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Images

Chapter34

Copyright©2014JohnWiley&Sons,Inc.Allrightsreserved.

34-1ImagesandPlaneMirrors

AnimageisareproducEonofanobjectvialight.Iftheimagecanformonasurface,itisarealimageandcanexistevenifnoobserverispresent.Iftheimagerequiresthevisualsystemofanobserver,itisavirtualimage.

Herearesomecommonexamplesofvirtualimage.

(a)ArayfromalowsecEonoftheskyrefractsthroughairthatisheatedbyaroad(withoutreachingtheroad).Anobserverwhointerceptsthelightperceivesittobefromapoolofwaterontheroad.(b)Bending(exaggerated)ofalightraydescendingacrossanimaginaryboundaryfromwarmairtowarmerair.(c)ShiQingofwavefrontsandassociatedbendingofaray,whichoccurbecausethelowerendsofwavefrontsmovefasterinwarmerair.(d)Bendingofarayascendingacrossanimaginaryboundarytowarmairfromwarmerair.

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Asshowninfigure(a),aplane(flat)mirrorcanformavirtualimageofalightsource(saidtobetheobject,O)byredirecEnglightraysemergingfromthesource.Theimagecanbeseenwherebackwardextensionsofreflectedrayspassthroughoneanother.Theobject’sdistancepfromthemirrorisrelatedtothe(apparent)imagedistanceifromthemirrorby

ObjectdistancepisaposiEvequanEty.ImagedistanceiforavirtualimageisanegaEvequanEty.

(a)

(b)

OnlyraysthatarefairlyclosetogethercanentertheeyeaQerreflecEonatamirror.FortheeyeposiEonshowninFig.(b),onlyasmallporEonofthemirrornearpointa(aporEonsmallerthanthepupiloftheeye)isusefulinformingtheimage.



34-2SphericalMirrors

AsphericalmirrorisintheshapeofasmallsecEonofasphericalsurfaceandcanbeconcave(theradiusofcurvaturerisaposiEvequanEty),convex(risanegaEvequanEty),orplane(flat,risinfinite).

Wemakeaconcavemirrorbycurvingthemirror’ssurfacesoitisconcave(“cavedin”totheobject)asinFig.(b).Wecanmakeaconvexmirrorbycurvingaplanemirrorsoitssurfaceisconvex(“flexedout”)asinFig.(c).Curvingthesurfaceinthisway(1)movesthecenterofcurvatureCtobehindthemirrorand(2)increasesthefieldofview.Italso(3)movestheimageoftheobjectclosertothemirrorand(4)shrinksit.TheseiteratedcharacterisEcsaretheexactoppositeforconcavemirror.




Ifparallelraysaresentintoa(spherical)concavemirrorparalleltothecentralaxis,thereflectedrayspassthroughacommonpoint(arealfocusF)atadistancef(aposiEvequanEty)fromthemirror(figurea).Iftheyaresenttowarda(spherical)convexmirror,backwardextensionsofthereflectedrayspassthroughacommonpoint(avirtualfocusF)atadistancef(anegaEvequanEty)fromthemirror(figureb).Formirrorsofbothtypes,thefocallengthfisrelatedtotheradiusofcurvaturerofthemirrorbywherer(andf)isposiEveforaconcavemirrorandnegaEveforaconvexmirror.




•  Aconcavemirrorcanformarealimage(iftheobjectisoutsidethefocalpoint)oravirtualimage(iftheobjectisinsidethefocalpoint).

•  Aconvexmirrorcanformonlyavirtualimage.•  ThemirrorequaEonrelatesanobjectdistancep,themirror’sfocallengthfandradiusofcurvaturer,andtheimagedistancei:

•  ThemagnitudeofthelateralmagnificaEonmofanobjectistheraEooftheimageheighth’toobjectheighth,

(a)AnobjectOinsidethefocalpointofaconcavemirror,anditsvirtualimageI.(b)TheobjectatthefocalpointF.(c)Theobjectoutsidethefocalpoint,anditsrealimageI.




1.  AraythatisiniEallyparalleltothecentralaxisreflectsthroughthefocalpointF(ray1inFig.a).

2.  AraythatreflectsfromthemirroraQerpassingthroughthefocalpointemergesparalleltothecentralaxis(Fig.a).

3.  AraythatreflectsfromthemirroraQerpassingthroughthecenterofcurvatureCreturnsalongitself(ray3inFig.b).

4.  Araythatreflectsfromthemirroratpointcisreflectedsymmetricallyaboutthataxis(ray4inFig.b).

TheimageofthepointisattheintersecEonofthetwospecialraysyouchoose.TheimageoftheobjectcanthenbefoundbylocaEngtheimagesoftwoormoreofitsoff-axispoints(say,thepointmostoffaxis)andthensketchingintherestoftheimage.YouneedtomodifythedescripEonsoftheraysslightlytoapplythemtoconvexmirrors,asinFigs.candd.

Loca7ngImagesbyDrawingRays



34-4ThinLenses

FormingaFocus.Figure(a)showsathinlenswithconvexrefracEngsurfaces,orsides.Whenraysthatareparalleltothecentralaxisofthelensaresentthroughthelens,theyrefracttwice,asisshownenlargedinFig.(b).ThisdoublerefracEoncausestheraystoconvergeandpassthroughacommonpointF2atadistanceffromthecenterofthelens.Hence,thislensisaconverginglens;further,arealfocalpoint(orfocus)existsatF2(becausetheraysreally

dopassthroughit),andtheassociatedfocallengthisf.WhenraysparalleltothecentralaxisaresentintheoppositedirecEonthroughthelens,wefindanotherrealfocalpointatF1ontheothersideofthelens.Forathinlens,thesetwofocalpointsareequidistantfromthelens.


34-4ThinLenses

34-4ThinLenses

FormingaFocus.Figure(c)showsathinlenswithconcavesides.Whenraysthatareparalleltothecentralaxisofthelensaresentthroughthislens,theyrefracttwice,asisshownenlargedinFig.(d);theseraysdiverge,neverpassingthroughanycommonpoint,andsothislensisadiverginglens.However,extensionsoftheraysdopassthroughacommonpointF2atadistanceffromthecenterofthelens.Hence,thelenshasavirtualfocalpointatF2.(Ifyoureye

interceptssomeofthedivergingrays,youperceiveabrightspottobeatF2,asifitisthesourceofthelight.)AnothervirtualfocusexistsontheoppositesideofthelensatF1,symmetricallyplacedifthelensisthin.Becausethefocalpointsofadiverginglensarevirtual,wetakethefocallengthftobenegaEve.


34-4ThinLenses

34-4ThinLenses

Loca7ngImagesofExtendedObjectsbyDrawingRays

1.  AraythatisiniEallyparalleltothecentralaxisofthelenswillpassthroughfocalpointF2(ray1inFig.a).

2.  AraythatiniEallypassesthroughfocalpointF1willemergefromthelensparalleltothecentralaxis(ray2inFig.a).

3.  AraythatisiniEallydirectedtowardthecenterofthelenswillemergefromthelenswithnochangeinitsdirecEon(ray3inFig.a)becausetherayencountersthetwosidesofthelenswheretheyarealmostparallel.

FigurebshowshowtheextensionsofthethreespecialrayscanbeusedtolocatetheimageofanobjectplacedinsidefocalpointF1ofaconverginglens.NotethatthedescripEonofray2requiresmodificaEon(itisnowaraywhosebackwardextensionpassesthroughF1).YouneedtomodifythedescripEonsofrays1and2tousethemtolocateanimageplaced(anywhere)infrontofadiverginglens.InFig.c,forexample,wefindthepointwhereray3intersectsthebackwardextensionsofrays1and2. ©2014JohnWiley&Sons,Inc.Allrights

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34-4ThinLenses

Interference

Chapter35

Copyright©2014JohnWiley&Sons,Inc.Allrightsreserved.

35-1LightasaWave

Thethree-dimensionaltransmissionofwaves,includinglight,mayoQenbepredictedbyHuygens’principle,whichstatesthat

Figure1showsthepropagaEonofaplanewaveinvacuum,asportrayedbyHuygens’principle.

TherefracEonofaplanewaveatanair–glassinterface,asportrayedbyHuygens’principle.Thewavelengthinglassissmallerthanthatinair.Forsimplicity,thereflectedwaveisnotshown.Parts(a)through(c)representthreesuccessivestagesoftherefracEon.

Figure1


35-1LightasaWave

35-1LightasaWave

TherefracEonofaplanewaveatanair–glassinterface,asportrayedbyHuygens’principle.Thewavelengthinglassissmallerthanthatinair.Forsimplicity,thereflectedwaveisnotshown.Parts(a)through(c)representthreesuccessivestagesoftherefracEon.ThelawofrefracEoncanbederivedfromHuygens’principlebyassumingthattheindexofrefracEonofanymediumis

n=c/v,inwhichvisthespeedoflightinthemediumandcisthespeedoflightinvacuum.ThewavelengthλnoflightinamediumdependsontheindexofrefracEonnofthemedium:Becauseofthisdependency,thephasedifferencebetweentwowavescanchangeiftheypassthroughdifferentmaterialswithdifferentindexesofrefracEon.

whereλisthewavelengthofvacuum


35-1LightasaWave

35-2Young’sInterference

Figure(a)showsthesituaEonschemaEcallyforanincidentplanewaveofwavelengthλencounteringaslitthathaswidtha=6.0λandextendsintoandoutofthepage.Thepartofthewavethatpassesthroughtheslitflaresoutonthefarside.Figures(b)(witha=3.0λ)and(c)(a=1.5λ)illustratethemainfeatureofdiffracEon:thenarrowertheslit,thegreaterthediffracEon.

TheflaringisconsistentwiththespreadingofwaveletsintheHuygensconstrucEon.Diffrac7onoccursforwavesofalltypes,notjustlightwaves.Figurebelowshowswavespassingthroughaslitflares.




FiguregivesthebasicarrangementofYoung’sexperiment.LightfromadistantmonochromaEcsourceilluminatesslitS0inscreenA.TheemerginglightthenspreadsviadiffracEontoilluminatetwoslitsS1andS2inscreenB.DiffracEonofthelightbythesetwoslitssendsoverlappingcircularwavesintotheregionbeyondscreenB,wherethewavesfromoneslitinterferewiththewavesfromtheotherslit.

Aphotographoftheinterferencepafernproducedbythearrangementshowninthefigure(right),butwithshortslits.(ThephotographisafrontviewofpartofscreenCoffigureonleQ.)ThealternaEngmaximaandminimaarecalledinterferencefringes(becausetheyresemblethedecoraEvefringesomeEmesusedonclothingandrugs).




(a)  WavesfromslitsS1andS2(whichextendintoandoutofthepage)combineatP,anarbitrarypointonscreenCatdistanceyfromthecentralaxis.TheangleθservesasaconvenientlocatorforP.

(b)  ForD>>d,wecanapproximateraysr1andr2asbeingparallel,atangleθtothecentralaxis.

ThecondiEonsformaximumandminimumintensityare



35-3InterferenceandDouble-SlitIntensity

IftwolightwavesthatmeetatapointaretointerferepercepEbly,bothmusthavethesamewavelengthandthephasedifferencebetweenthemmustremainconstantwithEme;thatis,thewavesmustbecoherent.

AplotofequaEonbelow,showingtheintensityofadouble-slitinterferencepafernasafuncEonofthephasedifferencebetweenthewaveswhentheyarrivefromthetwoslits.I0isthe(uniform)intensitythatwouldappearonthescreenifoneslitwerecovered.Theaverageintensityofthefringepafernis2I0,andthemaximumintensity(forcoherentlight)is4I0.

Asshowninfigure,inYoung’sinterferenceexperiment,twowaves,eachwithintensityI0,yieldaresultantwaveofintensityIattheviewingscreen,with

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35-3InterferenceandDouble-SlitIntensity

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