vision) is dueto a reduction in accommodation ability

The cornea and lens do not have sufficient focusing

power to bring nearby objects into focus on the retinaCondition can be corrected with converging lenses

In astigmatism , light from a point sourceproduces a line image on the retina

Produced when either the cornea or the lens or bothare not perfectly symmetricCan be corrected with lenses with different curvaturesin two mutually perpendicular directions

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Diopters

Optometrists and ophthalmologistsusually prescribe lenses measured in

dioptersThe power P of a lens in diopters equalsthe inverse of the focal length in meters

P = 1/

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Simple Magnifier

A simple magnifier consists of a singleconverging lens

This device is used to increase theapparent size of an objectThe size of an image formed on the

retina depends on the angle subtendedby the eye

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The Size of a Magnified Image

When an object isplaced at the near point,the angle subtended isa maximum

The near point is about25 cm

When the object isplaced near the focalpoint of a converginglens, the lens forms avirtual, upright, andenlarged image

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Angular Magnification, cont.The eye is most relaxed when the image is atinfinity

Although the eye can focus on an object anywherebetween the near point and infinity

For the image formed by a magnifying glassto appear at infinity, the object has to be at

the focal point of the lensThe angular magnification is min

25 cmo

m

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Magnification by a Lens

With a single lens, it is possible toachieve angular magnification up to

about 4 without serious aberrationsWith multiple lenses, magnifications of up to about 20 can be achieved

The multiple lenses can correct for aberrations

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Compound Microscope

A compoundmicroscope consists of two lenses

Gives greater magnification than asingle lensThe objective lens has a

short focal length,o< 1 cmThe eyepiece has afocal length, e of a few

cm

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Active Figure 36.44

(SLIDESHOW MODE ONLY)

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Compound Microscope, cont.The lenses are separated by a distance L

L is much greater than either focal length

The object is placed just outside the focalpoint of the objective

This forms a real, inverted imageThis image is located at or close to the focal pointof the eyepiece

This image acts as the object for the eyepieceThe image seen by the eye, I 2, is virtual, invertedand very much enlarged

Magnifications of the

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Magnifications of theCompound Microscope

The lateral magnification by the objective isM o = - L / o

The angular magnification by the eyepiece of the microscope ism e = 25 cm / e

The overall magnification of the microscope isthe product of the individual magnifications

25 cm o e o e

LM M m

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TelescopesTelescopes are designed to aid in viewingdistant objectsTwo fundamental types of telescopes

Refracting telescopes use a combination of lensesto form an imageReflecting telescopes use a curved mirror and alens to form an image

Telescopes can be analyzed by consideringthem to be two optical elements in a row

The image of the first element becomes the objectof the second element

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Refracting TelescopeThe two lenses are arrangedso that the objective forms areal, inverted image of adistant object

The image is near the focalpoint of the eyepieceThe two lenses areseparated by the distance o + e which corresponds tothe length of the tubeThe eyepiece forms anenlarged, inverted image of the first image

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Active Figure 36.45

(SLIDESHOW MODE ONLY)

Angular Magnification of a

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g gTelescope

The angular magnification depends on thefocal lengths of the objective and eyepiece

The negative sign indicates the image is invertedAngular magnification is particularly importantfor observing nearby objects

Nearby objects would include the sun or the moonVery distant objects still appear as a small point of light

o

o e

m

Disadvantages of Refracting

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g gTelescopes

Large diameters are needed to studydistant objects

Large lenses are difficult and expensiveto manufactureThe weight of large lenses leads to

sagging which produces aberrations

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Reflecting TelescopeHelps overcome some of the disadvantagesof refracting telescopes

Replaces the objective lens with a mirror

The mirror is often parabolic to overcomespherical aberrations

In addition, the light never passes throughglass

Except the eyepieceReduced chromatic aberrationsAllows for support and eliminates sagging

Reflecting Telescope,

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g p ,Newtonian Focus

The incoming rays arereflected from the mirror and converge towardpoint A

At A, an image would beformed

A small flat mirror, M,reflects the light toward

an opening in the sideand it passes into aneyepiece

This occurs before theimage is formed at A

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Examples of TelescopesReflecting Telescopes

Largest in the world are the 10-m diameter Kecktelescopes on Mauna Kea in Hawaii

Each contains 36 hexagonally shaped,computer-controlled mirrors that work together to form a large reflecting surface

Refracting Telescopes

Largest in the world is Yerkes Observatory inWilliams Bay, WisconsinHas a diameter of 1 m