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NCERTExemplarSolutions

Chapter9

RayOpticsandOpticalInstruments

MULTIPLECHOICEQUESTIONS-I

9.1Arayoflightincidentatanangleθonarefractingfaceofaprismemergesfromthe

otherfacenormally.Iftheangleoftheprismis5°andtheprismismadeofamaterial

ofrefractiveindex1.5,theangleofincidenceis

(a) 7.5°.

(b) 5°.

(c) 15°.

(d) 2.5°.

Ans.(a)

A=5o

μ=1.5

i2=0o

r2=0o

Exercise

r1+r2=A

r1=A−r2=5−0=5∘

=0.1305

=7.5∘

9.2Ashortpulseofwhitelightisincidentfromairtoaglassslabatnormalincidence.

Aftertravellingthroughtheslab,thefirstcolourtoemergeis

(a) blue.

(b) green.

(c) violet.

(d) red.

Ans.(d): c=v andvisconstantduringrefractionsoc .Thevelocityofredcolouris

maximuminglassasthe

9.3Anobjectapproachesaconvergentlensfromtheleftofthelenswithauniform

speed5m/sandstopsatthefocus.Theimage

(a) movesawayfromthelenswithanuniformspeed5m/s.

(b) movesawayfromthelenswithanuniformaccleration.

(c) movesawayfromthelenswithanon-uniformacceleration.

(d) movestowardsthelenswithanon-uniformacceleration.

Ans.(c):Whenanobjectapproachestowardsalenswithuniformspeed,itsimagemoves

awayfromthelenstoinfinitywithnon-uniformacceleration.

9.4Apassengerinanaeroplaneshall

(a) neverseearainbow.

(b) mayseeaprimaryandasecondaryrainbowasconcentriccircles.

(c) mayseeaprimaryandasecondaryrainbowasconcentricarcs.

(d) shallneverseeasecondaryrainbow.

Ans.(b):Apassengerinanaeroplanemayseeprimaryandsecondaryrainbowasconcentric

circles.

9.5Youaregivenfoursourcesoflighteachoneprovidingalightofasinglecolour–

red,blue,greenandyellow.Supposetheangleofrefractionforabeamofyellowlight

correspondingtoaparticularangleofincidenceattheinterfaceoftwomediais90°.

Whichofthefollowingstatementsiscorrectifthesourceofyellowlightisreplaced

withthatofotherlightswithoutchangingtheangleofincidence?

(a) Thebeamofredlightwouldundergototalinternalreflection.

(b) Thebeamofredlightwouldbendtowardsnormalwhileitgetsrefractedthroughthe

secondmedium.

(c) Thebeamofbluelightwouldundergototalinternalreflection.

(d) Thebeamofgreenlightwouldbendawayfromthenormalasitgetsrefractedthrough

thesecondmedium.

Ans.(c):Weknowthatifangleofrefractionis forthelengththenincidenceangleis

calledcriticalangle.Solightraysarepassingfromdensertorarermedium.

Assin so,

So,criticalangleforCv<Cg<CY<CR,i.e.,criticalangleofblueandgreenlightissmallerthan

thatofyellowanditisgreaterforredcolourlight.

Astheangleofrefractionforyellowlightis foraparticularincidentangle.This

incidenceangleiscriticalangleforyellowletitbeCY.AsCR>Cv.Soitwillnotgettotal

internalreflectionandCV<CY,Cg<CY.

Solightofblueandgreencolourgettotalinternalreflection.Socorrectansweris(c).

9.6Theradiusofcurvatureofthecurvedsurfaceofaplano-convexlensis20cm.Ifthe

refractiveindexofthematerialofthelensbe1.5,itwill

(a) actasaconvexlensonlyfortheobjectsthatlieonitscurvedside.

(b) actasaconcavelensfortheobjectsthatlieonitscurvedside.

(c) actasaconvexlensirrespectiveofthesideonwhichtheobjectlies.

(d) actasaconcavelensirrespectiveofsideonwhichtheobjectlies.

Ans.(c):IfobjectliesoncurvedsidethenR1=+20cmandR2= , 1=1, 2=1.5

F=+40cm

IfobjectliesonplanesideR1= andR2=-20cm, 1=1, 2=1.5

f=+40cm

So,lenswillalwaysactasaconvexlensirrespectiveofsideonwhichobjectslie.So,answer

is(c).

9.7Thephenomenainvolvedinthereflectionofradiowavesbyionosphereissimilarto

(a) reflectionoflightbyaplanemirror.

(b) totalinternalreflectionoflightinairduringamirage.

(c) dispersionoflightbywatermoleculesduringtheformationofarainbow.

(d) scatteringoflightbytheparticlesofair.

Ans.(b):Ionosphereistransparentopticalmediumandradiowaveisreflectedback.

Reflectionthroughtransparentsurfaceistotalinternalreflectionsothatinternalreflection

ofradiowavetakesplace.

9.8ThedirectionofrayoflightincidentonaconcavemirrorisshownbyPQwhile

directionsinwhichtheraywouldtravelafterreflectionisshownbyfourraysmarked

1,2,3and4(Fig9.1).Whichofthefourrayscorrectlyshowsthedirectionofreflected

ray?

(a) 1

(b) 2

(c) 3

(d) 4

Ans.(b):IncidencerayPQiscomingthroughprincipalfocusFsoitmustbeparallelto

principalaxis,i.e.either2or4.

Asitisaconcavemirrorso,raycannotgobehindthemirrorsoray(4)isdiscarded.

Soray2isthereflectedray.Itisverifiesanswer(b)

WecanverifyitagainbydrawingnormalQCandfindthe∠r=∠i.Soray(2)isthereflected

ray.

9.9Theopticaldensityofturpentineishigherthanthatofwaterwhileitsmassdensity

islower.Fig9.2.showsalayerofturpentinefloatingoverwaterinacontainer.For

whichoneofthefourraysincidentonturpentineinFig9.2,thepathshowniscorrect?

(a) 1

(b) 2

(c) 3

(d) 4

Ans.(b): a< T> w.Here,incidenceraypassesfromairtoturpentinetowater,i.e.,from

raretodenserthendensertorarersofirstitbendstowardsnormalthenawayfromnormal

sothepathshowniscorrectforray(2).

9.10Acarismovingwithataconstantspeedof60kmh–1onastraightroad.Lookingat

therearviewmirror,thedriverfindsthatthecarfollowinghimisatadistanceof100

mandisapproachingwithaspeedof5kmh–1.Inordertokeeptrackofthecarinthe

rear,thedriverbeginstoglancealternativelyattherearandsidemirrorofhiscar

afterevery2stilltheothercarovertakes.Ifthetwocarsweremaintainingtheir

speeds,whichofthefollowingstatement(s)is/arecorrect?

(a) Thespeedofthecarintherearis65km/h.

(b) Inthesidemirrorthecarintherearwouldappeartoapproachwithaspeedof5kmh–

1tothedriveroftheleadingcar.

(c) Intherearviewmirrorthespeedoftheapproachingcarwouldappeartodecreaseasthe

distancebetweenthecarsdecreases.

(d) Inthesidemirror,thespeedoftheapproachingcarwouldappeartoincreaseasthe

distancebetweenthecarsdecreases.

Ans.(d):Sowhenrearcarapproaches,initiallyitappearatrestasimageisformedatfocus.

Whencarapproachesnearerthisspeedwillappeartoincreasesoansweris(d).

9.11Therearecertainmaterialdevelopedinlaboratorieswhichhaveanegative

refractiveindex(Fig.9.3).Arayincidentfromair(medium1)intosuchamedium

(medium2)shallfollowapathgivenby

Ans.(a):Thenegativerefractiveindexmaterialsarethoseinwhichincidentmaterialsare

thoseinwhichincidentrayfromair(medium1)tothemrefractorbendsdifferentlyor

oppositeandsymmetrictonormaltothatofpositiverefractiveindexmedium.Soansweris

(a).

MULTIPLE CHOICE QUESTIONS-II

MORE THAN ONE OPTION

9.12Consideranextendedobjectimmersedinwatercontainedinaplanetrough.When

seenfromclosetotheedgeofthetroughtheobjectlooksdistortedbecause

(a) theapparentdepthofthepointsclosetotheedgearenearerthesurfaceofthewater

comparedtothepointsawayfromtheedge.

(b) theanglesubtendedbytheimageoftheobjectattheeyeissmallerthantheactualangle

subtendedbytheobjectinair.

(c) someofthepointsoftheobjectfarawayfromtheedgemaynotbevisiblebecauseoftotal

internalreflection.

(d) waterinatroughactsasalensandmagnifiestheobject.

Ans.(a),(b)and(c):weknowthatshifting(h)ofimageofanobjectimmersedinliquidfrom

objectisdirectlyproportionaltotherealdistanceofobjectfromthesurfaceofliquid

t=realdepthordistanceoftheobjectfromthesurfaceofliquidofrefractiveindex :Ifthe

objectisseenfromoneedgeoftroughtherelativedifferencesofdepth(distance)inH2O

betweentwoendsofobjectsislargerthanifitseenfromthetoporawayfromedge.By

aboveformulaordistortionofnearerendissmallerthanfarther,verifiestheoption(a).

Theanglesubtendedbyanobjectislargerthanitsimageinwater,asitsimageshiftsupward

verifiesoption(b).

Rayscomingoutfromobjecttoobserverpassesfromdensertorarermedium,andangleof

incidenceforraysfromfartherendBofobjectislargerthannearendA.Theincidence

anglesforrayscomingfromend3mayhaveincidenceanglemorethancriticalangleand

cancausetotalinternalreflection,sowillnotreachtoobserver,causenon-visibleoffarther

endverifiesoption(c).

9.13ArectangularblockofglassABCDhasarefractiveindex1.6.Apinisplaced

midwayonthefaceAB(Fig.9.4).WhenobservedfromthefaceAD,thepinshall-

(a) appeartobenearA.

(b) appeartobenearD.

(c) appeartobeatthecentreofAD.

(d) notbeseenatall.

Ans.(a)(b):

ApointPisonthemidpointoffaceAB.WhenseenthroughfaceAD,neartopointA,the

angleofincidence(i1)willbesmallerthancriticalangle .SoimageofPwillatP’

andimagecanbeseenatP’.P’isnearertobothAandDascomparedtoPverifiesoption(a)

and(b).

WhenseennearpointDthroughfaceAD,angleofincidencei2>ic.sototalinternalreflection

takesplaceandobjectcannotbeobserved.ButobjectcanbeseenwhenviewedneartoAso

option(d)notverified.

9.14Betweentheprimaryandsecondaryrainbows,thereisadarkbandknownas

Alexandar’sdarkband.Thisisbecause

(a) lightscatteredintothisregioninterferedestructively.

(b) thereisnolightscatteredintothisregion.

(c) lightisabsorbedinthisregion.

(d) anglemadeattheeyebythescatteredrayswithrespecttotheincidentlightofthesun

liesbetweenapproximately42°and50°.

Ans.(d):Alexandar’sdarkbandliesbetweentheprimaryandsecondaryrainbow.This

formsduetothelightscatteredintothisregioninterferedestructively.Becausetheprimary

andsecondaryrainbowssubtendangles( to )and( to )respectivelyatthe

observer’seyewithrespecttoincidentlightray,sothescatteredrayswithrespecttothe

incidentrayofthesunliesbetweenapproximately to .

9.15Amagnifyingglassisused,astheobjecttobeviewedcanbebroughtclosertothe

eyethanthenormalnearpoint.Thisresultsin

(a) alargerangletobesubtendedbytheobjectattheeyeandhenceviewedingreaterdetail.

(b) theformationofavirtualerectimage.

(c) increaseinthefieldofview.

(d) infinitemagnificationatthenearpoint.

Ans.(a)and(b):Inmagnifyingglass,theobjectisplacedwithinthefocallengthandthe

imageformedismagnifiedanderect.As(A’B’)imageismagnifiedsoitsubtendslargerangle

attheeyethanobject(AB),socanbeseenmoreclearly.

9.16Anastronomicalrefractivetelescopehasanobjectiveoffocallength20mandan

eyepieceoffocallength2cm.

(a) Thelengthofthetelescopetubeis20.02m.

(b) Themagnificationis1000.

(c) Theimageformedisinverted.

(d) Anobjectiveofalargeraperturewillincreasethebrightnessandreducechromatic

aberrationoftheimage.

Ans.(a),(b)and(c):Thelengthofthetelescope

L=fo+fe=20+00.02=20.02m

and =1000

Thefinalimageformedintelescope(Refracting)isinverted,virtualandsmallerthanobject.

VERY SHORT ANSWER TYPE

QUESTIONS

9.17Willthefocallengthofalensforredlightbemore,sameorlessthanthatforblue

light?

Ans.Asweknowthat( v> R),so willbelargeforbluelightandsmallerforRed

colouredlight.Sofwillbelargerforredcolouredlight.

9.18Thenearvisionofanaveragepersonis25cm.Toviewanobjectwithanangular

magnificationof10,whatshouldbethepowerofthemicroscope?

Ans.ToseethefinalimageatdistinctvisionD=25cm

foreyeslensu=-fandv=-25

=+2.25cm=0.025m

=40D

9.19Anunsymmetricaldoubleconvexthinlensformstheimageofapointobjectonits

axis.Willthepositionoftheimagechangeifthelensisreversed?

Ans.Positionofimageinconvexlenschangeseitherbychanginguoff.Here,uisconstant

and .Here, andcurvatureofthelensaresamesofisconstant.So

onreversingthelens,positionandnatureoftheimagewillnotchange.

9.20Threeimmiscibleliquidsofdensitiesd1>d2>d3andrefractive

indicesμ1>μ2>μ3areputinabeaker.Theheightofeachliquidcolumnis .Adotis

madeatthebottomofthebeaker.Fornearnormalvision,findtheapparentdepthof

thedot.

Ans.Theliquidsareimmiscible,soliquidsarrangedfrombottomtotopd1,d2andd3with

theirrefractiveindices 1, 2, 3respectivelyseparatedbylayersl1,l2asshowninfigure.

Considerthelayerl1.LetobjectisatitsbottomatPthendistance(x1)ofimageofPbyliquid

ofrefractiveindex 1fromlayerl1

ThisimageI1actsasobjectforliquidof 2andformtheimageatI2.

Thedistancex2belowlevell2

Realdepth

AsthepointPisseenfromoutside

[ 0=1refractiveindexofair]

X3isapparentdepth.

9.21Foraglassprism(μ= )theangleofminimumdeviationisequaltotheangleof

theprism.Findtheangleoftheprism.

Ans.Therequiredrelationforminimumangleofdeviation

[ =A(given)]

A=60°isangleofprism.8

SHORT ANSWER TYPE

QUESTIONS

9.22AshortobjectoflengthLisplacedalongtheprincipalaxisofaconcavemirror

awayfromfocus.Theobjectdistanceisu.Ifthemirrorhasafocallengthf,whatwillbe

thelengthoftheimage?YoumaytakeL<<|v-f|.

Ans.Asthemeandistanceofobjectfrommirrorisu

u1= and

Lettheimageofthetwoendsofobjectformatdistancev1andv2(v1>v2).Solengthofimage

onprincipalaxisisL’=(v1-v2)

or

SoL’=v1-v2=

L<<(u-f) <<<(u-f)

Soneglectingtheterms

Itisthelengthofimagef.

9.23Acirculardiscofradius‘R’isplacedco-axiallyandhorizontallyinsideanopaque

hemisphericalbowlofradius‘a’(Fig.9.5).Thefaredgeofthediscisjustvisiblewhen

viewedfromtheedgeofthebowl.Thebowlisfilledwithtransparentliquidof

refractiveindexμandthenearedgeofthediscbecomesjustvisible.Howfarbelowthe

topofthebowlisthediscplaced?

Ans.InfigureAMandBMaretheraysfromtheendsofdiscABreachingatoneendofbowl

atM.MNistangentatM,soMN ABi.e.,∠N=

TakingincidencerayBMandrefractedrayMD

BN=CN-CB=OM-CB=a-R

∠r=∠α=∠AMN

Sinr=cos( - )=

ForincidencerayBMtothehorizontallevelofliquidMP,MNwillbenormalatM.∠iand∠rwillbeincidenceandrefractedangleswhenrayBMpassesfromliquid( )toair.BySnell’s

law,asraypassesfromliquidtoair

[ 0forair=1][ 1= forliquid]

Itisrequiredexpression.

9.24Athinconvexlensoffocallength25cmiscutintotwopieces0.5cmabovethe

principalaxis.Thetoppartisplacedat(0,0)andanobjectplacedat(–50cm,0).Find

thecoordinatesoftheimage.

Ans.Objectisplaced0.5cmaboveprincipalaxis

u =-50cmf=+25cm,v=?

v=50cm

=-1

Sothesizeofimageisequaltothatofobject,misnegativesoimageisinverted.

Soimageisat(50cm,-1cm)and0.5cmbelowtheX–X’axis.

9.25Inmanyexperimentalset-upsthesourceandscreenarefixedatadistance

sayDandthelensismovable.Showthattherearetwopositionsforthelensforwhich

animageisformedonthescreen.Findthedistancebetweenthesepointsandtheratio

oftheimagesizesforthesetwopoints.

Ans.u=-(D-v)

v=v

fD=vD-v2

v2–vD+fD=0

…I

…II

FromII,whenthepositionofobject [infrontofl2(lowersign)]

ThenfromI,thepositionofimage [lowersign]

Similarlywhenpositionofobject [fromIIuppersign]

thenthepositionofimage [fromIuppersign]

Thedistancebetweentwopositionsoflendd=v1-v2

isthedistancebetweentwopositionsoflenses.

InfirstcaseofL1

Insecondcase

istherequiredratioofsizeofimagesintwocases.

9.26Ajarofheighthisfilledwithatransparentliquidofrefractiveindexμ(Fig.9.6).At

thecentreofthejaronthebottomsurfaceisadot.Findtheminimumdiameterofa

disc,suchthatwhenplacedonthetopsurfacesymmetricallyaboutthecentre,thedot

isinvisible.

Ans.ThepointOwillbeinvisibleifthelightraycomingfromOdoesnotcomeoutoritget

grazethesurfaceofliquidorgetstotalinternalreflectionasinfigurehere.

RayOAisincidentatAwithcriticalangleiCandit’sangleofrefractionwillbe .

Forotherrayifincidentangleismorethanicitwillgettotalinternalreflectionfromthe

surfacesowillnotcomeoutfromliquid.

Sodiscofdiameterdisrequiredtostoptheraysfrom‘O’outofliquid.

or d=2htanic

[Fromfigure]

B2=H2-P2

9.27Amyopicadulthasafarpointat0.1m.Hispowerofaccommodationis4diopters.

(i) Whatpowerlensesarerequiredtoseedistantobjects?

(ii) Whatishisnearpointwithoutglasses?

(iii) Whatishisnearpointwithglasses?(Taketheimagedistancefromthelensoftheeyeto

theretinatobe2cm.)

Ans.(i)Poweroflensrequiredtoseeclearlytheobjectplacedatinfinity.U=-∞

f=-10cm=-0.1m

m

m

P=-10Diopter

(ii) Whennocorrectivelensused:Letpowersofeyewhenobjectisatfarpoint,nearpoint

arePfandPnrespectivelyandpower

Pn=Pf+Pa

Whenobjectisatfarpointitsclearimageisformedatretina2cmfromeyelens

u=-10cm=-0.1m,V=2cm=0.02m

Iffisfocallengthofeyelensfocusedatfarpointthen

Pf=60D

Pn=Pf+Pa=60+4=64D

Letthenearpointbexn

u=-xn(v=2cm=0.02m)

=Power(Pn)

Nearpointwithoutglass (Approx.)

(iii) Whenusedcorrectivelens:Whencorrectivelensisusedtheneyecanseetheobjectat

infinity.PowerofeyelensinthissituationisP∞

u=∞andv=2cm=0.02m

If =Powerofeyeatnearpointwhencorrectivelensisused

Letnearpointinthissituationis

(alldistanceareinm)

LONG ANSWER TYPE

QUESTIONS

LONGANSWERTYPEQUESTIONS

9.28Showthatforamaterialwithrefractiveindex ,lightincidentatanyangle

shallbeguidedalongalengthperpendiculartotheincidentface.

Ans.Considerarectangularslabofrefractiveindex .Anincidencerayincidenceat

angleionfacePQatincidencepoint.RefractedrayBCstrikeatfaceQRwhichis

perpendiculartoPQwithincidenceangleicsothatrefractedrayCDpassesnormaltothe

facePQasperrequiredinquestion.Soicmustbecriticalangle.

(Snell’slawatc)

Sin

[ r+90+ic=180]

(squaringbothsides)

...(I)

(bySnell’slaw)

Or (squaringbothsides)

…(II)

Put(II)in(I)

[onmultiplyingbyμ2onbothsides]

Forsmallestanglei.e.,

Takingsquareroot

Henceproved.

9.29Themixtureapureliquidandasolutioninalongverticalcolumn(i.e,horizontal

dimensions<<verticaldimensions)producesdiffusionofsoluteparticlesandhencea

refractiveindexgradientalongtheverticaldimension.Arayoflightenteringthe

columnatrightanglestotheverticalisdeviatedfromitsoriginalpath.Findthe

deviationintravellingahorizontaldistanced<<h,theheightofthecolumn.

Ans.Consideralongverticalcolumnoftransparentliquidofinfiniteheight(h)andthickness

(dx).ConsiderarayABthatentersatanangleθintoliquidofheightyincolumnofliquid

andemergesatanangle( +d )atheight(y+dy).FromSnell’slaw,

Asd ,dθareverysmalltendstozero

sind =d ,cosd 1 (y)= (μatheightYconstant)

Again

Fromfigure

Byintegrationonbothsides

[ andydoesnotchangehorizontally d?,dyare

constanthorizontally.]

9.30Iflightpassesnearamassiveobject,thegravitationalinteractioncausesabending

oftheray.Thiscanbethoughtofashappeningduetoachangeintheeffective

refractiveindexofthemediumgivenby

whereristhedistanceofthepointofconsiderationfromthecentreofthemassofthe

massivebody,Gistheuniversalgravitationalconstant,Mthemassofthebodyandc

thespeedoflightinvacuum.Consideringasphericalobjectfindthedeviationofthe

rayfromtheoriginalpathasitgrazestheobject.

Ans.Considertwosphericalsurfaceatrand(r+dr)distancefromthecentreofmassive

objectofmassmandradiusR.

ArayABCDincidentatBandContwosurfaceatrand(r+dr)then,bySnell’slaw

[ d 0thencosd =1andd =0.

isverysmallasdμandd areverysmall

Dividingbothsidebydr

...(I)

(given)

…(II)

AstheGisverysmallandc2isverylargeso

r2=x2+R2(Fromfigure)

Differentiatew.r.t.rbothsides

2rdr=2xdx

.

x=Rtan

dx=Rsec2 .d

isrequiredrotation.

9.31AninfinitelylongcylinderofradiusRismadeofanunusualexoticmaterialwith

refractiveindex–1(Fig.9.7).Thecylinderisplacedbetweentwoplaneswhosenormals

arealongtheydirection.ThecenterofthecylinderOliesalongthey-axis.Anarrow

laserbeamisdirectedalongtheydirectionfromthelowerplate.Thelasersourceisat

ahorizontaldistancexfromthediameterintheydirection.Findtherangeofxsuch

thatlightemittedfromthelowerplanedoesnotreachtheupperplane.

Ans.Asthecylinderismadeofrefractiveindex(-1)andisplacedinairof =1so,whenray

ABisincidentatBtocylinder, rwillbenegativei.e.,refractiveraywillgetreflectedfrom

normal.SimilarthinghappensatincidencepointCandangleofrefractionandincidentatB

andCwillbeequal( r)asOB=OC=RandofrefractionatCis .

asreflectiontakesplace

Thetotaldeviationofout-comingrayfromtheincomingray4 1.Raysshallnotreachthe

receivingplaneif anglesmeasuredclockwisefromtheyaxis.

Or (ondividingby4toallsides)

or

Thusforlightemittedfromthesourceshallnotreachthereceivingplaneif

.

9.32(i)Considerathinlensplacedbetweenasource(S)andanobserver(O)(Fig.9.8).

Letthethicknessofthelensvaryas wherebistheverticaldistance

fromthepole.w0isaconstant.UsingFermat’sprinciplei.e.thetimeoftransitforaray

betweenthesourceandobserverisanextremum,findtheconditionthatallparaxial

raysstartingfromthesourcewillconvergeatapoint‘O’ontheaxis.Findthefocal

length.

(ii) Agravitationallensmaybeassumedtohaveavaryingwidthoftheform

=

Showthatanobserverwillseeanimageofapointobjectasaringaboutthecenterof

thelenswithanangularradius

Ans.ThetimetakenbyrayfromSto

(assumingb<<<u)

Similarly,timerequiredbyrayfromP1to

Timerequiredbyraytotravelthroughthelens

ThustotaltimebyrayfromStoO=t=t1+t2+t3

If

ByFermat’sPrinciple

ButtimefromStoOremainsconstant,so,

Or

Thus,theconvergencelensisformed .

ItisindependentofBandhenceallparaxialraysfromSwillconvergeatO(i.e.,forraysb

<<<uandb<<v).

Since itisequivalenttofocallengthf

Here

AstimefromStoOisconstant,so,

So

Thusalltherayspassingataheightbshallcontributetotheimage.Theraypathmakesan

angle