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Book5intheLightandMatterseriesoffreeintroductoryphysicstextbooks www.lightandmatter.com

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The LightandMatter seriesof introductoryphysicstextbooks: 1NewtonianPhysics 2ConservationLaws 3VibrationsandWaves 4ElectricityandMagnetism 5Optics 6TheModernRevolutioninPhysics

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BenjaminCrowell www.lightandmatter.com

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Fullerton,California www.lightandmatter.com copyright1999-2006BenjaminCrowell edition2.2 rev.April15,2008 ThisbookislicensedundertheCreativeCommonsAttribution-ShareAlikelicense,version1.0, http://creativecommons.org/licenses/by-sa/1.0/,except forthosephotographsanddrawingsofwhichIamnot theauthor,aslistedinthephotocredits.Ifyouagree tothelicense,itgrantsyoucertainprivilegesthatyou wouldnototherwisehave,suchastherighttocopythe book,ordownloadthedigitalversionfreeofchargefrom www.lightandmatter.com.Atyouroption,youmayalso copythisbookundertheGNUFreeDocumentation Licenseversion1.2,http://www.gnu.org/licenses/fdl.txt, withnoinvariantsections,nofront-covertexts,andno back-covertexts. ISBN0-9704670-5-2

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BriefContents 1TheRayModelofLight11 2ImagesbyReection29 3Images,Quantitatively43 4Refraction59 5WaveOptics77

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Contents 1TheRayModelofLight 1.1TheNatureofLight.......12 Thecauseandeectrelationshipinvision, 12.|Lightisathing,andittravelsfrom onepointtoanother.,13.|Lightcantravel throughavacuum.,14. 1.2InteractionofLightwithMatter...15 Absorptionoflight,15.|Howweseenonluminousobjects,15.|Numericalmeasurementofthebrightnessoflight,17. 1.3TheRayModelofLight.....17 Modelsoflight,17.|Raydiagrams,19. 1.4GeometryofSpecularReection.20 Reversibilityoflightrays,22. 1.5 ? ThePrincipleofLeastTimefor Reection.............24 Summary.............26 Problems.............27 2ImagesbyReection 2.1AVirtualImage.........30 2.2CurvedMirrors.........33 2.3ARealImage..........34 2.4ImagesofImages........35 Summary.............39 Problems.............40 3Images,Quantitatively 3.1ARealImageFormedbyaConvergingMirror.............44 Locationoftheimage,44.|Magnication, 47. 3.2OtherCasesWithCurvedMirrors.47 3.3 ? Aberrations..........51 Summary.............55 Problems.............57 4Refraction 4.1Refraction...........60 Refraction,60.|Refractivepropertiesof media,61.|Snell'slaw,62.|Theindex ofrefractionisrelatedtothespeedof light.,63.|AmechanicalmodelofSnell's law,64.|AderivationofSnell'slaw,64.| Colorandrefraction,65.|Howmuchlight isreected,andhowmuchistransmitted?, 65. 4.2Lenses............67 4.3 ? TheLensmaker'sEquation...68 8

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4.4 ? ThePrincipleofLeastTimefor Refraction.............69 Summary.............70 Problems.............71 5WaveOptics 5.1Diffraction...........78 5.2ScalingofDiffraction.......79 5.3TheCorrespondencePrinciple..80 5.4Huygens'Principle.......81 5.5Double-SlitDiffraction......82 5.6Repetition...........86 5.7Single-SlitDiffraction......87 5.8 R ? ThePrincipleofLeastTime..89 Summary.............91 Problems.............93 Appendix1:Exercises 97 Appendix2:PhotoCredits 107 Appendix3:HintsandSolutions 108 9

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Chapter1 TheRayModelofLight AdsforoneMacintoshcomputerbraggedthatitcoulddoanarithmeticcalculationinlesstimethanittookforthelighttogetfromthe screentoyoureye.Wendthisimpressivebecauseofthecontrast betweenthespeedoflightandthespeedsatwhichweinteractwith physicalobjectsinourenvironment.Perhapsitshouldn'tsurprise us,then,thatNewtonsucceededsowellinexplainingthemotionof objects,butwasfarlesssuccessfulwiththestudyoflight. ThesebooksarebilledastheLightandMatterseries,butonly now,inthefthofthesixvolumes,arewereadytofocusonlight. Ifyouarereadingtheseriesinorder,thenyouknowthattheclimax ofourstudyofelectricityandmagnetismwasdiscoverythatlight isanelectromagneticwave.Knowingthis,however,isnotthesame asknowingeverythingabouteyesandtelescopes.Infact,thefull descriptionoflightasawavecanberathercumbersome.Wewill insteadspendmostofthisbookmakinguseofasimplermodel oflight,theraymodel,whichdoesanejobinmostpractical situations.Notonlythat,butwewillevenbacktrackalittleand 11

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startwithadiscussionofbasicideasaboutlightandvisionthat predatedthediscoveryofelectromagneticwaves. 1.1TheNatureofLight Thecauseandeffectrelationshipinvision Despiteitstitle,thischapterisfarfromyourrstlookatlight. Thatfamiliaritymightseemlikeanadvantage,butmostpeoplehave neverthoughtcarefullyaboutlightandvision.Evensmartpeople whohavethoughthardaboutvisionhavecomeupwithincorrect ideas.TheancientGreeks,ArabsandChinesehadtheoriesoflight andvision,allofwhichweremostlywrong,andallofwhichwere acceptedforthousandsofyears. Onethingtheancientsdidgetrightisthatthereisadistinction betweenobjectsthatemitlightandobjectsthatdon't.Whenyou seealeafintheforest,it'sbecausethreedierentobjectsaredoing theirjobs:theleaf,theeye,andthesun.Butluminousobjects likethesun,aame,orthelamentofalightbulbcanbeseenby theeyewithoutthepresenceofathirdobject.Emissionoflight isoften,butnotalways,associatedwithheat.Inmoderntimes, wearefamiliarwithavarietyofobjectsthatglowwithoutbeing heated,includinguorescentlightsandglow-in-the-darktoys. Howdoweseeluminousobjects?TheGreekphilosophersPythagorasb.ca.560BCandEmpedoclesofAcragasb.ca.492 BC,whounfortunatelywereveryinuential,claimedthatwhen youlookedatacandleame,theameandyoureyewereboth sendingoutsomekindofmysteriousstu,andwhenyoureye'sstu collidedwiththecandle'sstu,thecandlewouldbecomeevidentto yoursenseofsight. BizarreastheGreekcollisionofstutheory"mightseem,it hadacoupleofgoodfeatures.Itexplainedwhyboththecandle andyoureyehadtobepresentforyoursenseofsighttofunction. Thetheorycouldalsoeasilybeexpandedtoexplainhowwesee nonluminousobjects.Ifaleaf,forinstance,happenedtobepresent atthesiteofthecollisionbetweenyoureye'sstuandthecandle's stu,thentheleafwouldbestimulatedtoexpressitsgreennature, allowingyoutoperceiveitasgreen. Modernpeoplemightfeeluneasyaboutthistheory,sinceitsuggeststhatgreennessexistsonlyforourseeingconvenience,implying ahumanprecedenceovernaturalphenomena.Nowadays,people wouldexpectthecauseandeectrelationshipinvisiontobethe otherwayaround,withtheleafdoingsomethingtooureyerather thanoureyedoingsomethingtotheleaf.Buthowcanyoutell? Themostcommonwayofdistinguishingcausefromeectistodeterminewhichhappenedrst,buttheprocessofseeingseemsto occurtooquicklytodeterminetheorderinwhichthingshappened. 12 Chapter1TheRayModelofLight

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a / Lightfromacandleisbumped offcoursebyapieceofglass. Insertingtheglasscausesthe apparentlocationofthecandle toshift.Thesameeffectcan beproducedbytakingoffyour eyeglassesandlookingatwhich youseeneartheedgeofthe lens,butaatpieceofglass worksjustaswellasalensfor thispurpose. Certainlythereisnoobvioustimelagbetweenthemomentwhen youmoveyourheadandthemomentwhenyourreectioninthe mirrormoves. Today,photographyprovidesthesimplestexperimentalevidence thatnothinghastobeemittedfromyoureyeandhittheleafinorder tomakeitgreenify."Acameracantakeapictureofaleafeven iftherearenoeyesanywherenearby.Sincetheleafappearsgreen regardlessofwhetheritisbeingsensedbyacamera,youreye,or aninsect'seye,itseemstomakemoresensetosaythattheleaf's greennessisthecause,andsomethinghappeninginthecameraor eyeistheeect. Lightisathing,andittravelsfromonepointtoanother. Anotherissuethatfewpeoplehaveconsiderediswhetheracandle'samesimplyaectsyoureyedirectly,orwhetheritsendsout lightwhichthengetsintoyoureye.Again,therapidityoftheeect makesitdiculttotellwhat'shappening.Ifsomeonethrowsarock atyou,youcanseetherockonitswaytoyourbody,andyoucan tellthatthepersonaectedyoubysendingamaterialsubstance yourway,ratherthanjustharmingyoudirectlywithanarmmotion,whichwouldbeknownasactionatadistance."Itisnoteasy todoasimilarobservationtoseewhetherthereissomestu"that travelsfromthecandletoyoureye,orwhetheritisacaseofaction atadistance. Newtonianphysicsincludesbothactionatadistancee.g.the earth'sgravitationalforceonafallingobjectandcontactforces suchasthenormalforce,whichonlyallowdistantobjectstoexert forcesoneachotherbyshootingsomesubstanceacrossthespace betweentheme.g.,agardenhosesprayingoutwaterthatexertsa forceonabush. Onepieceofevidencethatthecandlesendsoutstuthattravels toyoureyeisthatasingurea,interveningtransparentsubstances canmakethecandleappeartobeinthewronglocation,suggesting thatlightisathingthatcanbebumpedocourse.Manypeoplewoulddismissthiskindofobservationasanopticalillusion, however.Someopticalillusionsarepurelyneurologicalorpsychologicaleects,althoughsomeothers,includingthisone,turnoutto becausedbythebehavioroflightitself. Amoreconvincingwaytodecideinwhichcategorylightbelongs istondoutifittakestimetogetfromthecandletoyoureye;in Newtonianphysics,actionatadistanceissupposedtobeinstantaneous.Thefactthatwespeakcasuallytodayofthespeedof light"impliesthatatsomepointinhistory,somebodysucceededin showingthatlightdidnottravelinnitelyfast.Galileotried,and failed,todetectanitespeedforlight,byarrangingwithaperson inadistanttowertosignalbackandforthwithlanterns.Galileo Section1.1TheNatureofLight 13

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b / AnimageofJupiterand itsmoonIoleftfromtheCassini probe. c / TheearthismovingtowardJupiterandIo.Sincethe distanceisshrinking,itistaking lessandlesstimeforthelightto gettousfromIo,andIoappears tocircleJupitermorequicklythan normal.Sixmonthslater,the earthwillbeontheoppositeside ofthesun,andrecedingfrom JupiterandIo,soIowillappear torevolvearoundJupitermore slowly. uncoveredhislantern,andwhentheotherpersonsawthelight,he uncoveredhislantern.Galileowasunabletomeasureanytimelag thatwassignicantcomparedtothelimitationsofhumanreexes. Therstpersontoprovethatlight'sspeedwasnite,andto determineitnumerically,wasOleRoemer,inaseriesofmeasurementsaroundtheyear1675.RoemerobservedIo,oneofJupiter's moons,overaperiodofseveralyears.SinceIopresumablytookthe sameamountoftimetocompleteeachorbitofJupiter,itcouldbe thoughtofasaverydistant,veryaccurateclock.Apracticalandaccuratependulumclockhadrecentlybeeninvented,soRoemercould checkwhethertheratioofthetwoclocks'cycles,about42.5hours to1orbit,stayedexactlyconstantorchangedalittle.Iftheprocess ofseeingthedistantmoonwasinstantaneous,therewouldbeno reasonforthetwotogetoutofstep.Evenifthespeedoflightwas nite,youmightexpectthattheresultwouldbeonlytoosetone cyclerelativetotheother.Theearthdoesnot,however,stayata constantdistancefromJupiteranditsmoons.Sincethedistanceis changinggraduallyduetothetwoplanets'orbitalmotions,anite speedoflightwouldmaketheIoclock"appeartorunfasterasthe planetsdrewneareachother,andmoreslowlyastheirseparation increased.RoemerdidndavariationintheapparentspeedofIo's orbits,whichcausedIo'seclipsesbyJupiterthemomentswhenIo passedinfrontoforbehindJupitertooccurabout7minutesearly whentheearthwasclosesttoJupiter,and7minuteslatewhenit wasfarthest.Basedonthesemeasurements,Roemerestimatedthe speedoflighttobeapproximately2 10 8 m/s,whichisintheright ballparkcomparedtomodernmeasurementsof3 10 8 m/s.I'mnot surewhetherthefairlylargeexperimentalerrorwasmainlydueto impreciseknowledgeoftheradiusoftheearth'sorbitorlimitations inthereliabilityofpendulumclocks. Lightcantravelthroughavacuum. Manypeopleareconfusedbytherelationshipbetweensound andlight.Althoughweusedierentorganstosensethem,thereare somesimilarities.Forinstance,bothlightandsoundaretypically emittedinalldirectionsbytheirsources.Musiciansevenusevisual metaphorsliketonecolor,"orabrighttimbre"todescribesound. Onewaytoseethattheyareclearlydierentphenomenaistonote theirverydierentvelocities.Sure,bothareprettyfastcomparedto ayingarroworagallopinghorse,butaswehaveseen,thespeedof lightissogreatastoappearinstantaneousinmostsituations.The speedofsound,however,caneasilybeobservedjustbywatchinga groupofschoolchildrenahundredfeetawayastheyclaptheirhands toasong.Thereisanobviousdelaybetweenwhenyouseetheir palmscometogetherandwhenyouheartheclap. Thefundamentaldistinctionbetweensoundandlightisthat soundisanoscillationinairpressure,soitrequiresairorsome 14 Chapter1TheRayModelofLight

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othermediumsuchaswaterinwhichtotravel.Today,weknow thatouterspaceisavacuum,sothefactthatwegetlightfromthe sun,moonandstarsclearlyshowsthatairisnotnecessaryforthe propagationoflight. DiscussionQuestions A Ifyouobservethunderandlightning,youcantellhowfarawaythe stormis.Doyouneedtoknowthespeedofsound,oflight,orofboth? B WhenphenomenalikeX-raysandcosmicrayswererstdiscovered, suggestawayonecouldhavetestedwhethertheywereformsoflight. C WhydidRoemeronlyneedtoknowtheradiusoftheearth'sorbit, notJupiter's,inordertondthespeedoflight? 1.2InteractionofLightwithMatter Absorptionoflight Thereasonwhythesunfeelswarmonyourskinisthatthe sunlightisbeingabsorbed,andthelightenergyisbeingtransformed intoheatenergy.Thesamehappenswitharticiallight,sothenet resultofleavingalightturnedonistoheattheroom.Itdoesn't matterwhetherthesourceofthelightishot,likethesun,aame, oranincandescentlightbulb,orcool,likeauorescentbulb.If yourhousehaselectricheat,thenthereisabsolutelynopointin fastidiouslyturningolightsinthewinter;thelightswillhelpto heatthehouseatthesamedollarrateastheelectricheater. Thisprocessofheatingbyabsorptionisentirelydierentfrom heatingbythermalconduction,aswhenanelectricstoveheats spaghettisaucethroughapan.Heatcanonlybeconductedthrough matter,butthereisvacuumbetweenusandthesun,orbetweenus andthelamentofanincandescentbulb.Also,heatconductioncan onlytransferheatenergyfromahotterobjecttoacolderone,buta cooluorescentbulbisperfectlycapableofheatingsomethingthat hadalreadystartedoutbeingwarmerthanthebulbitself. Howweseenonluminousobjects Notallthelightenergythathitsanobjectistransformedinto heat.Someisreected,andthisleadsustothequestionofhow weseenonluminousobjects.Ifyouasktheaveragepersonhowwe seealightbulb,themostlikelyanswerisThelightbulbmakes light,whichhitsoureyes."Butifyouaskhowweseeabook,they arelikelytosayThebulblightsuptheroom,andthatletsme seethebook."Allmentionoflightactuallyenteringoureyeshas mysteriouslydisappeared. Mostpeoplewoulddisagreeifyoutoldthemthatlightwasreectedfromthebooktotheeye,becausetheythinkofreectionas somethingthatmirrorsdo,notsomethingthatabookdoes.They associatereectionwiththeformationofareectedimage,which Section1.2InteractionofLightwithMatter 15

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d / Twoself-portraitsofthe author,onetakeninamirrorand onewithapieceofaluminumfoil. e / Specularanddiffusereection. doesnotseemtoappearinapieceofpaper. Imaginethatyouarelookingatyourreectioninanicesmooth pieceofaluminumfoil,freshotheroll.Youperceiveaface,nota pieceofmetal.Perhapsyoualsoseethebrightreectionofalamp overyourshoulderbehindyou.Nowimaginethatthefoilisjust alittlebitlesssmooth.Thedierentpartsoftheimagearenow alittlebitoutofalignmentwitheachother.Yourbraincanstill recognizeafaceandalamp,butit'salittlescrambled,likeaPicasso painting.Nowsupposeyouuseapieceofaluminumfoilthathas beencrumpledupandthenattenedoutagain.Thepartsofthe imagearesoscrambledthatyoucannotrecognizeanimage.Instead, yourbraintellsyouyou'relookingatarough,silverysurface. Mirror-likereectionataspecicangleisknownasspecular reection,andrandomreectioninmanydirectionsiscalleddiuse reection.Diusereectionishowweseenonluminousobjects. Specularreectiononlyallowsustoseeimagesofobjectsother thantheonedoingthereecting.Intoppartofgured,imagine thattheraysoflightarecomingfromthesun.Ifyouarelooking downatthereectingsurface,thereisnowayforyoureye-brain systemtotellthattheraysarenotreallycomingfromasundown belowyou. Figurefshowsanotherexampleofhowwecan'tavoidtheconclusionthatlightbouncesoofthingsotherthanmirrors.The lampisoneIhaveinmyhouse.Ithasabrightbulb,housedina completelyopaquebowl-shapedmetalshade.Theonlywaylight cangetoutofthelampisbygoingupoutofthetopofthebowl. ThefactthatIcanreadabookinthepositionshowninthegure meansthatlightmustbebouncingooftheceiling,thenbouncing oofthebook,thennallygettingtomyeye. ThisiswheretheshortcomingsoftheGreektheoryofvision becomeglaringlyobvious.IntheGreektheory,thelightfromthe bulbandmymysteriouseyerays"arebothsupposedtogotothe book,wheretheycollide,allowingmetoseethebook.Butwenow haveatotaloffourobjects:lamp,eye,book,andceiling.Where doestheceilingcomein?Doesitalsosendoutitsownmysterious ceilingrays,"contributingtoathree-waycollisionatthebook? Thatwouldjustbetoobizarretobelieve! Thedierencesamongwhite,black,andthevariousshadesof grayinbetweenisamatterofwhatpercentageofthelightthey absorbandwhatpercentagetheyreect.That'swhylight-colored clothingismorecomfortableinthesummer,andlight-coloredupholsteryinacarstayscoolerthatdarkupholstery. 16 Chapter1TheRayModelofLight

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f / Lightbouncesoffofthe ceiling,thenoffofthebook. g / DiscussionquestionC. Numericalmeasurementofthebrightnessoflight Wehavealreadyseenthatthephysiologicalsensationofloudness relatestothesound'sintensitypowerperunitarea,butisnot directlyproportionaltoit.IfsoundAhasanintensityof1nW = m 2 soundBis10nW = m 2 ,andsoundCis100nW = m 2 ,thentheincrease inloudnessfromCtoBisperceivedtobethesameastheincrease fromAtoB,nottentimesgreater.Thatis,thesensationofloudness islogarithmic. Thesameistrueforthebrightnessoflight.Brightnessisrelatedtopowerperunitarea,butthepsychologicalrelationshipis alogarithmiconeratherthanaproportionality.Fordoingphysics, it'sthepowerperunitareathatwe'reinterestedin.Therelevant unitisW = m 2 .Onewaytodeterminethebrightnessoflightisto measuretheincreaseintemperatureofablackobjectexposedto thelight.Thelightenergyisbeingconvertedtoheatenergy,and theamountofheatenergyabsorbedinagivenamountoftimecan berelatedtothepowerabsorbed,usingtheknownheatcapacity oftheobject.Morepracticaldevicesformeasuringlightintensity, suchasthelightmetersbuiltintosomecameras,arebasedonthe conversionoflightintoelectricalenergy,butthesemetershaveto becalibratedsomehowagainstheatmeasurements. DiscussionQuestions A Thecurtainsinaroomaredrawn,butasmallgapletslightthrough, illuminatingaspotontheoor.Itmayormaynotalsobepossibletosee thebeamofsunshinecrossingtheroom,dependingontheconditions. What'sgoingon? B Laserbeamsaremadeoflight.Insciencectionmovies,laser beamsareoftenshownasbrightlinesshootingoutofalasergunona spaceship.Whyisthisscienticallyincorrect? C Adocumentarylm-makerwenttoHarvard's1987graduationceremonyandaskedthegraduates,oncamera,toexplainthecauseofthe seasons.Onlytwooutof23wereabletogiveacorrectexplanation,but younowhavealltheinformationneededtogureitoutforyourself,assumingyoudidn'talreadyknow.Thegureshowstheearthinitswinter andsummerpositionsrelativetothesun.Hint:Considertheunitsused tomeasurethebrightnessoflight,andrecallthatthesunislowerinthe skyinwinter,soitsraysarecominginatashallowerangle. 1.3TheRayModelofLight Modelsoflight NotehowI'vebeencasuallydiagrammingthemotionoflight withpicturesshowinglightraysaslinesonthepage.Moreformally, thisisknownastheraymodeloflight.Theraymodeloflight seemsnaturalonceweconvinceourselvesthatlighttravelsthrough space,andobservephenomenalikesunbeamscomingthroughholes inclouds.Havingalreadybeenintroducedtotheconceptoflight Section1.3TheRayModelofLight 17

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asanelectromagneticwave,youknowthattheraymodelisnotthe ultimatetruthaboutlight,buttheraymodelissimpler,andinany casesciencealwaysdealswithmodelsofreality,nottheultimate natureofreality.Thefollowingtablesummarizesthreemodelsof light. h / Threemodelsoflight. Theraymodelisagenericone.Byusingitwecandiscussthe pathtakenbythelight,withoutcommittingourselvestoanyspecic descriptionofwhatitisthatismovingalongthatpath.Wewill usethenicesimpleraymodelformostofthisbook,andwithitwe cananalyzeagreatmanydevicesandphenomena.Notuntilthe lastchapterwillweconcernourselvesspecicallywithwaveoptics, althoughintheinterveningchaptersIwillsometimesanalyzethe samephenomenonusingboththeraymodelandthewavemodel. Notethatthestatementsabouttheapplicabilityofthevarious modelsareonlyroughguides.Forinstance,waveinterferenceeects areoftendetectable,ifsmall,whenlightpassesaroundanobstacle thatisquiteabitbiggerthanawavelength.Also,thecriterionfor whenweneedtheparticlemodelreallyhasmoretodowithenergy scalesthandistancescales,althoughthetwoturnouttoberelated. Thealertreadermayhavenoticedthatthewavemodelisrequiredatscalessmallerthanawavelengthoflightontheorderofa micrometerforvisiblelight,andtheparticlemodelisdemandedon theatomicscaleorloweratypicalatombeingananometerorsoin size.Thisimpliesthatatthesmallestscalesweneed both thewave modelandtheparticlemodel.Theyappearincompatible,sohow canwesimultaneouslyuseboth?Theansweristhattheyarenot asincompatibleastheyseem.Lightisbothawaveandaparticle, 18 Chapter1TheRayModelofLight

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butafullunderstandingofthisapparentlynonsensicalstatementis atopicforthefollowingbookinthisseries. i / Examplesofraydiagrams. Raydiagrams Withoutevenknowinghowtousetheraymodeltocalculate anythingnumerically,wecanlearnagreatdealbydrawingray diagrams.Forinstance,ifyouwanttounderstandhoweyeglasses helpyoutoseeinfocus,araydiagramistherightplacetostart. Manystudentsunder-utilizeraydiagramsinopticsandinsteadrely onrotememorizationorpluggingintoformulas.Thetroublewith memorizationandplug-insisthattheycanobscurewhat'sreally goingon,anditiseasytogetthemwrong.Oftenthebestplanisto doaraydiagramrst,thendoanumericalcalculation,thencheck thatyournumericalresultsareinreasonableagreementwithwhat youexpectedfromtheraydiagram. j / 1.Correct.2.Incorrect:impliesthatdiffusereectiononly givesonerayfromeachreecting point.3.Correct,butunnecessarilycomplicated Figurejshowssomeguidelinesforusingraydiagramseectively. Thelightraysbendwhentheypassoutthroughthesurfaceofthe wateraphenomenonthatwe'lldiscussinmoredetaillater.The raysappeartohavecomefromapointabovethegoldsh'sactual location,aneectthatisfamiliartopeoplewhohavetriedspearshing. Astreamoflightisnotreallyconnedtoanitenumberof narrowlines.Wejustdrawitthatway.Inj/1,ithasbeen necessarytochooseanitenumberofraystodrawve, ratherthanthetheoreticallyinnitenumberofraysthatwill divergefromthatpoint. Section1.3TheRayModelofLight 19

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Thereisatendencytoconceptualizeraysincorrectlyasobjects.InhisOptics,Newtongoesoutofhiswaytocaution thereaderagainstthis,sayingthatsomepeopleconsider... therefractionof...raystobethebendingorbreakingofthem intheirpassingoutofonemediumintoanother."Butaray isarecordofthepathtraveledbylight,notaphysicalthing thatcanbebentorbroken. Intheory,raysmaycontinueinnitelyfarintothepastand future,butweneedtodrawlinesofnitelength.Inj/1,a judiciouschoicehasbeenmadeastowheretobeginandend therays.Thereisnopointincontinuingtheraysanyfarther thanshown,becausenothingnewandexcitingisgoingto happentothem.Thereisalsonogoodreasontostartthem earlier,beforebeingreectedbythesh,becausethedirection ofthediuselyreectedraysisrandomanyway,andunrelated tothedirectionoftheoriginal,incomingray. Whenrepresentingdiusereectioninaraydiagram,many studentshaveamentalblockagainstdrawingmanyraysfanningoutfromthesamepoint.Often,asinexamplej/2,the problemisthemisconceptionthatlightcanonlybereected inonedirectionfromonepoint. Anotherdicultyassociatedwithdiusereection,example j/3,isthetendencytothinkthatinadditiontodrawingmany rayscomingoutofonepoint,weshouldalsobedrawingmany rayscomingfrommanypoints.Inj/1,drawingmanyrays comingoutofonepointgivesusefulinformation,tellingus, forinstance,thattheshcanbeseenfromanyangle.Drawing manysetsofrays,asinj/3,doesnotgiveusanymoreuseful information,andjustcluttersupthepictureinthisexample. Theonlyreasontodrawsetsofraysfanningoutfrommore thanonepointwouldbeifdierentthingswerehappeningto thedierentsets. DiscussionQuestion A Supposeanintelligenttool-usingshisspear-huntingforhumans. Drawaraydiagramtoshowhowtheshhastocorrectitsaim.Note thatalthoughtheraysarenowpassingfromtheairtothewater,thesame rulesapply:theraysareclosertobeingperpendiculartothesurfacewhen theyareinthewater,andraysthathittheair-waterinterfaceatashallow anglearebentthemost. 1.4GeometryofSpecularReection Tochangethemotionofamaterialobject,weuseaforce.Isthere anywaytoexertaforceonabeamoflight?Experimentsshow thatelectricandmagneticeldsdonotdeectlightbeams,soapparentlylighthasnoelectriccharge.Lightalsohasnomass,so 20 Chapter1TheRayModelofLight

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k / Thegeometryofspecular reection. untilthetwentiethcenturyitwasbelievedtobeimmunetogravity aswell.Einsteinpredictedthatlightbeamswouldbeveryslightly deectedbystronggravitationalelds,andhewasprovedcorrect byobservationsofraysofstarlightthatcameclosetothesun,but obviouslythat'snotwhatmakesmirrorsandlenseswork! Ifweinvestigatehowlightisreectedbyamirror,wewillnd thattheprocessishorricallycomplex,butthenalresultissurprisinglysimple.Whatactuallyhappensisthatthelightismade ofelectricandmagneticelds,andtheseeldsacceleratetheelectronsinthemirror.Energyfromthelightbeamismomentarily transformedintoextrakineticenergyoftheelectrons,butbecause theelectronsareacceleratingtheyre-radiatemorelight,convertingtheirkineticenergybackintolightenergy.Wemightexpect thistoresultinaverychaoticsituation,butamazinglyenough,the electronsmovetogethertoproduceanew,reectedbeamoflight, whichobeystwosimplerules: Theangleofthereectedrayisthesameasthatoftheincident ray. Thereectedrayliesintheplanecontainingtheincidentray andthenormalperpendicularline.Thisplaneisknownas theplaneofincidence. Thetwoanglescanbedenedeitherwithrespecttothenormal, likeanglesBandCinthegure,orwithrespecttothereecting surface,likeanglesAandD.Thereisaconventionofseveralhundred years'standingthatonemeasurestheangleswithrespecttothe normal,buttheruleaboutequalanglescanlogicallybestatedeither asB=CorasA=D. Thephenomenonofreectionoccursonlyattheboundarybetweentwomedia,justlikethechangeinthespeedoflightthat passesfromonemediumtoanother.Aswehaveseeninbook3of thisseries,thisisthewayallwavesbehave. Mostpeoplearesurprisedbythefactthatlightcanbereected backfromalessdensemedium.Forinstance,ifyouaredivingand youlookupatthesurfaceofthewater,youwillseeareectionof yourself. Section1.4GeometryofSpecularReection 21

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self-checkA Eachofthesediagramsissupposedtoshowtwodifferentraysbeing reectedfromthesamepointonthesamemirror.Whicharecorrect, andwhichareincorrect? Answer,p.108 Reversibilityoflightrays Thefactthatspecularreectiondisplaysequalanglesofincidenceandreectionmeansthatthereisasymmetry:iftherayhad comeinfromtherightinsteadoftheleftinthegureabove,theangleswouldhavelookedexactlythesame.Thisisnotjustapointless detailaboutspecularreection.It'samanifestationofaverydeep andimportantfactaboutnature,whichisthatthelawsofphysics donotdistinguishbetweenpastandfuture.Cannonballsandplanetshavetrajectoriesthatareequallynaturalinreverse,andsodo lightrays.Thistypeofsymmetryiscalledtime-reversalsymmetry. Typically,time-reversalsymmetryisacharacteristicofanyprocessthatdoesnotinvolveheat.Forinstance,theplanetsdonot experienceanyfrictionastheytravelthroughemptyspace,sothere isnofrictionalheating.Weshouldthusexpectthetime-reversed versionsoftheirorbitstoobeythelawsofphysics,whichtheydo. Incontrast,abookslidingacrossatabledoesgenerateheatfrom frictionasitslowsdown,anditisthereforenotsurprisingthatthis typeofmotiondoesnotappeartoobeytime-reversalsymmetry.A booklyingstillonaattableisneverobservedtospontaneously startsliding,suckingupheatenergyandtransformingitintokinetic energy. Similarly,theonlysituationwe'veobservedsofarwherelight doesnotobeytime-reversalsymmetryisabsorption,whichinvolves heat.Yourskinabsorbsvisiblelightfromthesunandheatsup, butweneverobservepeople'sskintoglow,convertingheatenergy intovisiblelight.People'sskindoesglowininfraredlight,but thatdoesn'tmeanthesituationissymmetric.Evenifyouabsorb infrared,youdon'temitvisiblelight,becauseyourskinisn'thot enoughtoglowinthevisiblespectrum. Theseapparentheat-relatedasymmetriesarenotactualasymmetriesinthelawsofphysics.Theinterestedreadermaywishto learnmoreaboutthisfromtheoptionalthermodynamicschapterof book2inthisseries. Raytracingonacomputerexample1 Anumberoftechniquescanbeusedforcreatingarticialvisual scenesincomputergraphics.Figurelshowssuchascene,which 22 Chapter1TheRayModelofLight

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wascreatedbythebrute-forcetechniqueofsimplyconstructing averydetailedraydiagramonacomputer.Thistechniquerequiresagreatdealofcomputation,andisthereforetooslowto beusedforvideogamesandcomputer-animatedmovies.One trickforspeedingupthecomputationistoexploitthereversibility oflightrays.Ifonewastotraceeveryrayemittedbyeveryilluminatedsurface,onlyatinyfractionofthosewouldactuallyend uppassingintothevirtualcamera,andthereforealmostallof thecomputationaleffortwouldbewasted.Onecaninsteadstart arayatthecamera,traceitbackwardintime,andseewhereit wouldhavecomefrom.Withthistechnique,thereisnowasted effort. l / Thisphotorealisticimageofanonexistentcountertopwasproducedcompletelyonacomputer,bycomputingacomplicatedray diagram. Section1.4GeometryofSpecularReection 23

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m / DiscussionquestionB. n / DiscussionquestionC. o / Thesolidlinesarephysicallypossiblepathsforlightrays travelingfromAtoBandfrom AtoC.Theyobeytheprinciple ofleasttime.Thedashedlines donotobeytheprincipleof leasttime,andarenotphysically possible. DiscussionQuestions A Ifalightrayhasavelocityvectorwithcomponents c x and c y ,what willhappenwhenitisreectedfromasurfacethatliesalongthe y axis? Makesureyouranswerdoesnotimplyachangeintheray'sspeed. B GeneralizingyourreasoningfromdiscussionquestionA,whatwill happentothevelocitycomponentsofalightraythathitsacorner,as showninthegure,andundergoestworeections? C Threepiecesofsheetmetalarrangedperpendicularlyasshownin thegureformwhatisknownasaradarcorner.Let'sassumethatthe radarcornerislargecomparedtothewavelengthoftheradarwaves,so thattheraymodelmakessense.Iftheradarcornerisbathedinradar rays,atleastsomeofthemwillundergothreereections.Makingafurthergeneralizationofyourreasoningfromthetwoprecedingdiscussion questions,whatwillhappentothethreevelocitycomponentsofsucha ray?Whatwouldtheradarcornerbeusefulfor? 1.5 ? ThePrincipleofLeastTimeforReection Wehadtochoosebetweenanunwieldyexplanationofreectionat theatomiclevelandasimplergeometricdescriptionthatwasnotas fundamental.Thereisathirdapproachtodescribingtheinteraction oflightandmatterwhichisverydeepandbeautiful.Emphasized bythetwentieth-centuryphysicistRichardFeynman,itiscalledthe principleofleasttime,orFermat'sprinciple. Let'sstartwiththemotionoflightthatisnotinteractingwith matteratall.Inavacuum,alightraymovesinastraightline.This canberephrasedasfollows:ofalltheconceivablepathslightcould followfromPtoQ,theonlyonethatisphysicallypossibleisthe paththattakestheleasttime. Whataboutreection?Iflightisgoingtogofromonepointto another,beingreectedontheway,thequickestpathisindeedthe onewithequalanglesofincidenceandreection.Ifthestartingand endingpointsareequallyfarfromthereectingsurface,o,it'snot hardtoconvinceyourselfthatthisistrue,justbasedonsymmetry. Thereisalsoatrickyandsimpleproof,showningurep,forthe moregeneralcasewherethepointsareatdierentdistancesfrom thesurface. 24 Chapter1TheRayModelofLight

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p / PathsAQBandAPBare twoconceivablepathsthataray couldfollowtogetfromAtoB withonereection,butonlyAQB isphysicallypossible.Wewish toprovethatthepathAQB,with equalanglesofincidenceand reection,isshorterthanany otherpath,suchasAPB.The trickistoconstructathirdpoint, C,lyingasfarbelowthesurface asBliesaboveit.Thenpath AQCisastraightlinewhose lengthisthesameasAQB's,and pathAPChasthesamelengthas pathAPB.SinceAQCisstraight, itmustbeshorterthananyother pathsuchasAPCthatconnects AandC,andthereforeAQBmust beshorterthananypathsuchas APB. q / Lightisemittedatthecenter ofanellipticalmirror.Thereare fourphysicallypossiblepathsby whicharaycanbereectedand returntothecenter. Notonlydoestheprincipleofleasttimeworkforlightina vacuumandlightundergoingreection,wewillalsoseeinalater chapterthatitworksforthebendingoflightwhenitpassesfrom onemediumintoanother. Althoughitisbeautifulthattheentireraymodeloflightcan bereducedtoonesimplerule,theprincipleofleasttime,itmay seemalittlespookytospeakasiftherayoflightisintelligent, andhascarefullyplannedaheadtondtheshortestroutetoits destination.Howdoesitknowinadvancewhereit'sgoing?What ifwemovedthemirrorwhilethelightwasenroute,soconditions alongitsplannedpathwerenotwhatitexpected?"Theanswer isthattheprincipleofleasttimeisreallyashortcutfornding certainresultsofthewavemodeloflight,whichisthetopicofthe lastchapterofthisbook. Thereareacoupleofsubtlepointsabouttheprincipleofleast time.First,thepathdoesnothavetobethequickestofallpossiblepaths;itonlyneedstobequickerthananypaththatdiers innitesimallyfromit.Ingurep,forinstance,lightcouldgetfrom AtoBeitherbythereectedpathAQBorsimplybygoingstraight fromAtoB.AlthoughAQBisnottheshortestpossiblepath,it cannotbeshortenedbychangingitinnitesimally,e.g.,bymoving Qalittletotherightorleft.Ontheotherhand,pathAPBisphysicallyimpossible,becauseitispossibletoimproveonitbymoving pointPinnitesimallytotheright. It'snotquiterighttocallthistheprincipleof least time.Ingureq,forexample,thefourphysicallypossiblepathsbywhicharay canreturntothecenterconsistoftwoshortest-timepathsandtwo longest-timepaths.Strictlyspeaking,weshouldrefertothe principleofleastorgreatesttime ,butmostphysicistsomittheniceties, andassumethatotherphysicistsunderstandthatbothmaximaand minimaarepossible. Section1.5 ? ThePrincipleofLeastTimeforReection 25

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Summary SelectedVocabulary absorption....whathappenswhenlighthitsmatterandgives upsomeofitsenergy reection.....whathappenswhenlighthitsmatterand bounceso,retainingatleastsomeofitsenergy specularreection........ reectionfromasmoothsurface,inwhichthe lightrayleavesatthesameangleatwhichit camein diusereectionreectionfromaroughsurface,inwhichasinglerayoflightisdividedupintomanyweaker reectedraysgoinginmanydirections normal......thelineperpendiculartoasurfaceatagiven point Notation c ..........thespeedoflight Summary Wecanunderstandmanyphenomenainvolvinglightwithout havingtousesophisticatedmodelssuchasthewavemodelorthe particlemodel.Instead,wesimplydescribelightaccordingtothe pathittakes,whichwecallaray.Theraymodeloflightisuseful whenlightisinteractingwithmaterialobjectsthataremuchlarger thanawavelengthoflight.Sinceawavelengthofvisiblelightisso shortcomparedtothehumanscaleofexistence,theraymodelis usefulinmanypracticalcases. Weseethingsbecauselightcomesfromthemtooureyes.Objectsthatglowmaysendlightdirectlytooureyes,butweseean objectthatdoesn'tglowvialightfromanothersourcethathasbeen reectedbytheobject. Manyoftheinteractionsoflightandmattercanbeunderstood byconsideringwhathappenswhenlightreachestheboundarybetweentwodierentsubstances.Inthissituation,partofthelightis reectedbouncesbackandpartpassesonintothenewmedium. Thisisnotsurprising|itistypicalbehaviorforawave,andlightis awave.Lightenergycanalsobeabsorbedbymatter,i.e.,converted intoheat. Asmoothsurfaceproducesspecularreection,inwhichthereectedrayexitsatthesameanglewithrespecttothenormalas thatoftheincomingray.Aroughsurfacegivesdiusereection, whereasinglerayoflightisdividedupintomanyweakerreected raysgoinginmanydirections. 26 Chapter1TheRayModelofLight

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Drawaraydiagramshowingwhyasmalllightsourcea candle,sayproducessharpershadowsthanalargeonee.g.along uorescentbulb. 2 AGlobalPositioningSystemGPSreceiverisadevicethat letsyougureoutwhereyouarebyreceivingtimedradiosignals fromsatellites.Itworksbymeasuringthetraveltimeforthesignals, whichisrelatedtothedistancebetweenyouandthesatellite.By ndingtherangestoseveraldierentsatellitesinthisway,itcan pindownyourlocationinthreedimensionstowithinafewmeters. Howaccuratedoesthemeasurementofthetimedelayhavetobeto determineyourpositiontothisaccuracy? 3 Estimatethefrequencyofanelectromagneticwavewhose wavelengthissimilarinsizetoanatomaboutanm.Referring backtoyourelectricityandmagnetismtext,inwhatpartofthe electromagneticspectrumwouldsuchawavelieinfrared,gammarays,...? 4 TheStealthbomberisdesignedwithat,smoothsurfaces. Whywouldthismakeitdiculttodetectviaradar? 5 Thegureonthenextpageshowsacurvedparabolicmirror, withthreeparallellightrayscomingtowardit.Onerayisapproachingalongthemirror'scenterline.aTracethedrawingaccurately, andcontinuethelightraysuntiltheyareabouttoundergotheir secondreection.Togetgoodenoughaccuracy,you'llneedtophotocopythepageordownloadthebookandprintthepageand drawinthenormalateachplacewherearayisreected.What doyounotice?bMakeupanexampleofapracticaluseforthis device.cHowcouldyouusethismirrorwithasmalllightbulbto produceaparallelbeamoflightraysgoingototheright? 6 ThenativesofplanetWumpusplaypoolusinglightrayson aneleven-sidedtablewithmirrorsforbumpers,showninthegure onthenextpage.Tracethisshotaccuratelywitharulertoreveal thehiddenmessage.Togetgoodenoughaccuracy,you'llneedto photocopythepageordownloadthebookandprintthepageand drawinthenormalateachplacewheretheraystrikesabumper. Problems 27

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Problem5. Problem6. 28 Chapter1TheRayModelofLight

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Narcissus,byMichelangeloCaravaggio,ca.1598. Chapter2 ImagesbyReection InfantsarealwaysfascinatedbytheanticsoftheBabyintheMirror. Nowifyouwanttoknowsomethingaboutmirrorimagesthatmost peopledon'tunderstand,trythis.Firstbringthispagecloserand closertoyoureyes,untilyoucannolongerfocusonitwithout straining.Thengointhebathroomandseehowcloseyoucan getyourfacetothesurfaceofthemirrorbeforeyoucannolonger easilyfocusontheimageofyourowneyes.Youwillndthat theshortestcomfortableeye-mirrordistanceismuchlessthanthe shortestcomfortableeye-paperdistance.Thisdemonstratesthat theimageofyourfaceinthemirroractsasifithaddepthand 29

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a / Animageformedbya mirror. existedinthespace behind themirror.Iftheimagewaslikeaat pictureinabook,thenyouwouldn'tbeabletofocusonitfrom suchashortdistance. Inthischapterwewillstudytheimagesformedbyatand curvedmirrorsonaqualitative,conceptualbasis.Althoughthis typeofimageisnotascommonlyencounteredineverydaylifeas imagesformedbylenses,imagesformedbyreectionaresimpler tounderstand,sowediscussthemrst.Inchapter3wewillturn toamoremathematicaltreatmentofimagesmadebyreection. Surprisingly,thesameequationscanalsobeappliedtolenses,which arethetopicofchapter4. 2.1AVirtualImage Wecanunderstandamirrorimageusingaraydiagram.Figure ashowsseverallightrays,1,thatoriginatedbydiusereectionat theperson'snose.Theybounceothemirror,producingnewrays, 2.Toanyonewhoseeyeisintherightpositiontogetoneofthese rays,theyappeartohavecomefromabehindthemirror,3,where theywouldhaveoriginatedfromasinglepoint.Thispointiswhere thetipoftheimage-person'snoseappearstobe.Asimilaranalysis appliestoeveryotherpointontheperson'sface,soitlooksas thoughtherewasanentirefacebehindthemirror.Thecustomary wayofdescribingthesituationrequiressomeexplanation: Customarydescriptioninphysics: Thereisanimageoftheface behindthemirror. Translation: Thepatternofrayscomingfromthemirrorisexactly thesameasitwouldbeiftherewasafacebehindthemirror. Nothingisreallybehindthemirror. Thisisreferredtoasa virtual image,becausetheraysdonot actuallycrossatthepointbehindthemirror.Theyonlyappearto haveoriginatedthere. self-checkA Imaginethatthepersoningureamoveshisfacedownquiteabita coupleoffeetinreallife,orafewinchesonthisscaledrawing.Drawa newraydiagram.Willtherestillbeanimage?Ifso,whereisitvisible from? Answer,p.108 Thegeometryofspecularreectiontellsusthatrays1and2 areatequalanglestothenormaltheimaginaryperpendicularline piercingthemirroratthepointofreection.Thismeansthatray 2'simaginarycontinuation,3,formsthesameanglewiththemirror asray3.Sinceeachrayoftype3formsthesameangleswiththe 30 Chapter2ImagesbyReection

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c / Thepraxinoscope. mirrorasitspartneroftype1,weseethatthedistanceoftheimage fromthemirroristhesameastheactualfacefromthemirror,and liesdirectlyacrossfromit.Theimagethereforeappearstobethe samesizeastheactualface. b / Example1. Aneyeexamexample1 Figurebshowsatypicalsetupinanoptometrist'sexamination room.Thepatient'svisionissupposedtobetestedatadistance of6metersfeetintheU.S.,butthisdistanceislargerthan theamountofspaceavailableintheroom.Thereforeamirroris usedtocreateanimageoftheeyechartbehindthewall. ThePraxinoscopeexample2 Figurecshowsanold-fashioneddevicecalledapraxinoscope, whichdisplaysananimatedpicturewhenspun.Theremovable stripofpaperwiththepicturesprintedonithastwicetheradius oftheinnercirclemadeofatmirrors,soeachpicture'svirtual imageisatthecenter.Asthewheelspins,eachpicture'simage isreplacedbythenext,andsoon. Section2.1AVirtualImage 31

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DiscussionQuestion A Thegureshowsanobjectthatisofftoonesideofamirror.Draw araydiagram.Isanimageformed?Ifso,whereisit,andfromwhich directionswoulditbevisible? 32 Chapter2ImagesbyReection

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e / Theimageismagnied bythesamefactorindepthand initsotherdimensions. d / Animageformedbya curvedmirror. 2.2CurvedMirrors Animageinaatmirrorisapretechnologicalexample:even animalscanlookattheirreectionsinacalmpond.Wenowpass toourrstnontrivialexampleofthemanipulationofanimageby technology:animageinacurvedmirror.Beforewedivein,let's considerwhythisisanimportantexample.Ifitwasjustaquestionofmemorizingabunchoffactsaboutcurvedmirrors,thenyou wouldrightlyrebelagainstaneorttospoilthebeautyofyourliberallyeducatedbrainbyforce-feedingyoutechnologicaltrivia.The reasonthisisanimportantexampleisnotthatcurvedmirrorsare soimportantinandofthemselves,butthattheresultswederivefor curvedbowl-shapedmirrorsturnouttobetrueforalargeclassof otheropticaldevices,includingmirrorsthatbulgeoutwardrather thaninward,andlensesaswell.Amicroscopeoratelescopeissimplyacombinationoflensesormirrorsorboth.Whatyou'rereally learningabouthereisthebasicbuildingblockofallopticaldevices frommovieprojectorstooctopuseyes. Becausethemirroringurediscurved,itbendstheraysback closertogetherthanaatmirrorwould:wedescribeitas converging Notethatthetermreferstowhatitdoestothelightrays,nottothe physicalshapeofthemirror'ssurface.Thesurfaceitselfwouldbe describedas concave .Thetermisnotallthathardtoremember, becausethehollowed-outinteriorofthemirrorislikeacave.It issurprisingbuttruethatalltherayslike3reallydoconvergeon apoint,formingagoodimage.Wewillnotprovethisfact,butit istrueforanymirrorwhosecurvatureisgentleenoughandthat issymmetricwithrespecttorotationabouttheperpendicularline passingthroughitscenternotasymmetriclikeapotatochip.The old-fashionedmethodofmakingmirrorsandlensesisbygrinding themingritbyhand,andthisautomaticallytendstoproducean almostperfectsphericalsurface. Bendingaraylike2inwardimpliesbendingitsimaginarycontinuation3outward,inthesamewaythatraisingoneendofaseesaw causestheotherendtogodown.Theimagethereforeformsdeeper behindthemirror.Thisdoesn'tjustshowthatthereisextradistancebetweentheimage-noseandthemirror;italsoimpliesthat theimageitselfisbiggerfromfronttoback.Ithasbeen magnied inthefront-to-backdirection. Itiseasytoprovethatthesamemagnicationalsoappliestothe image'sotherdimensions.ConsiderapointlikeEinguree.The trickisthatoutofalltheraysdiuselyreectedbyE,wepickthe onethathappenstoheadforthemirror'scenter,C.Theequal-angle propertyofspecularreectionplusalittlestraightforwardgeometry easilyleadsustotheconclusionthattrianglesABCandCDEare thesameshape,withABCbeingsimplyascaled-upversionofCDE. ThemagnicationofdepthequalstheratioBC/CD,andtheupdownmagnicationisAB/DE.Arepetitionofthesameproofshows Section2.2CurvedMirrors 33

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thatthemagnicationinthethirddimensionoutofthepageis alsothesame.Thismeansthattheimage-headissimplyalarger versionoftherealone,withoutanydistortion.Thescalingfactor iscalledthemagnication, M .Theimageinthegureismagnied byafactor M =1.9. Notethatwedidnotexplicitlyspecifywhetherthemirrorwas asphere,aparaboloid,orsomeothershape.However,weassumed thatafocusedimagewouldbeformed,whichwouldnotnecessarily betrue,forinstance,foramirrorthatwasasymmetricorverydeeply curved. 2.3ARealImage Ifwestartbyplacinganobjectveryclosetothemirror,f/1,and thenmoveitfartherandfartheraway,theimageatrstbehaves aswewouldexpectfromoureverydayexperiencewithatmirrors, recedingdeeperanddeeperbehindthemirror.Atacertainpoint, however,adramaticchangeoccurs.Whentheobjectismorethan acertaindistancefromthemirror,f/2,theimageappearsupsidedownandin front ofthemirror. f / 1.Avirtualimage.2.Arealimage.Asyou'llverifyinhomework problem6,theimageisupsidedown Here'swhat'shappened.Themirrorbendslightraysinward,but whentheobjectisveryclosetoit,asinf/1,therayscomingfroma 34 Chapter2ImagesbyReection

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g / ANewtoniantelescope beingusedwithacamera. givenpointontheobjectaretoostronglydivergingspreadingfor themirrortobringthembacktogether.Onreection,theraysare stilldiverging,justnotasstronglydiverging.Butwhentheobject issucientlyfaraway,f/2,themirrorisonlyinterceptingtherays thatcameoutinanarrowcone,anditisabletobendtheseenough sothattheywillreconverge. Notethattheraysshowninthegure,whichbothoriginatedat thesamepointontheobject,reunitewhentheycross.Thepoint wheretheycrossistheimageofthepointontheoriginalobject. Thistypeofimageiscalleda realimage ,incontradistinctiontothe virtualimageswe'vestudiedbefore.Theuseofthewordreal"is perhapsunfortunate.Itsoundsasthoughwearesayingtheimage wasanactualmaterialobject,whichofcourseitisnot. Thedistinctionbetweenarealimageandavirtualimageisan importantone,becausearealimagecanprojectedontoascreenor photographiclm.Ifapieceofpaperisinsertedinguref/2at thelocationoftheimage,theimagewillbevisibleonthepaper providedtheobjectisbrightandtheroomisdark.Youreyeuses alenstomakearealimageontheretina. self-checkB Sketchanothercopyofthefaceinguref/1,evenfartherfromthemirror, anddrawaraydiagram.Whathashappenedtothelocationofthe image? Answer,p.108 2.4ImagesofImages Ifyouarewearingglassesrightnow,thenthelightraysfromthe pagearebeingmanipulatedrstbyyourglassesandthenbythelens ofyoureye.Youmightthinkthatitwouldbeextremelydicult toanalyzethis,butinfactitisquiteeasy.Inanyseriesofoptical elementsmirrorsorlensesorboth,eachelementworksontherays furnishedbythepreviouselementinexactlythesamemannerasif theimageformedbythepreviouselementwasanactualobject. Figuregshowsanexampleinvolvingonlymirrors.TheNewtoniantelescope,inventedbyIsaacNewton,consistsofalargecurved mirror,plusasecond,atmirrorthatbringsthelightoutofthe tube.Inverylargetelescopes,theremaybeenoughroomtoput acameraorevenapersoninsidethetube,inwhichcasethesecondmirrorisnotneeded.Thetubeofthetelescopeisnotvital;it ismainlyastructuralelement,althoughitcanalsobehelpfulfor blockingoutstraylight.Thelenshasbeenremovedfromthefront ofthecamerabody,andisnotneededforthissetup.Notethatthe twosamplerayshavebeendrawnparallel,becauseanastronomical telescopeisusedforviewingobjectsthatareextremelyfaraway. Thesetwoparallel"linesactuallymeetatacertainpoint,saya crateronthemoon,sotheycan'tactuallybeperfectlyparallel,but Section2.4ImagesofImages 35

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h / ANewtoniantelescope beingusedforvisualratherthan photographicobserving.Inreal life,aneyepiecelensisnormally usedforadditionalmagnication, butthissimplersetupwillalso work. theyareparallelforallpracticalpurposessincewewouldhaveto followthemupwardforaquarterofamillionmilestogettothe pointwheretheyintersect. ThelargecurvedmirrorbyitselfwouldformanimageI,butthe smallatmirrorcreatesanimageoftheimage,I 0 .Therelationship betweenIandI 0 isexactlythesameasitwouldbeifIwasanactual objectratherthananimage:IandI 0 areatequaldistancesfrom theplaneofthemirror,andthelinebetweenthemisperpendicular totheplaneofthemirror. Onesurprisingwrinkleisthatwhereasaatmirrorusedbyitself formsavirtualimageofanobjectthatisreal,herethemirroris formingarealimageofvirtualimageI.Thisshowshowpointlessit wouldbetotrytomemorizelistsoffactsaboutwhatkindsofimages areformedbyvariousopticalelementsundervariouscircumstances. Youarebetterosimplydrawingaraydiagram. Althoughthemainpointherewastogiveanexampleofanimageofanimage,gurehshowsaninterestingcasewhereweneed tomakethedistinctionbetween magnication and angularmagnication .Ifyouarelookingatthemoonthroughthistelescope, thentheimagesIandI 0 aremuch smaller thantheactualmoon. Otherwise,forexample,imageIwouldnottinsidethetelescope! However,theseimagesareveryclosetoyoureyecomparedtothe actualmoon.Thesmallsizeoftheimagehasbeenmorethancompensatedforbytheshorterdistance.Theimportantthinghereis theamountof angle withinyoureldofviewthattheimagecovers, anditisthisanglethathasbeenincreased.Thefactorbywhichit isincreasediscalledthe angularmagnication M a i / Theangularsizeoftheower dependsonitsdistancefromthe eye. 36 Chapter2ImagesbyReection

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DiscussionQuestions A Locatetheimagesofyouthatwillbeformedifyoustandbetween twoparallelmirrors. B Locatetheimagesformedbytwoperpendicularmirrors,asinthe gure.Whathappensifthemirrorsarenotperfectlyperpendicular? Section2.4ImagesofImages 37

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C Locatetheimagesformedbytheperiscope. 38 Chapter2ImagesbyReection

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Summary SelectedVocabulary realimage....aplacewhereanobjectappearstobe,becausetheraysdiuselyreectedfromany givenpointontheobjecthavebeenbentso thattheycomebacktogetherandthenspread outagainfromthenewpoint virtualimage..likearealimage,buttheraysdon'tactually crossagain;theyonlyappeartohavecome fromthepointontheimage converging....describesanopticaldevicethatbringslight raysclosertotheopticalaxis diverging..... bendslightraysfartherfromtheopticalaxis magnication..thefactorbywhichanimage'slinearsizeis increasedordecreased angularmagnication....... thefactorbywhichanimage'sapparentangularsizeisincreasedordecreased concave......describesasurfacethatishollowedoutlikea cave convex...... describesasurfacethatbulgesoutward Notation M .........themagnicationofanimage M a ........theangularmagnicationofanimage Summary Alargeclassofopticaldevices,includinglensesandatand curvedmirrors,operatesbybendinglightraystoformanimage.A realimageisoneforwhichtheraysactuallycrossateachpointof theimage.Avirtualimage,suchastheoneformedbehindaat mirror,isoneforwhichtheraysonlyappeartohavecrossedata pointontheimage.Arealimagecanbeprojectedontoascreen;a virtualonecannot. Mirrorsandlenseswillgenerallymakeanimagethatiseither smallerthanorlargerthantheoriginalobject.Thescalingfactor iscalledthemagnication.Inmanysituations,theangularmagnicationismoreimportantthantheactualmagnication. Summary 39

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Amaniswalkingat1.0m/sdirectlytowardsaatmirror. Atwhatspeedishisseparationfromhisimagedecreasing? p 2 Ifamirroronawallisonlybigenoughforyoutoseeyourselffromyourheaddowntoyourwaist,canyouseeyourentire bodybybackingup?Testthisexperimentallyandcomeupwithan explanationforyourobservations,includingaraydiagram. Notethatwhenyoudotheexperiment,it'seasytoconfuseyourself ifthemirrorisevenatinybitoofvertical.Onewaytocheck yourselfistoarticiallylowerthetopofthemirrorbyputtinga pieceoftapeorapost-itnotewhereitblocksyourviewofthetop ofyourhead.Youcanthencheckwhetheryouareabletoseemore ofyourselfbothabove and belowbybackingup. 3 Inthischapterwe'veonlydoneexamplesofmirrorswith hollowed-outshapescalledconcavemirrors.Nowdrawaraydiagramforacurvedmirrorthathasabulgingoutwardshapecalleda convexmirror.aHowdoestheimage'sdistancefromthemirror comparewiththeactualobject'sdistancefromthemirror?From thiscomparison,determinewhetherthemagnicationisgreater thanorlessthanone.bIstheimagereal,orvirtual?Could thismirrorevermaketheothertypeofimage? 4 Asdiscussedinquestion3,therearetwotypesofcurved mirrors,concaveandconvex.Makealistofallthepossiblecombinationsoftypesofimagesvirtualorrealwithtypesofmirrors concaveandconvex.Notallofthefourcombinationsarephysicallypossible.Nowforeachone,useraydiagramstodetermine whetherincreasingthedistanceoftheobjectfromthemirrorleads toanincreaseoradecreaseinthedistanceoftheimagefromthe mirror. DrawBIGraydiagrams!Eachdiagramshoulduseupabouthalfa pageofpaper. Sometips:Todrawaraydiagram,youneedtworays.Foroneof these,picktheraythatcomesstraightalongthemirror'saxis,since itsreectioniseasytodraw.Afteryoudrawthetworaysandlocate theimagefortheoriginalobjectposition,pickanewobjectposition thatresultsinthesametypeofimage,andstartanewraydiagram, inadierentcolorofpen,rightontopoftherstone.Forthetwo newrays,picktheonesthatjusthappentohitthemirroratthe sametwoplaces;thismakesitmucheasiertogettheresultright withoutdependingonextremeaccuracyinyourabilitytodrawthe 40 Chapter2ImagesbyReection

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Problem7. reectedrays. 5 Iftheuserofanastronomicaltelescopemovesherheadcloser toorfartherawayfromtheimagesheislookingat,doesthemagnicationchange?Doestheangularmagnicationchange?Explain. Forsimplicity,assumethatnoeyepieceisbeingused. 6 Ingurefinonpage34,onlytheimageofmyforeheadwas locatedbydrawingrays.Eitherphotocopythegureordownload thebookandprintouttherelevantpage.Onthiscopyofthegure, makeanewsetofrayscomingfrommychin,andlocateitsimage. Tomakeiteasiertojudgetheanglesaccurately,drawraysfromthe chinthathappentohitthemirroratthesamepointswherethetwo raysfromtheforeheadwereshownhittingit.Bycomparingthe locationsofthechin'simageandtheforehead'simage,verifythat theimageisactuallyupside-down,asshownintheoriginalgure. 7 Thegureshowsfourpointswhererayscross.Ofthese,which areimagepoints?Explain. 8 Here'sagamemykidsliketoplay.Isitnexttoasunny window,andthesunreectsfromtheglassonmywatch,makinga diskoflightonthewalloroor,whichtheypretendtochaseasI moveitaround.Isthespotadiskbecausethat'stheshapeofthe sun,orbecauseit'stheshapeofmywatch?Inotherwords,would asquarewatchmakeasquarespot,ordowejusthaveacircular imageofthecircularsun,whichwillbecircularnomatterwhat? Problems 41

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42 Chapter2ImagesbyReection

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BreakfastTable,byWillemClasz.deHeda,17thcentury.Thepaintingshowsavarietyofimages,someof themdistorted,resultingbothfromreectionandfromrefractionch.4. Chapter3 Images,Quantitatively Itsoundsabitoddwhenascientistreferstoatheoryasbeautiful,"buttothoseintheknowitmakesperfectsense.Onemark ofabeautifultheoryisthatitsurprisesusbybeingsimple.The mathematicaltheoryoflensesandcurvedmirrorsgivesusjustsuch asurprise.Weexpectthesubjecttobecomplexbecausethereare somanycases:aconvergingmirrorformingarealimage,adiverginglensthatmakesavirtualimage,andsoonforatotalofsix possibilities.Ifwewanttopredictthelocationoftheimagesinall thesesituations,wemightexpecttoneedsixdierentequations, andsixmoreforpredictingmagnications.Instead,itturnsout thatwecanusejustoneequationforthelocationoftheimageand oneequationforitsmagnication,andthesetwoequationswork inallthedierentcaseswithnochangesexceptforplusandminus signs.ThisisthekindofthingthephysicistEugeneWignerreferred toastheunreasonableeectivenessofmathematics."Sometimes 43

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a / Therelationshipbetween theobject'spositionandthe image'scanbeexpressedin termsoftheangles o and i wecanndadeeperreasonforthiskindofunexpectedsimplicity, butsometimesitalmostseemsasifGodwentoutofHerwayto makethesecretsofuniversesusceptibletoattackbythehuman thought-toolcalledmath. 3.1ARealImageFormedbyaConverging Mirror Locationoftheimage Wewillnowderivetheequationforthelocationofarealimage formedbyaconvergingmirror.Weassumeforsimplicitythatthe mirrorisspherical,butactuallythisisn'tarestrictiveassumption, becauseanyshallow,symmetriccurvecanbeapproximatedbya sphere.Theshapeofthemirrorcanbespeciedbygivingthe locationofitscenter,C.Adeeplycurvedmirrorisaspherewitha smallradius,soCisclosetoit,whileaweaklycurvedmirrorhas Cfartheraway.GiventhepointOwheretheobjectis,wewishto ndthepointIwheretheimagewillbeformed. Tolocateanimage,weneedtotrackaminimumoftworays comingfromthesamepoint.Sincewehaveprovedintheprevious chapterthatthistypeofimageisnotdistorted,wecanuseanon-axis point,O,ontheobject,asingurea/1.Theresultswederivewill alsoholdforo-axispoints,sinceotherwisetheimagewouldhave tobedistorted,whichweknowisnottrue.Weletoneoftheraysbe theonethatisemittedalongtheaxis;thisrayisespeciallyeasyto trace,becauseitbouncesstraightbackalongtheaxisagain.Asour secondray,wechooseonethatstrikesthemirroratadistanceof1 fromtheaxis.Onewhat?"askstheastutereader.Theansweris thatitdoesn'treallymatter.Whenamirrorhasshallowcurvature, allthereectedrayshitthesamepoint,so1couldbeexpressed inanyunitsyoulike.Itcould,forinstance,be1cm,unlessyour mirrorissmallerthan1cm! Theonlywaytondoutanythingmathematicalabouttherays istousethesolemathematicalfactwepossessconcerningspecular reection:theincidentandreectedraysformequalangleswith respecttothenormal,whichisshownasadashedline.Therefore thetwoanglesshowningurea/2arethesame,andskippingsome straightforwardgeometry,thisleadstothevisuallyreasonableresult thatthetwoanglesingurea/3arerelatedasfollows: i + o =constant Notethat i and o ,whicharemeasuredfromtheimageandthe object,notfromtheeyeliketheangleswereferredtoindiscussing angularmagnicationonpage36.Forexample,moveOfarther fromthemirror.Thetopangleingurea/2isincreased,sothe bottomanglemustincreasebythesameamount,causingtheimage 44 Chapter3Images,Quantitatively

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b / Thegeometricalinterpretationofthefocalangle. c / Example1,analternative testforndingthefocalangle. Themirroristhesameasin gureb. point,I,tomoveclosertothemirror.Intermsoftheanglesshownin gurea/3,themoredistantobjecthasresultedinasmallerangle o whilethecloserimagecorrespondstoalarger i ;Oneangleincreases bythesameamountthattheotherdecreases,sotheirsumremains constant.Thesechangesaresummarizedingurea/4. Thesum i + o isaconstant.Whatdoesthisconstantrepresent?Geometrically,weinterpretitasdoubletheanglemadeby thedashedradiusline.Optically,itisameasureofthestrengthof themirror,i.e.,howstronglythemirrorfocuseslight,andsowecall itthefocalangle, f i + o = f Suppose,forexample,thatwewishtouseaquickanddirtyoptical testtodeterminehowstrongaparticularmirroris.Wecanlay itontheoorasshowningurec,anduseittomakeanimage ofalampmountedontheceilingoverhead,whichweassumeis veryfarawaycomparedtotheradiusofcurvatureofthemirror, sothatthemirrorinterceptsonlyaverynarrowconeofraysfrom thelamp.Thisconeissonarrowthatitsraysarenearlyparallel, and o isnearlyzero.Therealimagecanbeobservedonapieceof paper.Bymovingthepapernearerandfarther,wecanbringthe imageintofocus,atwhichpointweknowthepaperislocatedat theimagepoint.Since o 0,wehave i f ,andwecanthen determinethismirror'sfocalangleeitherbymeasuring i directly withaprotractor,orindirectlyviatrigonometry.Astrongmirror willbringtheraystogethertoformanimageclosetothemirror, andtheserayswillformablunt-angledconewithalarge i and f Analternativeopticaltestexample1 Figurecshowsanalternativeopticaltest.Ratherthanplacing theobjectatinnityasingureb,weadjustitsothattheimage isrightontopoftheobject.PointsOandIcoincide,andtherays arereectedrightbackontopofthemselves.Ifwemeasurethe angle showningurec,howcanwendthefocalangle? Theobjectandimageanglesarethesame;theanglelabeled inthegureequalsbothofthem.Wethereforehave i + o = = f .Comparingguresbandc,itisindeedplausiblethatthe anglesarerelatedbyafactoroftwo. Atthispoint,wecouldconsiderourworktobedone.Typically, weknowthestrengthofthemirror,andwewanttondtheimage locationforagivenobjectlocation.Giventhemirror'sfocalangle andtheobjectlocation,wecandetermine o bytrigonometry,subtracttond i = f )]TJ/F20 10.9091 Tf 11.073 0 Td [( o ,andthendomoretrigtondtheimage location. Thereis,however,ashortcutthatcansaveusfromdoingso muchwork.Figurea/3showstworighttriangleswhoselegsof length1coincideandwhoseacuteanglesare o and i .Thesecan berelatedbytrigonometrytotheobjectandimagedistancesshown Section3.1ARealImageFormedbyaConvergingMirror 45

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d / Theobjectandimagedistances e / Mirror1isweakerthan mirror2.Ithasashallower curvature,alongerfocallength, andasmallerfocalangle.It reectsraysatanglesnotmuch differentthanthosethatwouldbe producedwithaatmirror. ingured: tan o =1 =d o tan i =1 =d i Eversincechapter2,we'vebeenassumingsmallangles.Forsmall angles,wecanusethesmall-angleapproximationtan x x for x inradians,givingsimply o =1 =d o i =1 =d i Welikewisedeneadistancecalledthefocallength, f accordingto f =1 =f .Ingureb, f isthedistancefromthemirrortotheplace wheretherayscross.Wecannowreexpresstheequationrelating theobjectandimagepositionsas 1 f = 1 d i + 1 d o Figureesummarizestheinterpretationofthefocallengthandfocal angle. 1 Whichformisbetter, f = i + o or1 =f =1 =d i +1 =d o ?The angularformhasinitsfavoritssimplicityanditsstraightforward visualinterpretation,buttherearetworeasonswhywemightprefer thesecondversion.First,thenumericalvaluesoftheanglesdepend onwhatwemeanbyoneunit"forthedistanceshownas1in gurea/1.Second,itisusuallyeasiertomeasuredistancesrather thanangles,sothedistanceformismoreconvenientfornumber crunching.Neitherformissuperioroverall,andwewilloftenneed tousebothtosolveanygivenproblem. 2 Asearchlightexample2 Supposeweneedtocreateaparallelbeamoflight,asinasearchlight.Whereshouldweplacethelightbulb?Aparallelbeamhas zeroanglebetweenitsrays,so i =0.Toplacethelightbulb correctly,however,weneedtoknowadistance,notanangle: thedistance d o betweenthebulbandthemirror.Theproblem involvesamixtureofdistancesandangles,soweneedtoget everythingintermsofoneortheotherinordertosolveit.Since thegoalistondadistance,let'sgureouttheimagedistance 1 Thereisastandardpieceofterminologywhichisthatthefocalpoint"is thepointlyingontheopticalaxisatadistancefromthemirrorequaltothefocal length.Thistermisn'tparticularlyhelpful,becauseitnamesalocationwhere nothingnormallyhappens.Inparticular,itis not normallytheplacewherethe rayscometoafocus!|thatwouldbethe image point.Inotherwords,we don'tnormallyhave d i = f ,unlessperhaps d o = 1 .Arecentonlinediscussion amongsomephysicsteachershttps://carnot.physics.bualo.edu/archives,Feb. 2006showedthatmanydislikedtheterminology,feltitwasmisleading,ordidn't knowitandwouldhavemisinterpreteditiftheyhadcomeacrossit.Thatis,it appearstobewhatgrammarianscallaskunkedterm"|awordthatbothers halfthepopulationwhenit'susedincorrectly,andtheotherhalfwhenit'sused correctly. 2 IwouldliketothankFouadAjamiforpointingoutthepedagogicaladvantagesofusingbothequationssidebyside. 46 Chapter3Images,Quantitatively

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correspondingtothegivenangle i =0.Thesearerelatedby d i =1 = i ,sowehave d i = 1 .Yes,dividingbyzerogivesinnity.Don'tbeafraidofinnity.Innityisausefulproblem-solving device.Solvingthedistanceequationfor d o ,wehave d o = = f )]TJ/F39 10.9091 Tf 10.909 0 Td [(1 = d i )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 = = f )]TJ/F39 10.9091 Tf 10.909 0 Td [(0 )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 = f Thebulbhastobeplacedatadistancefromthemirrorequalto itsfocalpoint. Dioptersexample3 Anequationlike d i =1 = i reallydoesn'tmakesenseintermsof units.Anglesareunitless,sinceradiansaren'treallyunits,so theright-handsideisunitless.Wecan'thavealeft-handside withunitsofdistanceiftheright-handsideofthesameequation isunitless.Thisisanartifactofmycavalierstatementthatthe conicalbundlesofraysspreadouttoadistanceof1fromtheaxis wheretheystrikethemirror,withoutspecifyingtheunitsusedto measurethis1.Inreallife,optometristsdenethethingwe're calling i =1 = d i asthedioptricstrengthofalensormirror, andmeasureitinunitsofinversemetersm )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 ,alsoknownas dioptersD=1m )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 Magnication Wehavealreadydiscussedinthepreviouschapterhowtond themagnicationofavirtualimagemadebyacurvedmirror.The resultisthesameforarealimage,andweomittheproof,which isverysimilar.Inournewnotation,theresultis M = d i =d o .A numericalexampleisgiveninsection3.2. 3.2OtherCasesWithCurvedMirrors Theequation d i =caneasilyproduceanegativeresult,butwehave beenthinkingof d i asadistance,anddistancescan'tbenegative. Asimilarproblemoccurswith i = f )]TJ/F20 10.9091 Tf 11.577 0 Td [( o for o > f .What's goingonhere? Theinterpretationoftheangularequationisstraightforward. Aswebringtheobjectcloserandclosertotheimage, o getsbigger andbigger,andeventuallywereachapointwhere o = f and i =0.Thislargeobjectanglerepresentsabundleofraysforming aconethatisverybroad,sobroadthatthemirrorcannolonger bendthembacksothattheyreconvergeontheaxis.Theimage angle i =0representsanoutgoingbundleofraysthatareparallel. Theoutgoingraysnevercross,sothisisnotarealimage,unlesswe wanttobecharitableandsaythattherayscrossatinnity.Ifwe goonbringingtheobjectevencloser,wegetavirtualimage. Section3.2OtherCasesWithCurvedMirrors 47

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f / Agraphoftheimagedistance d i asafunctionoftheobjectdistance d o Toanalyzethedistanceequation,let'slookatagraphof d i as afunctionof d o .Thebranchontheupperrightcorrespondstothe caseofarealimage.Strictlyspeaking,thisistheonlypartofthe graphthatwe'veprovencorrespondstoreality,sinceweneverdid anygeometryforothercases,suchasvirtualimages.Asdiscussedin theprevioussection,making d o biggercauses d i tobecomesmaller, andvice-versa. Letting d o belessthan f isequivalentto o > f :avirtualimage isproducedonthefarsideofthemirror.Thisistherstexample ofWigner'sunreasonableeectivenessofmathematics"thatwe haveencounteredinoptics.Eventhoughourproofdependedon theassumptionthattheimagewasreal,theequationwederived turnsouttobeapplicabletovirtualimages,providedthatweeither interpretthepositiveandnegativesignsinacertainway,orelse modifytheequationtohavedierentpositiveandnegativesigns. self-checkA Interpretthethreeplaceswhere,inphysicallyrealisticpartsofthegraph, thegraphapproachesoneofthedashedlines.[Thiswillcomemore naturallyifyouhavelearnedtheconceptoflimitsinamathclass.] Answer,p.108 Aatmirrorexample4 Wecanevenapplytheequationtoaatmirror.Asaspheregets 48 Chapter3Images,Quantitatively

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biggerandbigger,itssurfaceismoreandmoregentlycurved. TheplanetEarthissolarge,forexample,thatwecannoteven perceivethecurvatureofitssurface.Torepresentaatmirror,we letthemirror'sradiusofcurvature,anditsfocallength,become innite.Dividingbyinnitygiveszero,sowehave 1 = d o = )]TJ/F39 10.9091 Tf 8.485 0 Td [(1 = d i or d o = )]TJ/F106 10.9091 Tf 8.485 0 Td [(d i Ifweinterprettheminussignasindicatingavirtualimageonthe farsideofthemirrorfromtheobject,thismakessense. Itturnsoutthatforanyofthesixpossiblecombinationsof realorvirtualimagesformedbyconvergingordiverginglensesor mirrors,wecanapplyequationsoftheform f = i + o and 1 f = 1 d i + 1 d o withonlyamodicationofplusorminussigns.Therearetwopossibleapproacheshere.Theapproachwehavebeenusingsofaris themorepopularapproachinAmericantextbooks:leavetheequationthesame,butattachinterpretationstotheresultingnegative orpositivevaluesofthevariables.Thetroublewiththisapproach isthatoneisthenforcedtomemorizetablesofsignconventions, e.g.thatthevalueof d i shouldbenegativewhentheimageisa virtualimageformedbyaconvergingmirror.Positiveandnegative signsalsohavetobememorizedforfocallengths.Ugh!It'shighly unlikelythatanystudenthaseverretainedtheselengthytablesin hisorhermindformorethanveminutesafterhandinginthenal examinaphysicscourse.Ofcourseonecanalwayslooksuchthings upwhentheyareneeded,buttheeectistoturnthewholething intoanexerciseinblindlypluggingnumbersintoformulas. Asyouhavegatheredbynow,thereisanothermethodwhichI thinkisbetter,andwhichI'llusethroughouttherestofthisbook. Inthismethod,alldistancesandanglesare positivebydenition andweputinpositiveandnegativesignsinthe equations depending onthesituation.IthoughtIwasthersttoinventthismethod,but I'vebeentoldthatthisisknownastheEuropeansignconvention, andthatit'sfairlycommoninEurope.Ratherthanmemorizing thesesigns,westartwiththegenericequations f = i o 1 f = 1 d i 1 d o Section3.2OtherCasesWithCurvedMirrors 49

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g / Example5. andthendeterminethesignsbyatwo-stepmethodthatdependson raydiagrams.Therearereallyonlytwosignstodetermine,notfour; thesignsinthetwoequationsmatchupinthewayyou'dexpect. Themethodisasfollows: 1.Useraydiagramstodecidewhether o and i varyinthesame wayorinoppositeways.Inotherwords,decidewhethermaking o greaterresultsinagreatervalueof i orasmallerone.Basedon this,decidewhetherthetwosignsintheangleequationarethesame oropposite.Ifthesignsareopposite,goontostep2todetermine whichispositiveandwhichisnegative. 2.Ifthesignsareopposite,weneedtodecidewhichisthe positiveoneandwhichisthenegative.Sincethefocalangleisnever negative,thesmalleranglemustbetheonewithaminussign. Instep1,manystudentshavetroubledrawingtheraydiagram correctly.Forsimplicity,youshouldalwaysdoyourdiagramfora pointontheobjectthatisontheaxisofthemirror,andletone ofyourraysbetheonethatisemittedalongtheaxisandreect straightbackonitself,asintheguresinsection3.1.Asshownin gurea/4insection3.1,therearefouranglesinvolved:twoatthe mirror,oneattheobject o ,andoneattheimage i .Makesure todrawinthenormaltothemirrorsothatyoucanseethetwo anglesatthemirror.Thesetwoanglesareequal,soasyouchange theobjectposition,theyfanoutorfanin,likeopeningorclosing abook.Onceyou'vedrawnthiseect,youshouldeasilybeableto tellwhether o and i changeinthesamewayorinoppositeways. Althoughfocallengthsarealwayspositiveinthemethodused inthisbook,youshouldbeawarethatdivergingmirrorsandlenses areassignednegativefocallengthsintheothermethod,soifyou seealenslabeled f = )]TJ/F15 10.9091 Tf 8.484 0 Td [(30cm,you'llknowwhatitmeans. Ananti-shopliftingmirrorexample5 Conveniencestoresofteninstalladivergingmirrorsothatthe clerkhasaviewofthewholestoreandcancatchshoplifters.Use araydiagramtoshowthattheimageisreduced,bringingmore intotheclerk'seldofview.Ifthefocallengthofthemirroris3.0 m,andthemirroris7.0mfromthefarthestwall,howdeepisthe imageofthestore? Asshowninraydiagramg/1, d i islessthan d o .Themagnication, M = d i = d o ,willbelessthanone,i.e.,theimageisactually reducedratherthanmagnied. Applythemethodoutlinedabovefordeterminingtheplusand minussigns.Step1:Theobjectisthepointontheopposite wall.Asanexperiment,g/2,movetheobjectcloser.Ididthese drawingsusingillustrationsoftware,butifyouweredoingthem byhand,you'dwanttomakethescalemuchlargerforgreater accuracy.Also,althoughIsplitguregintotwoseparatedrawings 50 Chapter3Images,Quantitatively

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inordertomakethemeasiertounderstand,you'relesslikelyto makeamistakeifyoudothemontopofeachother. Thetwoanglesatthemirrorfanoutfromthenormal.Increasing o hasclearlymade i largeraswell.Allfouranglesgotbigger.Theremustbeacancellationoftheeffectsofchangingthe twotermsontherightinthesameway,andtheonlywaytoget suchacancellationisifthetwotermsintheangleequationhave oppositesigns: f =+ i )]TJ/F73 10.9091 Tf 10.91 0 Td [( o or f = )]TJ/F73 10.9091 Tf 8.485 0 Td [( i + o Step2:Nowwhichisthepositivetermandwhichisnegative? Sincetheimageangleisbiggerthantheobjectangle,theangle equationmustbe f = i )]TJ/F73 10.9091 Tf 10.909 0 Td [( o inordertogiveapositiveresultforthefocalangle.Thesignsof thedistanceequationbehavethesameway: 1 f = 1 d i )]TJ/F39 10.9091 Tf 14.685 7.38 Td [(1 d o Solvingfor d i ,wend d i = 1 f + 1 d o )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 =2.1m. Theimageofthestoreisreducedbyafactorof2.1 = 7.0=0.3, i.e.,itissmallerby70%. Ashortcutforrealimagesexample6 Inthecaseofarealimage,thereisashortcutforstep1,the determinationofthesigns.Inarealimage,therayscrossat boththeobjectandtheimage.Wecanthereforetime-reversethe raydiagram,sothatalltheraysarecomingfromtheimageand reconvergingattheobject.Objectandimageswaproles.Due tothistime-reversalsymmetry,theobjectandimagecannotbe treateddifferentlyinanyoftheequations,andtheymusttherefore havethesamesigns.Theyarebothpositive,sincetheymustadd uptoapositiveresult. 3.3 ? Aberrations Animperfectionordistortioninanimageiscalledanaberration. Anaberrationcanbeproducedbyaawinalensormirror,but evenwithaperfectopticalsurfacesomedegreeofaberrationisunavoidable.Toseewhy,considerthemathematicalapproximation Section3.3 ? Aberrations 51

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h / Adivergingmirrorintheshape ofasphere.Theimageisreduced M < 1.Thisissimilar toexample5,butheretheimage isdistortedbecausethemirror's curveisnotshallow. we'vebeenmaking,whichisthatthedepthofthemirror'scurve issmallcomparedto d o and d i .Sinceonlyaatmirrorcansatisfythisshallow-mirrorconditionperfectly,anycurvedmirrorwill deviatesomewhatfromthemathematicalbehaviorwederivedby assumingthatcondition.Therearetwomaintypesofaberrationin curvedmirrors,andthesealsooccurwithlenses. Anobjectontheaxisofthelensormirrormaybeimaged correctly,buto-axisobjectsmaybeoutoffocusordistorted.In acamera,thistypeofaberrationwouldshowupasafuzzinessor warpingnearthesidesofthepicturewhenthecenterwasperfectly focused.Anexampleofthisisshowningurei,andinthatparticularexample,theaberrationisnotasignthattheequipmentwas oflowqualityorwasn'trightforthejobbutratheraninevitable resultoftryingtoattenapanoramicview;inthelimitofa360degreepanorama,theproblemwouldbesimilartotheproblemof representingtheEarth'ssurfaceonaatmap,whichcan'tbeaccomplishedwithoutdistortion. Theimagemaybesharpwhentheobjectisatcertaindistancesandblurrywhenitisatotherdistances.Theblurriness occursbecausetheraysdonotallcrossatexactlythesamepoint. Ifweknowinadvancethedistanceoftheobjectswithwhichthe mirrororlenswillbeused,thenwecanoptimizetheshapeofthe opticalsurfacetomakein-focusimagesinthatsituation.Forinstance,asphericalmirrorwillproduceaperfectimageofanobject thatisatthecenterofthesphere,becauseeachrayisreecteddirectlyontotheradiusalongwhichitwasemitted.Forobjectsat greaterdistances,however,thefocuswillbesomewhatblurry.In astronomytheobjectsbeingusedarealwaysatinnity,soaspher52 Chapter3Images,Quantitatively

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i / Thisphotowastakenusinga sh-eyelens,whichgivesanextremelylargeeldofview. icalmirrorisapoorchoiceforatelescope.Adierentshapea parabolaisbetterspecializedforastronomy. j / Sphericalmirrorsarethe cheapesttomake,butparabolic mirrorsarebetterformaking imagesofobjectsatinnity. Aspherehasequalcurvature everywhere,butaparabolahas tightercurvatureatitscenterand gentlercurvatureatthesides. Onewayofdecreasingaberrationistouseasmall-diametermirrororlens,orblockmostofthelightwithanopaquescreenwitha holeinit,sothatonlylightthatcomesinclosetotheaxiscanget through.Eitherway,weareusingasmallerportionofthelensor mirrorwhosecurvaturewillbemoreshallow,therebymakingthe shallow-mirrororthin-lensapproximationmoreaccurate.Your eyedoesthisbynarrowingdownthepupiltoasmallerhole.In acamera,thereiseitheranautomaticormanualadjustment,and narrowingtheopeningiscalledstoppingdown."Thedisadvantage ofstoppingdownisthatlightiswasted,sotheimagewillbedimmer Section3.3 ? Aberrations 53

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oralongerexposuremustbeused. k / Eventhoughthesphericalmirrorsolidlineisnotwelladapted forviewinganobjectatinnity, wecanimproveitsperformance greatlybystoppingitdown.Now theonlypartofthemirrorbeingusedisthecentralportion, whereitsshapeisvirtuallyindistinguishablefromaparabola dashedline. WhatIwouldsuggestyoutakeawayfromthisdiscussionforthe sakeofyourgeneralscienticeducationissimplyanunderstanding ofwhatanaberrationis,whyitoccurs,andhowitcanbereduced, notdetailedfactsaboutspecictypesofaberrations. l / TheHubbleSpaceTelescope wasplacedintoorbitwithfaulty opticsin1990.Itsmainmirrorwassupposedtohavebeen nearlyparabolic,sinceitisanastronomicaltelescope,meantfor producingimagesofobjectsatinnity.However,contractorPerkinElmerhaddeliveredafaulty mirror,whichproducedaberrations.Thelargephotoshowsastronautsputtingcorrectingmirrors inplacein1993.Thetwosmall photosshowimagesproducedby thetelescopebeforeandafterthe x. 54 Chapter3Images,Quantitatively

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Summary SelectedVocabulary focallength...apropertyofalensormirror,equaltothe distancefromthelensormirrortotheimage itformsofanobjectthatisinnitelyfaraway Notation f ..........thefocallength d o .........thedistanceoftheobjectfromthemirror d i .........thedistanceoftheimagefromthemirror f .........thefocalangle,denedas1 =f o .........theobjectangle,denedas1 =d o i .........theimageangle,denedas1 =d i OtherTerminologyandNotation f> 0.......describesaconverginglensormirror;inthis book,allfocallengthsarepositive,sothereis nosuchimplication f< 0.......describesadiverginglensormirror;inthis book,allfocallengthsarepositive M< 0...... indicatesaninvertedimage;inthisbook,all magnicationsarepositive Summary Everylensormirrorhasapropertycalledthefocallength,which isdenedasthedistancefromthelensormirrortotheimageit formsofanobjectthatisinnitelyfaraway.Astrongerlensor mirrorhasashorterfocallength. Therelationshipbetweenthelocationsofanobjectanditsimage formedbyalensormirrorcanalwaysbeexpressedbyequationsof theform f = i o 1 f = 1 d i 1 d o Thechoiceofplusandminussignsdependsonwhetherwearedealingwithalensoramirror,whetherthelensormirrorisconverging ordiverging,andwhethertheimageisrealorvirtual.Amethod fordeterminingtheplusandminussignsisasfollows: 1.Useraydiagramstodecidewhether i and o varyinthesame wayorinoppositeways.Basedonthis,decidewhetherthe twosignsintheequationarethesameoropposite.Ifthesigns areopposite,goontostep2todeterminewhichispositive andwhichisnegative. 2.Ifthesignsareopposite,weneedtodecidewhichisthepositive oneandwhichisthenegative.Sincethefocalangleisnever negative,thesmalleranglemustbetheonewithaminussign. Summary 55

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Oncethecorrectformoftheequationhasbeendetermined,the magnicationcanbefoundviatheequation M = d i d o Thisequationexpressestheideathattheentireimage-worldis shrunkconsistentlyinallthreedimensions. 56 Chapter3Images,Quantitatively

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Problem5. Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Applytheequation M = d i =d o tothecaseofaatmirror. 2 Usethemethoddescribedinthetexttoderivetheequation relatingobjectdistancetoimagedistanceforthecaseofavirtual imageproducedbyaconvergingmirror. Solution,p.110 3 aMakeupanumericalexampleofavirtualimageformedby aconvergingmirrorwithacertainfocallength,anddeterminethe magnication.Youwillneedtheresultofproblem2.Makesure tochoosevaluesof d o and f thatwouldactuallyproduceavirtual image,notarealone.Nowchangethelocationoftheobject alittle bit andredeterminethemagnication,showingthatitchanges.At mylocaldepartmentstore,thecosmeticsdepartmentsellsmirrors advertisedasgivingamagnicationof5times.Howwouldyou interpretthis? bSupposeaNewtoniantelescopeisbeingusedforastronomical observing.Assumeforsimplicitythatnoeyepieceisused,andassumeavalueforthefocallengthofthemirrorthatwouldbereasonableforanamateurinstrumentthatistotinacloset.Isthe angularmagnicationdierentforobjectsatdierentdistances? Forexample,youcouldconsidertwoplanets,oneofwhichistwice asfarastheother. 4 aFindacasewherethemagnicationofacurvedmirror isinnite.Isthe angular magnicationinnitefromanyrealistic viewingposition?bExplainwhyanarbitrarilylargemagnication can'tbeachievedbyhavingasucientlysmallvalueof d o 5 Thegureshowsadeviceforconstructingarealisticoptical illusion.Twomirrorsofequalfocallengthareputagainsteach otherwiththeirsilveredsurfacesfacinginward.Asmallobject placedinthebottomofthecavitywillhaveitsimageprojectedin theairabove.Thewayitworksisthatthetopmirrorproducesa virtualimage,andthebottommirrorthencreatesarealimageof thevirtualimage.aShowthatiftheimageistobepositioned asshown,atthemouthofthecavity,thenthefocallengthofthe mirrorsisrelatedtothedimension h viatheequation 1 f = 1 h + 1 h + 1 h )]TJ/F18 7.9701 Tf 12.461 4.296 Td [(1 f )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 bRestatetheequationintermsofasinglevariable x = h=f ,and showthattherearetwosolutionsfor x .Whichsolutionisphysically consistentwiththeassumptionsofthecalculation? ? Problems 57

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Problem8. 6 Aconcavesurfacethatreectssoundwavescanactjustlike aconvergingmirror.Supposethat,standingnearsuchasurface, youareabletondapointwhereyoucanplaceyourheadsothat yourownwhispersarefocusedbackonyourhead,sothatthey soundloudtoyou.Givenyourdistancetothesurface,whatisthe surface'sfocallength? 7 Findthefocallengthofthemirrorinproblem5ofchapter1. 8 Rankthefocallengthsofthemirrors,fromshortesttolongest. 9 aAconvergingmirrorisbeingusedtocreateavirtualimage. Whatistherangeofpossiblemagnications?bDothesamefor theothertypesofimagesthatcanbeformedbycurvedmirrors bothconverginganddiverging. 10 aAconvergingmirrorwithafocallengthof20cmisused tocreateanimage,usinganobjectatadistanceof10cm.Isthe imagereal,orisitvirtual?bHowabout f =20cmand d o =30 cm?cWhatifitwasa diverging mirrorwith f =20cmand d o =10cm?dAdivergingmirrorwith f =20cmand d o =30 cm? Solution,p.110 11 Adivergingmirroroffocallength f isxed,andfacesdown. Anobjectisdroppedfromthesurfaceofthemirror,andfallsaway fromitwithacceleration g .Thegoaloftheproblemistondthe maximumvelocityoftheimage. aDescribethemotionoftheimageverbally,andexplainwhywe shouldexpecttheretobeamaximumvelocity. bUseargumentsbasedonunitstodeterminetheformofthe solution,uptoanunknownunitlessmultiplicativeconstant. cCompletethesolutionbydeterminingtheunitlessconstant. R 58 Chapter3Images,Quantitatively

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Threestagesintheevolutionoftheeye.Theatwormhastwoeyepits.Thenautilus'seyesarepinhole cameras.Thehumaneyeincorporatesalens. Chapter4 Refraction Economistsnormallyconsiderfreemarketstobethenaturalwayof judgingthemonetaryvalueofsomething,butsocialscientistsalso usequestionnairestogaugetherelativevalueofprivileges,disadvantages,orpossessionsthatcannotbeboughtorsold.Theyask peopleto imagine thattheycouldtradeonethingforanotherand askwhichtheywouldchoose.Oneinterestingresultisthattheaveragelight-skinnedpersonintheU.S.wouldratherloseanarmthan suertheracisttreatmentroutinelyenduredbyAfrican-Americans. Evenmoreimpressiveisthevalueofsight.Manyprospectiveparentscanimaginewithouttoomuchfearhavingadeafchild,but wouldhaveafarmorediculttimecopingwithraisingablindone. Sogreatisthevalueattachedtosightthatsomehaveimbued itwithmysticalaspects.Moseshadvision,"GeorgeBushdidnot. Christianfundamentalistswhoperceiveaconictbetweenevolution andtheirreligionhaveclaimedthattheeyeissuchaperfectdevice thatitcouldneverhavearisenthroughaprocessashelter-skelteras evolution,orthatitcouldnothaveevolvedbecausehalfofaneye wouldbeuseless.Infact,thestructureofaneyeisfundamentally dictatedbyphysics,andithasarisenseparatelybyevolutionsome59

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a / Ahumaneye. b / Theanatomyoftheeye. c / Asimpliedopticaldiagramoftheeye.Lightraysare bentwhentheycrossfromthe airintotheeye.Alittleofthe incidentrays'energygoesinto thereectedraysratherthanthe onestransmittedintotheeye. wherebetweeneightand40times,dependingonwhichbiologistyou ask.Wehumanshaveaversionoftheeyethatcanbetracedback totheevolutionofalight-sensitiveeyespot"ontheheadofan ancientinvertebrate.Asunkenpitthendevelopedsothattheeye wouldonlyreceivelightfromonedirection,allowingtheorganism totellwherethelightwascomingfrom.Modernatwormshave thistypeofeye.Thetopofthepitthenbecamepartiallycovered, leavingahole,forevengreaterdirectionalityasinthenautilus. Atsomepointthecavitybecamelledwithjelly,andthisjellynallybecamealens,resultinginthegeneraltypeofeyethatwe sharewiththebonyshesandothervertebrates.Farfrombeing aperfectdevice,thevertebrateeyeismarredbyaseriousdesign awduetothelackofplanningorintelligentdesigninevolution: thenervecellsoftheretinaandthebloodvesselsthatservethem areallinfrontofthelight-sensitivecells,blockingpartofthelight. Squidsandothermolluscs,whoseeyesevolvedonaseparatebranch oftheevolutionarytree,haveamoresensiblearrangement,withthe light-sensitivecellsoutinfront. 4.1Refraction Refraction Thefundamentalphysicalphenomenonatworkintheeyeis thatwhenlightcrossesaboundarybetweentwomediasuchasair andtheeye'sjelly,partofitsenergyisreected,butpartpasses intothenewmedium.Intheraymodeloflight,wedescribethe originalrayassplittingintoareectedrayandatransmittedone theonethatgetsthroughtheboundary.Ofcoursethereected raygoesinadirectionthatisdierentfromthatoftheoriginalone, accordingtotherulesofreectionwehavealreadystudied.More surprisingly|andthisisthecrucialpointformakingyoureye focuslight|thetransmittedrayisbentsomewhataswell.This bendingphenomenoniscalled refraction .Theoriginoftheword isthesameasthatofthewordfracture,"i.e.,therayisbentor broken."Keepinmind,however,thatlightraysarenotphysical objectsthatcanreallybebroken."Refractionoccurswithall waves,notjustlightwaves. Theactualanatomyoftheeye,b,isquitecomplex,butinessence itisverymuchlikeeveryotheropticaldevicebasedonrefraction. Theraysarebentwhentheypassthroughthefrontsurfaceofthe eye,c.Raysthatenterfartherfromthecentralaxisarebentmore, withtheresultthatanimageisformedontheretina.Thereis onlyoneslightlynovelaspectofthesituation.Inmosthuman-built opticaldevices,suchasamovieprojector,thelightisbentasit passesintoalens,bentagainasitreemerges,andthenreachesa focusbeyondthelens.Intheeye,however,thescreen"isinside theeye,sotheraysareonlyrefractedonce,onenteringthejelly, 60 Chapter4Refraction

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d / Theincident,reected, andtransmittedrefractedrays alllieinaplanethatincludesthe normaldashedline. e / Theangles 1 and 2 are relatedtoeachother,andalso dependonthepropertiesofthe twomedia.Becauserefraction istime-reversalsymmetric,there isnoneedtolabeltherayswith arrowheads. f / Refractionhastime-reversal symmetry.Regardlessofwhether thelightisgoingintooroutofthe water,therelationshipbetween thetwoanglesisthesame,and therayisclosertothenormal whileinthewater. andneveremergeagain. Acommonmisconceptionisthatthelens"oftheeyeiswhat doesthefocusing.Allthetransparentpartsoftheeyearemade offairlysimilarstu,sothedramaticchangeinmediumiswhena raycrossesfromtheairintotheeyeattheoutsidesurfaceofthe cornea.Thisiswherenearlyalltherefractiontakesplace.Thelens mediumdiersonlyslightlyinitsopticalpropertiesfromtherest oftheeye,soverylittlerefractionoccursaslightentersandexits thelens.Thelens,whoseshapeisadjustedbymusclesattachedto it,isonlymeantforne-tuningthefocustoformimagesofnearor farobjects. Refractivepropertiesofmedia Whataretherulesgoverningrefraction?Therstthingtoobserveisthatjustaswithreection,thenew,bentpartoftheraylies inthesameplaneasthenormalperpendicularandtheincident ray,d. Ifyoutryshootingabeamoflightattheboundarybetween twosubstances,saywaterandair,you'llndthatregardlessofthe angleatwhichyousendinthebeam,thepartofthebeaminthe waterisalwaysclosertothenormalline,e.Itdoesn'tmatterifthe rayisenteringthewaterorleaving,sorefractionissymmetricwith respecttotime-reversal,f. If,insteadofwaterandair,youtryanothercombinationofsubstances,sayplasticandgasoline,againyou'llndthattheray's anglewithrespecttothenormalisconsistentlysmallerinoneand largerintheother.Also,wendthatifsubstanceAhasrayscloser tonormalthaninB,andBhasraysclosertonormalthaninC,then AhasraysclosertonormalthanC.Thismeansthatwecanrankorderallmaterialsaccordingtotheirrefractiveproperties.Isaac Newtondidso,includinginhislistmanyamusingsubstances,such asDanzigvitriol"andapseudo-topazius,beinganatural,pellucid,brittle,hairystone,ofayellowcolor."Severalgeneralrulescan beinferredfromsuchalist: Vacuumliesatoneendofthelist.Inrefractionacrossthe interfacebetweenvacuumandanyothermedium,theother mediumhasraysclosertothenormal. Amonggases,theraygetsclosertothenormalifyouincrease thedensityofthegasbypressurizingitmore. Therefractivepropertiesofliquidmixturesandsolutionsvary inasmoothandsystematicmannerastheproportionsofthe mixturearechanged. Densersubstancesusually,butnotalways,haverayscloserto thenormal. Section4.1Refraction 61

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g / Therelationshipbetween theanglesinrefraction. Thesecondandthirdrulesprovideuswithamethodformeasuringthedensityofanunknownsampleofgas,ortheconcentration ofasolution.Thelattertechniqueisverycommonlyused,andthe CRCHandbookofPhysicsandChemistry,forinstance,contains extensivetablesoftherefractivepropertiesofsugarsolutions,cat urine,andsoon. Snell'slaw ThenumericalrulegoverningrefractionwasdiscoveredbySnell, whomusthavecollectedexperimentaldatasomethinglikewhatis shownonthisgraphandthenattemptedbytrialanderrortond therightequation.Theequationhecameupwithwas sin 1 sin 2 =constant. Thevalueoftheconstantwoulddependonthecombinationofmedia used.Forinstance,anyoneofthedatapointsinthegraphwould havesucedtoshowthattheconstantwas1.3foranair-water interfacetakingairtobesubstance1andwatertobesubstance 2. SnellfurtherfoundthatifmediaAandBgaveaconstant K AB andmediaBandCgaveaconstant K BC ,thenrefractionataninterfacebetweenAandCwouldbedescribedbyaconstantequaltothe product, K AC = K AB K BC .Thisisexactlywhatonewouldexpect iftheconstantdependedontheratioofsomenumbercharacterizingonemediumtothenumbercharacteristicofthesecondmedium. Thisnumberiscalledthe indexofrefraction ofthemedium,written as n inequations.Sincemeasuringtheangleswouldonlyallowhim todeterminethe ratio oftheindicesofrefractionoftwomedia,Snell hadtopicksomemediumanddeneitashaving n =1.Hechose todenevacuumashaving n =1.Theindexofrefractionofair atnormalatmosphericpressureis1.0003,soformostpurposesitis agoodapproximationtoassumethatairhas n =1.Healsohad todecidewhichwaytodenetheratio,andhechosetodeneitso thatmediawiththeirraysclosertothenormalwouldhavelargerindicesofrefraction.Thishadtheadvantagethatdensermediawould typicallyhavehigherindicesofrefraction,andforthisreasonthe indexofrefractionisalsoreferredtoastheopticaldensity.Written intermsofindicesofrefraction,Snell'sequationbecomes sin 1 sin 2 = n 2 n 1 butrewritingitintheform 62 Chapter4Refraction

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h / Example1. n 1 sin 1 = n 2 sin 2 [relationshipbetweenanglesofraysattheinterfacebetweenmediawithindicesofrefraction n 1 and n 2 ;angles aredenedwithrespecttothenormal] makesuslesslikelytogetthe1'sand2'smixedup,sothistheway mostpeoplerememberSnell'slaw.Afewindicesofrefractionare giveninthebackofthebook. self-checkA Whatwouldthegraphlooklikefortwosubstanceswiththesame indexofrefraction? Basedonthegraph,whendoesrefractionatanair-waterinterface changethedirectionofaraymoststrongly? Answer,p.108 FindinganangleusingSnell'slawexample1 Asubmarineshinesitssearchlightuptowardthesurfaceofthe water.Whatistheangle showninthegure? ThetrickypartisthatSnell'slawreferstotheangleswithrespecttothenormal.Forgettingthisisaverycommonmistake. Thebeamisatanangleof30 withrespecttothenormalin thewater.Let'srefertotheairasmedium1andthewateras2. SolvingSnell'slawfor 1 ,wend 1 =sin )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 n 2 n 1 sin 2 Asmentionedabove,airhasanindexofrefractionverycloseto 1,andwater'sisabout1.3,sowend 1 =40 .Theangle is therefore50 Theindexofrefractionisrelatedtothespeedoflight. WhatneitherSnellnorNewtonknewwasthatthereisavery simpleinterpretationoftheindexofrefraction.Thismaycomeas arelieftothereaderwhoistakenabackbythecomplexreasoning involvingproportionalitiesthatledtoitsdenition.Laterexperimentsshowedthattheindexofrefractionofamediumwasinversely proportionaltothespeedoflightinthatmedium.Since c isdened asthespeedoflightinvacuum,and n =1isdenedastheindex ofrefractionofvacuum,wehave n = c v [ n =medium'sindexofrefraction, v =speedoflight inthatmedium, c =speedoflightinavacuum] Manytextbooksstartwiththisasthedenitionoftheindex ofrefraction,althoughthatapproachmakesthequantity'sname somewhatofamystery,andleavesstudentswonderingwhy c=v was usedratherthan v=c .Itshouldalsobenotedthatmeasuringangles ofrefractionisafarmorepracticalmethodfordetermining n than directmeasurementofthespeedoflightinthesubstanceofinterest. Section4.1Refraction 63

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i / Amechanicalmodelofrefraction. AmechanicalmodelofSnell'slaw Whyshouldrefractionberelatedtothespeedoflight?The mechanicalmodelshownintheguremayhelptomakethismore plausible.Supposemedium2isthick,stickymud,whichslowsdown thecar.Thecar'srightwheelhitsthemudrst,causingtheright sideofthecartoslowdown.Thiswillcausethecartoturntothe rightuntilismovesfarenoughforwardfortheleftwheeltocross intothemud.Afterthat,thetwosidesofthecarwillonceagainbe movingatthesamespeed,andthecarwillgostraight. Ofcourse,lightisn'tacar.Whyshouldabeamoflighthave anythingresemblingaleftwheel"andrightwheel?"Afterall, themechanicalmodelwouldpredictthatamotorcyclewouldgo straight,andamotorcycleseemslikeabetterapproximationtoa rayoflightthanacar.Thewholethingisjustamodel,nota descriptionofphysicalreality. j / AderivationofSnell'slaw. AderivationofSnell'slaw Howeverintuitivelyappealingthemechanicalmodelmaybe, lightisawave,andweshouldbeusingwavemodelstodescribe refraction.InfactSnell'slawcanbederivedquitesimplyfrom waveconcepts.Figurejshowstherefractionofawaterwave.The waterintheupperleftpartofthetankisshallower,sothespeed ofthewavesisslowerthere,andtheirwavelengthsisshorter.The reectedpartofthewaveisalsoveryfaintlyvisible. 64 Chapter4Refraction

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Intheclose-upviewontheright,thedashedlinesarenormals totheinterface.Thetwomarkedanglesontherightsideareboth equalto 1 ,andthetwoontheleftto 2 Trigonometrygives sin 1 = 1 =h and sin 2 = 2 =h Eliminating h bydividingtheequations,wend sin 1 sin 2 = 1 2 Thefrequenciesofthetwowavesmustbeequalorelsetheywould getoutofstep,soby v = f weknowthattheirwavelengthsare proportionaltotheirvelocities.Combining / v with v / 1 =n gives / 1 =n ,sowend sin 1 sin 2 = n 2 n 1 whichisoneformofSnell'slaw. Oceanwavesnearandfarfromshoreexample2 Oceanwavesareformedbywinds,typicallyontheopensea,and thewavefrontsareperpendiculartothedirectionofthewindthat formedthem.Atthebeach,however,youhaveundoubtedlyobservedthatwavestendcomeinwiththeirwavefrontsverynearly butnotexactlyparalleltotheshoreline.Thisisbecausethe speedofwaterwavesinshallowwaterdependsondepth:the shallowerthewater,theslowerthewave.Althoughthechange fromthefast-waveregiontotheslow-waveregionisgradualrather thanabrupt,thereisstillrefraction,andthewavemotionisnearly perpendiculartothenormalintheslowregion. Colorandrefraction Ingeneral,thespeedoflightinamediumdependsbothonthe mediumandonthewavelengthofthelight.Anotherwayofsayingit isthatamedium'sindexofrefractionvarieswithwavelength.This iswhyaprismcanbeusedtosplitupabeamofwhitelightintoa rainbow.Eachwavelengthoflightisrefractedthroughadierent angle. Howmuchlightisreected,andhowmuchistransmitted? Inbook3wedevelopedanequationforthepercentageofthe waveenergythatistransmittedandthepercentagereectedata boundarybetweenmedia.Thiswasonlydoneinthecaseofwaves inonedimension,however,andratherthandiscussthefullthreedimensionalgeneralizationitwillbemoreusefultogointosomequalitativeobservationsaboutwhathappens.First,reectionhappens Section4.1Refraction 65

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k / Totalinternalreectionin aber-opticcable. l / Asimplieddrawingofa surgicalendoscope.Therst lensformsarealimageat oneendofabundleofoptical bers.Thelightistransmitted throughthebundle,andisnally magniedbytheeyepiece. m / Endoscopicimagesofa duodenalulcer. onlyattheinterfacebetweentwomedia,andtwomediawiththe sameindexofrefractionactasiftheywereasinglemedium.Thus, attheinterfacebetweenmediawiththesameindexofrefraction, thereisnoreection,andtheraykeepsgoingstraight.Continuing thislineofthought,itisnotsurprisingthatweobserveverylittlereectionataninterfacebetweenmediawithsimilarindicesof refraction. Thenextthingtonoteisthatitispossibletohavesituations wherenopossibleanglefortherefractedraycansatisfySnell'slaw. SolvingSnell'slawfor 2 ,wend 2 =sin )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 n 1 n 2 sin 1 andif n 1 isgreaterthan n 2 ,thentherewillbelargevaluesof 1 forwhichthequantity n 1 =n 2 sin isgreaterthanone,meaning thatyourcalculatorwillashanerrormessageatyouwhenyou trytotaketheinversesine.Whatcanhappenphysicallyinsuch asituation?Theansweristhatallthelightisreected,sothere isnorefractedray.Thisphenomenonisknownas totalinternal reection ,andisusedintheber-opticcablesthatnowadayscarry almostalllong-distancetelephonecalls.Theelectricalsignalsfrom yourphonetraveltoaswitchingcenter,wheretheyareconverted fromelectricityintolight.Fromthere,thelightissentacrossthe countryinathintransparentber.Thelightisaimedstraightinto theendoftheber,andaslongasthebernevergoesthroughany turnsthataretoosharp,thelightwillalwaysencountertheedge oftheberatananglesucientlyobliquetogivetotalinternal reection.Iftheber-opticcableisthickenough,onecanseean imageatoneendofwhatevertheotherendispointedat. Alternatively,abundleofcablescanbeused,sinceasinglethick cableistoohardtobend.Thistechniqueforseeingaroundcorners isusefulformakingsurgerylesstraumatic.Insteadofcuttinga personwideopen,asurgeoncanmakeasmallkeyhole"incision andinsertabundleofber-opticcableknownasanendoscope intothebody. Sinceraysatsucientlylargeangleswithrespecttothenormal maybecompletelyreected,itisnotsurprisingthattherelative amountofreectionchangesdependingontheangleofincidence, andisgreatestforlargeanglesofincidence. 66 Chapter4Refraction

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DiscussionQuestions A Whatindexofrefractionshouldashhaveinordertobeinvisibleto othersh? B Doesasurgeonusinganendoscopeneedasourceoflightinside thebodycavity?Ifso,howcouldthisbedonewithoutinsertingalight bulbthroughtheincision? C Adensersampleofagashasahigherindexofrefractionthana lessdensesamplei.e.,asampleunderlowerpressure,butwhywould itnotmakesensefortheindexofrefractionofagastobeproportionalto density? D Theearth'satmospheregetsthinnerandthinnerasyougohigherin altitude.Ifarayoflightcomesfromastarthatisbelowthezenith,what willhappentoitasitcomesintotheearth'satmosphere? E Doestotalinternalreectionoccurwhenlightinadensermedium encountersalessdensemedium,ortheotherwayaround?Orcanit occurineithercase? 4.2Lenses Figuresn/1andn/2showexamplesoflensesformingimages.There isessentiallynothingforyoutolearnaboutimagingwithlenses thatistrulynew.Youalreadyknowhowtoconstructanduseray diagrams,andyouknowaboutrealandvirtualimages.Theconcept ofthefocallengthofalensisthesameasforacurvedmirror.The equationsforlocatingimagesanddeterminingmagnicationsare ofthesameform.It'sreallyjustaquestionofexingyourmental musclesonafewexamples.Thefollowingself-checksanddiscussion questionswillgetyoustarted. n / 1.Aconverginglensformsan imageofacandleame.2.Adiverginglens. self-checkB Inguresn/1andn/2,classifytheimagesasrealorvirtual. Section4.2Lenses 67

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q / Theprincipleofleasttime appliedtorefraction. p / Theradiiofcurvatureappearinginthelensmaker's equation. Glasshasanindexofrefractionthatisgreaterthanthatofair.Considerthetopmostrayinguren/1.Explainwhytheraymakesaslight leftturnuponenteringthelens,andanotherleftturnwhenitexits. Iftheameinguren/2wasmovedclosertothelens,whatwould happentothelocationoftheimage? Answer,p.108 DiscussionQuestions A Inguresn/1andn/2,thefrontandbacksurfacesareparallelto eachotheratthecenterofthelens.Whatwillhappentoaraythatenters nearthecenter,butnotnecessarilyalongtheaxisofthelens?Drawa BIGraydiagram,andshowaraythatcomesfromoffaxis. B Supposeyouwantedtochangethesetupinguren/1sothatthe locationoftheactualameinthegurewouldinsteadbeoccupiedbyan imageofaame.Wherewouldyouhavetomovethecandletoachieve this?Whataboutinn/2? C Therearethreequalitativelydifferenttypesofimageformationthat canoccurwithlenses,ofwhichguresn/1andn/2exhaustonlytwo. Figureoutwhatthethirdpossibilityis.Whichofthethreepossibilitiescan resultinamagnicationgreaterthanone? D Classifytheexamplesshowningureoaccordingtothetypesof imagesdelineatedindiscussionquestionC. E Inguresn/1andn/2,theonlyraysdrawnwerethosethathappened toenterthelenses.Discussthisinrelationtogureo. F Intheright-handsideofgureo,theimageviewedthroughthelens isinfocus,butthesideoftherosethatsticksoutfrombehindthelensis not.Why? 4.3 ? TheLensmaker'sEquation Thefocallengthofasphericalmirrorissimply r= 2,butwecannotexpectthefocallengthofalenstobegivenbypuregeometry, sinceitalsodependsontheindexofrefractionofthelens.Suppose wehavealenswhosefrontandbacksurfacesarebothspherical. Thisisnogreatlossofgenerality,sinceanysurfacewithasucientlyshallowcurvaturecanbeapproximatedwithasphere.Then ifthelensisimmersedinamediumwithanindexofrefractionof 1,itsfocallengthisgivenapproximatelyby f = n )]TJ/F15 10.9091 Tf 10.91 0 Td [(1 1 r 1 1 r 2 )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 where n istheindexofrefractionand r 1 and r 2 aretheradiiof curvatureofthetwosurfacesofthelens.Thisisknownasthe lensmaker'sequation.Inmyopinionitisnotparticularlyworthy ofmemorization.Thepositivesignisusedwhenbothsurfacesare curvedoutwardorbotharecurvedinward;otherwiseanegative signapplies.Theproofofthisequationisleftasanexerciseto thosereaderswhoaresucientlybraveandmotivated. 68 Chapter4Refraction

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o / Twoimagesofarosecreatedbythesamelensandrecordedwiththesamecamera. 4.4 ? ThePrincipleofLeastTimefor Refraction Wehaveseenpreviouslyhowtherulesgoverningstraight-line motionoflightandreectionoflightcanbederivedfromtheprincipleofleasttime.Whataboutrefraction?Inthegure,itisindeed plausiblethatthebendingoftherayservestominimizethetime requiredtogetfromapointAtopointB.Iftherayfollowedtheunbentpathshownwithadashedline,itwouldhavetotravelalonger distanceinthemediuminwhichitsspeedisslower.Bybending thecorrectamount,itcanreducethedistanceithastocoverinthe slowermediumwithoutgoingtoofaroutofitsway.Itistruethat Snell'slawgivesexactlythesetofanglesthatminimizesthetime requiredforlighttogetfromonepointtoanother.Theproofof thisfactisleftasanexercise. Section4.4 ? ThePrincipleofLeastTimeforRefraction 69

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Summary SelectedVocabulary refraction....thechangeindirectionthatoccurswhena waveencounterstheinterfacebetweentwomedia indexofrefraction........ anopticalpropertyofmatter;thespeedof lightinavacuumdividedbythespeedoflight inthesubstanceinquestion Notation n ..........theindexofrefraction Summary Refractionisachangeindirectionthatoccurswhenawaveencounterstheinterfacebetweentwomedia.Together,refractionand reectionaccountforthebasicprinciplesbehindnearlyalloptical devices. Snelldiscoveredtheequationforrefraction, n 1 sin 1 = n 2 sin 2 [anglesmeasuredwithrespecttothenormal] throughexperimentswithlightrays,longbeforelightwasproven tobeawave.Snell'slawcanbeprovenbasedonthegeometrical behaviorofwaves.Here n istheindexofrefraction.Snellinvented thisquantitytodescribetherefractivepropertiesofvarioussubstances,butitwaslaterfoundtoberelatedtothespeedoflightin thesubstance, n = c v where c isthespeedoflightinavacuum.Ingeneralamaterial's indexofrefractionisdierentfordierentwavelengthsoflight. Asdiscussedinthethirdbookofthisseries,anywaveispartiallytransmittedandpartiallyreectedattheboundarybetween twomediainwhichitsspeedsaredierent.Itisnotparticularlyimportanttoknowtheequationthattellswhatfractionistransmitted andthusrefracted,butimportanttechnologiessuchasberoptics arebasedonthefactthatthisfractionbecomes zero forsuciently obliqueangles.Thisphenomenonisreferredtoastotalinternal reection.ItoccurswhenthereisnoanglethatsatisesSnell'slaw. 70 Chapter4Refraction

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Supposeaconverginglensisconstructedofatypeofplastic whoseindexofrefractionislessthanthatofwater.Howwillthe lens'sbehaviorbedierentifitisplacedunderwater? 2 Therearetwomaintypesoftelescopes,refractingusing lensesandreectingusingmirrors.Sometelescopesuseamixtureofthetwotypesofelements:thelightrstencountersalarge curvedmirror,andthengoesthroughaneyepiecethatisalens. Whatimplicationswouldthecolor-dependenceoffocallengthhave fortherelativemeritsofthetwotypesoftelescopes?Whatwould happenwithwhitestarlight,forexample? 3 BasedonSnell'slaw,explainwhyraysoflightpassingthrough theedgesofaconverginglensarebentmorethanrayspassing throughpartsclosertothecenter.Itmightseemlikeitshould betheotherwayaround,sincetheraysattheedgepassthrough lessglass|shouldn'ttheybeaectedless?Inyouranswer: Includearaydiagramshowingahugeclose-upviewofthe relevantpartofthelens. Makeuseofthefactthatthefrontandbacksurfacesaren't alwaysparallel;alensinwhichthefrontandbacksurfaces are alwaysparalleldoesn'tfocuslightatall,soifyourexplanation doesn'tmakeuseofthisfact,yourargumentmustbeincorrect. Makesureyourargumentstillworkseveniftheraysdon't comeinparalleltotheaxis. 4 Whenyoutakepictureswithacamera,thedistancebetween thelensandthelmhastobeadjusted,dependingonthedistance atwhichyouwanttofocus.Thisisdonebymovingthelens.If youwanttochangeyourfocussothatyoucantakeapictureof somethingfartheraway,whichwaydoyouhavetomovethelens? Explainusingraydiagrams.[BasedonaproblembyEricMazur.] 5 aLightisbeingreecteddiuselyfromanobject1.000m underwater.Thelightthatcomesuptothesurfaceisrefractedat thewater-airinterface.Iftherefractedraysallappeartocomefrom thesamepoint,thentherewillbeavirtualimageoftheobjectinthe water,abovetheobject'sactualposition,whichwillbevisibletoan observerabovethewater.Considerthreerays,A,BandC,whose Problems 71

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Problem6. Problem8. anglesinthewaterwithrespecttothenormalare i =0.000 1.000 and20.000 respectively.Findthedepthofthepointat whichtherefractedpartsofAandBappeartohaveintersected, anddothesameforAandC.Showthattheintersectionsareat nearlythesamedepth,butnotquite.[Check:Thedierencein depthshouldbeabout4cm.] bSincealltherefractedraysdonotquiteappeartohavecome fromthesamepoint,thisistechnicallynotavirtualimage.In practicalterms,whateectwouldthishaveonwhatyousee? cInthecasewheretheanglesareallsmall,usealgebraandtrigto showthattherefractedraysdoappeartocomefromthesamepoint, andndanequationforthedepthofthevirtualimage.Donotput inanynumericalvaluesfortheanglesorfortheindicesofrefraction |justkeepthemassymbols.Youwillneedtheapproximation sin tan ,whichisvalidforsmallanglesmeasuredinradians. ? 6 Thedrawingshowstheanatomyofthehumaneye,attwicelife size.Findtheradiusofcurvatureoftheoutersurfaceofthecornea bymeasurementsonthegure,andthenderivethefocallengthof theair-corneainterface,wherealmostallthefocusingoflightoccurs. Youwillneedtousephysicalreasoningtomodifythelensmaker's equationforthecasewherethereisonlyasinglerefractingsurface. Assumethattheindexofrefractionofthecorneaisessentiallythat ofwater. ? 7 Whenswimmingunderwater,whyisyourvisionmademuch clearerbywearinggoggleswithatpiecesofglassthattrapair behindthem?[Hint:Youcansimplifyyourreasoningbyconsidering thespecialcasewhereyouarelookingatanobjectfaraway,and alongtheopticaxisoftheeye.] 8 Thegureshowsfourlenses.Lens1hastwosphericalsurfaces. Lens2isthesameaslens1butturnedaround.Lens3ismadeby cuttingthroughlens1andturningthebottomaround.Lens4is madebycuttingacentralcircleoutoflens1andrecessingit. aAparallelbeamoflightenterslens1fromtheleft,parallelto itsaxis.ReasoningbasedonSnell'slaw,willthebeamemerging fromthelensbebentinward,oroutward,orwillitremainparallel totheaxis?Explainyourreasoning.Aspartofyouranswer,make ahugedrawingofonesmallpartofthelens,andapplySnell'slaw atbothinterfaces.Recallthatraysarebentmoreiftheycometo theinterfaceatalargeranglewithrespecttothenormal. bWhatwillhappenwithlenses2,3,and4?Explain.Drawings arenotnecessary. 9 ProvethattheprincipleofleasttimeleadstoSnell'slaw. ? 72 Chapter4Refraction

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Problem13. 10 Anobjectismorethanonefocallengthfromaconverging lens.aDrawaraydiagram.bUsingreasoninglikethatdevelopedinchapter3,determinethepositiveandnegativesignsinthe equation1 =f = 1 =d i 1 =d o .cTheimagesoftheroseinsection 4.2weremadeusingalenswithafocallengthof23cm.Ifthelens isplaced80cmfromtherose,locatetheimage. p 11 Anobjectislessthanonefocallengthfromaconverginglens. aDrawaraydiagram.bUsingreasoninglikethatdevelopedin chapter3,determinethepositiveandnegativesignsintheequation 1 =f = 1 =d i 1 =d o .cTheimagesoftheroseinsection4.2were madeusingalenswithafocallengthof23cm.Ifthelensisplaced 10cmfromtherose,locatetheimage. p 12 Nearsightedpeoplewearglasseswhoselensesarediverging. aDrawaraydiagram.Forsimplicitypretendthatthereisno eyebehindtheglasses.bUsingreasoninglikethatdevelopedin chapter3,determinethepositiveandnegativesignsintheequation 1 =f = 1 =d i 1 =d o .cIfthefocallengthofthelensis50.0cm, andthepersonislookingatanobjectatadistanceof80.0cm, locatetheimage. p 13 Twostandardfocallengthsforcameralensesare50mm standardand28mmwide-angle.Toseehowthefocallengths relatetotheangularsizeoftheeldofview,itishelpfultovisualize thingsasrepresentedinthegure.Insteadofshowingmanyrays comingfromthesamepointonthesameobject,aswenormallydo, thegureshowstworaysfromtwodierentobjects.Althoughthe lenswillinterceptinnitelymanyraysfromeachofthesepoints,we haveshownonlytheonesthatpassthroughthecenterofthelens, sothattheysuernoangulardeection.Anyangulardeectionat thefrontsurfaceofthelensiscanceledbyanoppositedeectionat theback,sincethefrontandbacksurfacesareparallelatthelens's center.Whatisspecialaboutthesetworaysisthattheyareaimed attheedgesofone35-mm-wideframeoflm;thatis,theyshow thelimitsoftheeldofview.Throughoutthisproblem,weassume that d o ismuchgreaterthan d i .aComputetheangularwidth ofthecamera'seldofviewwhenthesetwolensesareused.b Usesmall-angleapproximationstondasimpliedequationforthe angularwidthoftheeldofview, ,intermsofthefocallength, f ,andthewidthofthelm, w .Yourequationshouldnothave anytrigfunctionsinit.Comparetheresultsofthisapproximation withyouranswersfromparta.cSupposethatweareholding constanttheapertureamountofsurfaceareaofthelensbeing usedtocollectlight.Whenswitchingfroma50-mmlenstoa28mmlens,howmanytimeslongerorshortermusttheexposurebe inordertomakeaproperlydevelopedpicture,i.e.,onethatisnot under-oroverexposed?[BasedonaproblembyArnoldArons.] Solution,p.110 Problems 73

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14 Anearsightedpersonisonewhoseeyesfocuslighttoo strongly,andwhoisthereforeunabletorelaxthelensinsideher eyesucientlytoformanimageonherretinaofanobjectthatis toofaraway. aDrawaraydiagramshowingwhathappenswhentheperson tries,withuncorrectedvision,tofocusatinnity. bWhattypeoflensesdoherglasseshave?Explain. cDrawaraydiagramshowingwhathappenswhenshewears glasses.Locateboththeimageformedbytheglassesandthenalimage. dSupposeshesometimesusescontactlensesinsteadofherglasses. Doesthefocallengthofhercontactshavetobelessthan,equalto, orgreaterthanthatofherglasses?Explain. 15 Diamondhasanindexofrefractionof2.42,andpartofthe reasondiamondssparkleisthatthisencouragesalightraytoundergomanytotalinternalreectionsbeforeitemerges.Calculatethe criticalangleatwhichtotalinternalreectionoccursindiamond. Explaintheinterpretationofyourresult:Isitmeasuredfromthe normal,orfromthesurface?Isitaminimum,oramaximum?How wouldthecriticalanglehavebeendierentforasubstancesuchas glassorplastic,withalowerindexofrefraction? 16 Fred'seyesareabletofocusonthingsascloseas5.0cm. Fredholdsamagnifyingglasswithafocallengthof3.0cmata heightof2.0cmaboveaatworm.aLocatetheimage,andnd themagnication.bWithoutthemagnifyingglass,fromwhat distancewouldFredwanttoviewtheatwormtoseeitsdetails aswellaspossible?Withthemagnifyingglass?cComputethe angularmagnication. Problem17. 17 Panel1ofthegureshowstheopticsinsideapairofbinoculars.Theyareessentiallyapairoftelescopes,oneforeacheye. 74 Chapter4Refraction

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Buttomakethemmorecompact,andallowtheeyepiecestobethe rightdistanceapartforahumanface,theyincorporateasetofeight prisms,whichfoldthelightpath.Inaddition,theprismsmakethe imageupright.Panel2showsoneoftheseprisms,knownasaPorro prism.Thelightentersalonganormal,undergoestwototalinternal reectionsatanglesof45degreeswithrespecttothebacksurfaces, andexitsalonganormal.TheimageoftheletterRhasbeenipped acrossthehorizontal.Panel3showsapairoftheseprismsglued together.Theimagewillbeippedacrossboththehorizontaland thevertical,whichmakesitorientedtherightwayfortheuserof thebinoculars. aFindtheminimumpossibleindexofrefractionfortheglassused intheprisms. bForamaterialofthisminimalindexofrefraction,ndthefractionoftheincominglightthatwillbelosttoreectioninthefour Porroprismsonaeachsideofapairofbinoculars.Seechapter 4or VibrationsandWaves ,orsection6.2of SimpleNature .In real,high-qualitybinoculars,theopticalsurfacesoftheprismshave antireectivecoatings,butcarryoutyourcalculationforthecase wherethereisnosuchcoating. cDiscussthereasonswhyadesignerofbinocularsmightormight notwanttouseamaterialwithexactlytheindexofrefractionfound inparta. ? 18 Itwouldbeannoyingifyoureyeglassesproducedamagnied orreducedimage.Provethatwhentheeyeisveryclosetoalens, andthelensproducesavirtualimage,theangularmagnicationis alwaysapproximatelyequalto1regardlessofwhetherthelensis divergingorconverging. 19 Atypicalmirrorconsistsofapaneofglassofthickness t andindexofrefraction n ,silvered"onthebackwithareective coating.Let d o and d i bemeasuredfromthebackofthemirror. Showthat d i = d o )]TJ/F15 10.9091 Tf 11.227 0 Td [(2 )]TJ/F15 10.9091 Tf 11.227 0 Td [(1 =n t .Usetheresultof,andmakethe approximationemployedin,problem5c.Asacheckonyourresult, considerseparatelythespecialvaluesof n and t thatwouldrecover thecasewithoutanyglass. Problems 75

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ThisimageofthePleiadesstarclustershowshaloesaroundthestarsduetothewavenatureoflight. Chapter5 WaveOptics Electronmicroscopescanmakeimagesofindividualatoms,butwhy willavisible-lightmicroscopeneverbeableto?Stereospeakers createtheillusionofmusicthatcomesfromabandarrangedin yourlivingroom,butwhydoesn'tthestereoillusionworkwithbass notes?Whyarecomputerchipmanufacturersinvestingbillionsof dollarsinequipmenttoetchchipswithx-raysinsteadofvisible light? Theanswerstoallofthesequestionshavetodowiththesubject ofwaveoptics.Sofarthisbookhasdiscussedtheinteractionof lightwaveswithmatter,anditspracticalapplicationstooptical deviceslikemirrors,butwehaveusedtheraymodeloflightalmost exclusively.Hardlyeverhaveweexplicitlymadeuseofthefactthat lightisanelectromagneticwave.Wewereabletogetawaywiththe 77

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a / Inthisviewfromoverhead,a straight,sinusoidalwaterwave encountersabarrierwithtwo gapsinit.Strongwavevibration occursatanglesXandZ,but thereisnoneatallatangleY. Thegurehasbeenretouched fromarealphotoofwaterwaves. Inreality,thewavesbeyondthe barrierwouldbemuchweaker thantheonesbeforeit,andthey wouldthereforebedifcultto see. b / Thisdoesn'thappen. simpleraymodelbecausethechunksofmatterwewerediscussing, suchaslensesandmirrors,werethousandsoftimeslargerthana wavelengthoflight.Wenowturntophenomenaanddevicesthat canonlybeunderstoodusingthewavemodeloflight. 5.1Diffraction Figureashowsatypicalprobleminwaveoptics,enactedwith waterwaves.Itmayseemsurprisingthatwedon'tgetasimple patternlikegureb,butthepatternwouldonlybethatsimple ifthewavelengthwashundredsoftimesshorterthanthedistance betweenthegapsinthebarrierandthewidthsofthegaps. Waveopticsisabroadsubject,butthisexamplewillhelpus topickoutareasonablesetofrestrictionstomakethingsmore manageable: Werestrictourselvestocasesinwhichawavetravelsthrough auniformmedium,encountersacertainareainwhichthemedium hasdierentproperties,andthenemergesontheothersideintoa seconduniformregion. Weassumethattheincomingwaveisanicetidysine-wave patternwithwavefrontsthatarelinesor,inthreedimensions, planes. Ingureawecanseethatthewavepatternimmediately beyondthebarrierisrathercomplex,butfartheronitsortsitself outintoasetofwedgesseparatedbygapsinwhichthewateris still.Wewillrestrictourselvestostudyingthesimplerwavepatterns thatoccurfartheraway,sothatthemainquestionofinterestishow intensetheoutgoingwaveisatagivenangle. Thekindofphenomenondescribedbyrestrictioniscalled diraction .Diractioncanbedenedasthebehaviorofawave whenitencountersanobstacleoranonuniformityinitsmedium. Ingeneral,diractioncausesawavetobendaroundobstaclesand makepatternsofstrongandweakwavesradiatingoutbeyondthe obstacle.Understandingdiractionisthecentralproblemofwave optics.Ifyouunderstanddiraction,eventhesubsetofdiraction problemsthatfallwithinrestrictionsand,therestofwave opticsisicingonthecake. Diractioncanbeusedtondthestructureofanunknown diractingobject:eveniftheobjectistoosmalltostudywith ordinaryimaging,itmaybepossibletoworkbackwardfromthe diractionpatterntolearnabouttheobject.Thestructureofa crystal,forexample,canbedeterminedfromitsx-raydiraction pattern. Diractioncanalsobeabadthing.Inatelescope,forexample, lightwavesarediractedbyallthepartsoftheinstrument.Thiswill 78 Chapter5WaveOptics

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c / Apractical,low-techsetupfor observingdiffractionoflight. d / Thebottomgureissimplyacopyofthemiddleportion ofthetopone,scaledupbya factoroftwo.Alltheanglesare thesame.Physically,theangular patternofthediffractionfringes can'tbeanydifferentifwescale both and d bythesamefactor, leaving = d unchanged. causetheimageofastartoappearfuzzyevenwhenthefocushas beenadjustedcorrectly.Byunderstandingdiraction,onecanlearn howatelescopemustbedesignedinordertoreducethisproblem |essentially,itshouldhavethebiggestpossiblediameter. Therearetwowaysinwhichrestriction2mightcommonlybe violated.First,thelightmightbeamixtureofwavelengths.Ifwe simplywanttoobserveadiractionpatternortousediractionas atechniqueforstudyingtheobjectdoingthediractinge.g.,ifthe objectistoosmalltoseewithamicroscope,thenwecanpassthe lightthroughacoloredlterbeforediractingit. Asecondissueisthatlightfromsourcessuchasthesunora lightbulbdoesnotconsistofaniceneatplanewave,exceptover verysmallregionsofspace.Dierentpartsofthewaveareoutof stepwitheachother,andthewaveisreferredtoas incoherent .One wayofdealingwiththisisshowningurec.Afterlteringtoselect acertainwavelengthofredlight,wepassthelightthroughasmall pinhole.Theregionofthelightthatisinterceptedbythepinholeis sosmallthatonepartofitisnotoutofstepwithanother.Beyond thepinhole,lightspreadsoutinasphericalwave;thisisanalogous towhathappenswhenyouspeakintooneendofapapertowelroll andthesoundwavesspreadoutinalldirectionsfromtheotherend. Bythetimethesphericalwavegetstothedoubleslitithasspread outandreduceditscurvature,sothatwecannowthinkofitasa simpleplanewave. Ifthisseemslaborious,youmayberelievedtoknowthatmodern technologygivesusaneasierwaytoproduceasingle-wavelength, coherentbeamoflight:thelaser. Thepartsofthenalimageonthescreenincarecalleddiractionfringes.Thecenterofeachfringeisapointofmaximumbrightness,andhalfwaybetweentwofringesisaminimum. DiscussionQuestion A Whywouldx-raysratherthanvisiblelightbeusedtondthestructure ofacrystal?Soundwavesareusedtomakeimagesoffetusesinthe womb.Whatwouldinuencethechoiceofwavelength? 5.2ScalingofDiffraction Thischapterhasoptics"initstitle,soitisnominallyaboutlight, butwestartedoutwithanexampleinvolvingwaterwaves.Water wavesarecertainlyeasiertovisualize,butisthisalegitimatecomparison?Infacttheanalogyworksquitewell,despitethefactthat alightwavehasawavelengthaboutamilliontimesshorter.This isbecausediractioneectsscaleuniformly.Thatis,ifweenlarge orreducethewholediractionsituationbythesamefactor,includingboththewavelengthsandthesizesoftheobstaclesthewave encounters,theresultisstillavalidsolution. Section5.2ScalingofDiffraction 79

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e / ChristiaanHuygens1695. Thisisunusuallysimplebehavior!Intherstbookofthisseries wesawmanyexamplesofmorecomplexscaling,suchastheimpossibilityofbacteriathesizeofdogs,ortheneedforanelephantto eliminateheatthroughitsearsbecauseofitssmallsurface-to-volume ratio,whereasatinyshrew'slife-stylecentersaroundconservingits bodyheat. Ofcoursewaterwavesandlightwavesdierinmanyways,not justinscale,butthegeneralfactsyouwilllearnaboutdiraction areapplicabletoallwaves.Insomewaysitmighthavebeenmore appropriatetoinsertthischapterattheendofbook3,Vibrations andWaves,butmanyoftheimportantapplicationsaretolight waves,andyouwouldprobablyhavefoundthesemuchmoredicult withoutanybackgroundinoptics. Anotherwayofstatingthesimplescalingbehaviorofdiraction isthatthediractionangleswegetdependonlyontheunitlessratio /d,where isthewavelengthofthewaveand d issomedimension ofthediractingobjects,e.g.,thecenter-to-centerspacingbetween theslitsingurea.If,forinstance,wescaleupboth and d bya factorof37,theratio =d willbeunchanged. 5.3TheCorrespondencePrinciple Theonlyreasonwedon'tusuallynoticediractionoflightineverydaylifeisthatwedon'tnormallydealwithobjectsthatare comparableinsizetoawavelengthofvisiblelight,whichisabouta millionthofameter.Doesthismeanthatwaveopticscontradicts rayoptics,orthatwaveopticssometimesgiveswrongresults?No. Ifyouholdthreengersoutinthesunlightandcastashadowwith them, either waveopticsorrayopticscanbeusedtopredictthe straightforwardresult:ashadowpatternwithtwobrightlineswhere thelighthasgonethroughthegapsbetweenyourngers.Waveopticsisamoregeneraltheorythanrayoptics,soinanycasewhere rayopticsisvalid,thetwotheorieswillagree.Thisisanexample ofageneralideaenunciatedbythephysicistNielsBohr,calledthe correspondenceprinciple: whenawsinaphysicaltheoryleadto thecreationofanewandmoregeneraltheory,thenewtheorymust stillagreewiththeoldtheorywithinitsmorerestrictedareaofapplicability.Afterall,atheoryisonlycreatedasawayofdescribing experimentalobservations.Iftheoriginaltheoryhadnotworkedin anycasesatall,itwouldneverhavebecomeaccepted. Inthecaseofoptics,thecorrespondenceprincipletellsusthat when =d issmall,boththerayandthewavemodeloflightmust giveapproximatelythesameresult.Supposeyouspreadyourngers andcastashadowwiththemusingacoherentlightsource.The quantity =d isabout10-4,sothetwomodelswillagreeveryclosely. Tobespecic,theshadowsofyourngerswillbeoutlinedbya seriesoflightanddarkfringes,buttheanglesubtendedbyafringe 80 Chapter5WaveOptics

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f / Double-slitdiffraction. g / Awavefrontcanbeanalyzed bytheprincipleofsuperposition, breakingitdownintomanysmall parts. h / Ifitwasbyitself,eachof thepartswouldspreadoutasa circularripple. i / Addinguptheripplesproducesanewwavefront. willbeontheorderof10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(4 radians,sotheywillbeinvisibleand washedoutbythenaturalfuzzinessoftheedgesofsun-shadows, causedbythenitesizeofthesun. self-checkA Whatkindofwavelengthwouldanelectromagneticwavehavetohave inordertodiffractdramaticallyaroundyourbody?Doesthiscontradict thecorrespondenceprinciple? Answer,p.108 5.4Huygens'Principle Returningtotheexampleofdouble-slitdiraction,f,notethe strongvisualimpressionoftwooverlappingsetsofconcentricsemicircles.Thisisanexampleof Huygens'principle ,namedaftera Dutchphysicistandastronomer.Therstsyllablerhymeswith boy."Huygens'principlestatesthatanywavefrontcanbebroken downintomanysmallside-by-sidewavepeaks,g,whichthenspread outascircularripples,h,andbytheprincipleofsuperposition,the resultofaddingupthesesetsofripplesmustgivethesameresult asallowingthewavetopropagateforward,i.Inthecaseofsound orlightwaves,whichpropagateinthreedimensions,theripples" areactuallysphericalratherthancircular,butwecanoftenimagine thingsintwodimensionsforsimplicity. Indouble-slitdiractiontheapplicationofHuygens'principleis visuallyconvincing:itisasthoughallthesetsofrippleshavebeen blockedexceptfortwo.Itisarathersurprisingmathematicalfact, however,thatHuygens'principlegivestherightresultinthecaseof anunobstructedlinearwave,handi.Atheoreticallyinnitenumber ofcircularwavepatternssomehowconspiretoaddtogetherand producethesimplelinearwavemotionwithwhichwearefamiliar. SinceHuygens'principleisequivalenttotheprincipleofsuperposition,andsuperpositionisapropertyofwaves,whatHuygens hadcreatedwasessentiallytherstwavetheoryoflight.However, heimaginedlightasaseriesofpulses,likehandclaps,ratherthan asasinusoidalwave. Thehistoryisinteresting.IsaacNewtonlovedtheatomictheory ofmattersomuchthathesearchedenthusiasticallyforevidencethat lightwasalsomadeoftinyparticles.Thepathsofhislightparticles wouldcorrespondtoraysinourdescription;theonlysignicant dierencebetweenaraymodelandaparticlemodeloflightwould occurifonecouldisolateindividualparticlesandshowthatlight hadagraininess"toit.Newtonneverdidthis,soalthoughhe thoughtofhismodelasaparticlemodel,itismoreaccuratetosay hewasoneofthebuildersoftheraymodel. Almostallthatwasknownaboutreectionandrefractionof lightcouldbeinterpretedequallywellintermsofaparticlemodel orawavemodel,butNewtonhadonereasonforstronglyopposing Section5.4Huygens'Principle 81

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j / ThomasYoung k / Double-slitdiffraction. l / UseofHuygens'principle. m / Constructiveinterference alongthecenter-line. Huygens'wavetheory.Newtonknewthatwavesexhibiteddiraction,butdiractionoflightisdiculttoobserve,soNewtonbelievedthatlightdidnotexhibitdiraction,andthereforemustnot beawave.AlthoughNewton'scriticismswerefairenough,thedebatealsotookontheovertonesofanationalisticdisputebetween EnglandandcontinentalEurope,fueledbyEnglishresentmentover Leibniz'ssupposedplagiarismofNewton'scalculus.Newtonwrote abookonoptics,andhisprestigeandpoliticalprominencetended todiscouragequestioningofhismodel. ThomasYoung-1829wasthepersonwhonally,ahundredyearslater,didacarefulsearchforwaveinterferenceeects withlightandanalyzedtheresultscorrectly.Heobserveddoubleslitdiractionoflightaswellasavarietyofotherdiractioneffects,allofwhichshowedthatlightexhibitedwaveinterferenceeffects,andthatthewavelengthsofvisiblelightwaveswereextremely short.ThecrowningachievementwasthedemonstrationbytheexperimentalistHeinrichHertzandthetheoristJamesClerkMaxwell thatlightwasan electromagnetic wave.Maxwellissaidtohaverelatedhisdiscoverytohiswifeonestarryeveningandtoldherthat shewastheonlyotherpersonintheworldwhoknewwhatstarlight was. 5.5Double-SlitDiffraction Let'snowanalyzedouble-slitdiraction,k,usingHuygens'principle.Themostinterestingquestionishowtocomputetheangles suchasXandZwherethewaveintensityisatamaximum,and thein-betweenangleslikeYwhereitisminimized.Let'smeasure allourangleswithrespecttotheverticalcenterlineofthegure, whichwastheoriginaldirectionofpropagationofthewave. Ifweassumethatthewidthoftheslitsissmallontheorder ofthewavelengthofthewaveorless,thenwecanimagineonlya singlesetofHuygensripplesspreadingoutfromeachone,l.White linesrepresentpeaks,blackonestroughs.Theonlydimensionofthe diractingslitsthathasanyeectonthegeometricpatternofthe overlappingripplesisthenthecenter-to-centerdistance, d ,between theslits. Weknowfromourdiscussionofthescalingofdiractionthat theremustbesomeequationthatrelatesananglelike Z tothe ratio =d d $ Z Iftheequationfor Z dependedonsomeotherexpressionsuchas + d or 2 =d ,thenitwouldchangewhenwescaled and d bythe samefactor,whichwouldviolatewhatweknowaboutthescaling ofdiraction. Alongthecentralmaximumline,X,wealwayshavepositive 82 Chapter5WaveOptics

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n / Thewavestraveldistances L and L 0 fromthetwoslitstoget tothesamepointinspace,atan angle fromthecenterline. o / Aclose-upviewofgure n,showinghowthepathlength difference L )]TJ/F106 9.9627 Tf 11.188 0 Td [(L 0 isrelatedto d andtotheangle wavescoincidingwithpositiveonesandnegativewavescoinciding withnegativeones.Ihavearbitrarilychosentotakeasnapshotof thepatternatamomentwhenthewavesemergingfromtheslitare experiencingapositivepeak.Thesuperpositionofthetwosetsof ripplesthereforeresultsinadoublingofthewaveamplitudealong thisline.Thereisconstructiveinterference.Thisiseasytoexplain, becausebysymmetry,eachwavehashadtotravelanequalnumber ofwavelengthstogetfromitsslittothecenterline,m:Because bothsetsofrippleshavetenwavelengthstocoverinordertoreach thepointalongdirectionX,theywillbeinstepwhentheygetthere. AtthepointalongdirectionYshowninthesamegure,one wavehastraveledtenwavelengths,andisthereforeatapositive extreme,buttheotherhastraveledonlynineandahalfwavelengths, soitatanegativeextreme.Thereisperfectcancellation,sopoints alongthislineexperiencenowavemotion. Butthedistancetraveleddoesnothavetobeequalinorderto getconstructiveinterference.AtthepointalongdirectionZ,one wavehasgoneninewavelengthsandtheotherten.Theyareboth atapositiveextreme. self-checkB AtapointhalfawavelengthbelowthepointmarkedalongdirectionX, carryoutasimilaranalysis. Answer,p.109 Tosummarize,wewillhaveperfectconstructiveinterferenceat anypointwherethedistancetooneslitdiersfromthedistanceto theotherslitbyanintegernumberofwavelengths.Perfectdestructiveinterferencewilloccurwhenthenumberofwavelengthsofpath lengthdierenceequalsanintegerplusahalf. Nowwearereadytondtheequationthatpredictstheangles ofthemaximaandminima.Thewavestraveldierentdistances togettothesamepointinspace,n.Weneedtondwhetherthe wavesareinphaseinsteporoutofphaseatthispointinorderto predictwhethertherewillbeconstructiveinterference,destructive interference,orsomethinginbetween. Oneofourbasicassumptionsinthischapteristhatwewillonly bedealingwiththediractedwaveinregionsveryfarawayfromthe objectthatdiractsit,sothetriangleislongandskinny.Mostrealworldexampleswithdiractionoflight,infact,wouldhavetriangles withevenskinnerproportionsthanthisone.Thetwolongsidesare thereforeverynearlyparallel,andwearejustiedindrawingthe righttriangleshowningureo,labelingonelegoftherighttriangle asthedierenceinpathlength, L )]TJ/F20 10.9091 Tf 9.453 0 Td [(L 0 ,andlabelingtheacuteangle as .Inrealitythisangleisatinybitgreaterthantheonelabeled inguren. Section5.5Double-SlitDiffraction 83

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p / Cutting d inhalfdoubles theanglesofthediffraction fringes. q / Double-slitdiffractionpatternsoflong-wavelengthredlight topandshort-wavelengthblue lightbottom. Thedierenceinpathlengthisrelatedto d and bytheequation L )]TJ/F20 10.9091 Tf 10.909 0 Td [(L 0 d =sin Constructiveinterferencewillresultinamaximumatanglesfor which L )]TJ/F20 10.9091 Tf 10.909 0 Td [(L 0 isanintegernumberofwavelengths, L )]TJ/F20 10.9091 Tf 10.909 0 Td [(L 0 = m .[conditionforamaximum; m isaninteger] Here m equals0forthecentralmaximum, )]TJ/F15 10.9091 Tf 8.484 0 Td [(1fortherstmaximum toitsleft,+2forthesecondmaximumontheright,etc.Putting alltheingredientstogether,wend m=d =sin ,or d = sin m .[conditionforamaximum; m isaninteger] Similarly,theconditionforaminimumis d = sin m .[conditionforaminimum; m isanintegerplus1 = 2] Thatis,theminimaareabouthalfwaybetweenthemaxima. Asexpectedbasedonscaling,thisequationrelatesanglestothe unitlessratio =d .Alternatively,wecouldsaythatwehaveproven thescalingpropertyinthespecialcaseofdouble-slitdiraction.It wasinevitablethattheresultwouldhavethesescalingproperties, sincethewholeproofwasgeometric,andwouldhavebeenequally validwhenenlargedorreducedonaphotocopyingmachine! Counterintuitively,thismeansthatadiractingobjectwith smallerdimensionsproducesabiggerdiractionpattern,p. Double-slitdiffractionofblueandredlightexample1 Bluelighthasashorterwavelengththanred.Foragivendoubleslitspacing d ,thesmallervalueof = d forleadstosmallervalues ofsin ,andthereforetoamorecloselyspacedsetofdiffraction fringes,g 84 Chapter5WaveOptics

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Thecorrespondenceprincipleexample2 Let'salsoconsiderhowtheequationsfordouble-slitdiffraction relatetothecorrespondenceprinciple.Whentheratio = d isvery small,weshouldrecoverthecaseofsimplerayoptics.Nowif = d issmall,sin mustbesmallaswell,andthespacingbetween thediffractionfringeswillbesmallaswell.Althoughwehavenot provenit,thecentralfringeisalwaysthebrightest,andthefringes getdimmeranddimmeraswegofartherfromit.Forsmallvalues of = d ,thepartofthediffractionpatternthatisbrightenoughto bedetectablecoversonlyasmallrangeofangles.Thisisexactly whatwewouldexpectfromrayoptics:therayspassingthrough thetwoslitswouldremainparallel,andwouldcontinuemoving inthe =0direction.Infacttherewouldbeimagesofthetwo separateslitsonthescreen,butouranalysiswasallintermsof angles,soweshouldnotexpectittoaddresstheissueofwhether thereisstructurewithinasetofraysthatarealltravelinginthe =0direction. Spacingofthefringesatsmallanglesexample3 Atsmallangles,wecanusetheapproximationsin ,which isvalidif ismeasuredinradians.Theequationfordouble-slit diffractionbecomessimply d = m whichcanbesolvedfor togive = m d Thedifferenceinanglebetweensuccessivefringesisthechange in thatresultsfromchanging m byplusorminusone, = d Forexample,ifwewrite 7 fortheangleoftheseventhbright fringeononesideofthecentralmaximumand 8 fortheneighboringone,wehave 8 )]TJ/F73 10.9091 Tf 10.909 0 Td [( 7 = 8 d )]TJ/F39 10.9091 Tf 12.105 7.38 Td [(7 d = d andsimilarlyforanyotherneighboringpairoffringes. Althoughtheequation =d =sin =m isonlyvalidforadouble slit,itiscanstillbeaguidetoourthinkingevenifweareobserving diractionoflightbyavirusoraea'sleg:itisalwaystruethat largevaluesof =d leadtoabroaddiractionpattern,and Section5.5Double-SlitDiffraction 85

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t / Adouble-slitdiffractionpattern top,andapatternmadebyve slitsbottom. r / Atripleslit. diractionpatternsarerepetitive. Inmanycasestheequationlooksjustlike =d =sin =m but withanextranumericalfactorthrownin,andwith d interpretedas someotherdimensionoftheobject,e.g.,thediameterofapieceof wire. 5.6Repetition Supposewereplaceadoubleslitwithatripleslit,r.Wecanthink ofthisasathirdrepetitionofthestructuresthatwerepresentin thedoubleslit.Willthisdevicebeanimprovementoverthedouble slitforanypracticalreasons? Theanswerisyes,ascanbeshownusinggures.Forease ofvisualization,Ihaveviolatedourusualruleofonlyconsidering pointsveryfarfromthediractingobject.Thescaleofthedrawing issuchthatawavelengthsisonecm.Ins/1,allthreewavestravel anintegernumberofwavelengthstoreachthesamepoint,sothere isabrightcentralspot,aswewouldexpectfromourexperience withthedoubleslit.Ingures/2,weshowthepathlengthsto anewpoint.ThispointisfartherfromslitAbyaquarterofa wavelength,andcorrespondinglyclosertoslitC.Thedistancefrom slitBhashardlychangedatall.Becausethepathslengthstraveled fromslitsAandCdierfromhalfawavelength,therewillbeperfect destructiveinterferencebetweenthesetwowaves.Thereisstillsome uncanceledwaveintensitybecauseofslitB,buttheamplitudewill bethreetimeslessthaningures/1,resultinginafactorof9 decreaseinbrightness.Thus,bymovingototherightalittle,we havegonefromthebrightcentralmaximumtoapointthatisquite dark. Nowlet'scomparewithwhatwouldhavehappenedifslitChad beencovered,creatingaplainolddoubleslit.Thewavescoming fromslitsAandBwouldhavebeenoutofphaseby0.23wavelengths, butthiswouldnothavecausedverysevereinterference.Thepoint ingures/2wouldhavebeenquitebrightlylitup. Tosummarize,wehavefoundthataddingathirdslitnarrows downthecentralfringedramatically.Thesameistrueforallthe otherfringesaswell,andsincethesameamountofenergyisconcentratedinnarrowerdiractionfringes,eachfringeisbrighterand easiertosee,t. Thisisanexampleofamoregeneralfactaboutdiraction:if 86 Chapter5WaveOptics

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u / Single-slitdiffractionof waterwaves. v / Single-slitdiffractionof redlight.Notethedoublewidth ofthecentralmaximum. w / Aprettygoodsimulation ofthesingle-slitpatternofgure u,madebyusingthreemotorsto produceoverlappingripplesfrom threeneighboringpointsinthe water. s / 1.Thereisabrightcentralmaximum.2.Atthispointjustoffthecentralmaximum,thepathlengthstraveled bythethreewaveshavechanged. somefeatureofthediractingobjectisrepeated,thelocationsof themaximaandminimaareunchanged,buttheybecomenarrower. Takingthisreasoningtoitslogicalconclusion,adiractingobjectwiththousandsofslitswouldproduceextremelynarrowfringes. Suchanobjectiscalledadiractiongrating. 5.7Single-SlitDiffraction Ifweuseonlyasingleslit,istherediraction?Iftheslitisnot widecomparedtoawavelengthoflight,thenwecanapproximate itsbehaviorbyusingonlyasinglesetofHuygensripples.There arenoothersetsofripplestoaddtoit,sotherearenoconstructive ordestructiveinterferenceeects,andnomaximaorminima.The resultwillbeauniformsphericalwaveoflightspreadingoutinall directions,likewhatwewouldexpectfromatinylightbulb.We couldcallthisadiractionpattern,butitisacompletelyfeaturelessone,anditcouldnotbeused,forinstance,todeterminethe wavelengthofthelight,asotherdiractionpatternscould. Allofthis,however,assumesthattheslitisnarrowcomparedto awavelengthoflight.If,ontheotherhand,theslitisbroader,there willindeedbeinterferenceamongthesetsofripplesspreadingout fromvariouspointsalongtheopening.Figureushowsanexample withwaterwaves,andgurevwithlight. self-checkC Howdoesthewavelengthofthewavescomparewiththewidthofthe slitingureu? Answer,p.109 Wewillnotgointothedetailsoftheanalysisofsingle-slitdiraction,butletusseehowitspropertiescanberelatedtothegeneral thingswe'velearnedaboutdiraction.Weknowbasedonscaling argumentsthattheangularsizesoffeaturesinthediractionpatternmustberelatedtothewavelengthandthewidth, a ,oftheslit Section5.7Single-SlitDiffraction 87

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y / Aradiotelescope. x / AnimageofthePleiades starcluster.Thecircularrings aroundthebrightstarsaredueto single-slitdiffractionatthemouth ofthetelescope'stube. bysomerelationshipoftheform a $ Thisisindeedtrue,andforinstancetheanglebetweenthemaximum ofthecentralfringeandthemaximumofthenextfringeononeside equals1.5 =a .Scalingargumentswillneverproducefactorssuchas the1.5,buttheytellusthattheanswermustinvolve =a ,soallthe familiarqualitativefactsaretrue.Forinstance,shorter-wavelength lightwillproduceamorecloselyspaceddiractionpattern. Animportantscienticexampleofsingle-slitdiractionisin telescopes.Imagesofindividualstars,asingurex,areagoodway toexaminediractioneects,becauseallstarsexceptthesunareso farawaythatnotelescope,evenatthehighestmagnication,can imagetheirdisksorsurfacefeatures.Thusanyfeaturesofastar's imagemustbeduepurelytoopticaleectssuchasdiraction.A prominentcrossappearsaroundthebrighteststar,anddimmerones surroundthedimmerstars.Somethinglikethisisseeninmosttelescopephotos,andindicatesthatinsidethetubeofthetelescope thereweretwoperpendicularstrutsorsupports.Lightdiracted aroundthesestruts.Youmightthinkthatdiractioncouldbeeliminatedentirelybygettingridofallobstructionsinthetube,butthe circlesaroundthestarsarediractioneectsarisingfromsingleslitdiractionatthemouthofthetelescope'stube!Actuallywe havenoteventalkedaboutdiractionthroughacircularopening, buttheideaisthesame.Sincetheangularsizesofthediracted imagesdependon /a,theonlywaytoimprovetheresolutionof theimagesistoincreasethediameter, a ,ofthetube.Thisisone ofthemainreasonsinadditiontolight-gatheringpowerwhythe besttelescopesmustbeverylargeindiameter. self-checkD Whatwouldthisimplyaboutradiotelescopesascomparedwithvisiblelighttelescopes? Answer,p. 109 Double-slitdiractioniseasiertounderstandconceptuallythan single-slitdiraction,butifyoudoadouble-slitdiractionexperimentinreallife,youarelikelytoencounteracomplicatedpattern likegurez/1,ratherthanthesimplerone,2,youwereexpecting. Thisisbecausetheslitsarefairlybigcomparedtothewavelength ofthelightbeingused.Wereallyhavetwodierentdistancesin ourpairofslits: d ,thedistancebetweentheslits,and w ,thewidth ofeachslit.Rememberthatsmallerdistancesontheobjectthe lightdiractsaroundcorrespondtolargerfeaturesofthediraction pattern.Thepattern1thushastwospacingsinit:ashortspacingcorrespondingtothelargedistance d ,andalongspacingthat relatestothesmalldimension w 88 Chapter5WaveOptics

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aa / Lightcouldtakemany differentpathsfromAtoB. z / 1.Adiffractionpatternformedbyarealdoubleslit.Thewidthofeachslitisfairlybigcomparedto thewavelengthofthelight.Thisisarealphoto.2.Thisidealizedpatternisnotlikelytooccurinreallife.Toget it,youwouldneedeachslittobesonarrowthatitswidthwascomparabletothewavelengthofthelight,but that'snotusuallypossible.Thisisnotarealphoto.3.Arealphotoofasingle-slitdiffractionpatterncausedby aslitwhosewidthisthesameasthewidthsoftheslitsusedtomakethetoppattern. DiscussionQuestion A Whyisitopticallyimpossibleforbacteriatoevolveeyesthatuse visiblelighttoformimages? 5.8 R ? ThePrincipleofLeastTime Insectionsection1.5andsection4.4,wesawhowintheraymodel oflight,bothrefractionandreectioncanbedescribedinanelegantandbeautifulwaybyasingleprinciple,theprincipleofleast time.Wecannowjustifytheprincipleofleasttimebasedonthe wavemodeloflight.Consideranexampleinvolvingreection,aa. StartingatpointA,Huygens'principleforwavestellsusthatwe canthinkofthewaveasspreadingoutinalldirections.Supposewe imagineallthepossiblewaysthataraycouldtravelfromAtoB. Weshowthisbydrawing25possiblepaths,ofwhichthecentralone istheshortest.Sincetheprincipleofleasttimeconnectsthewave modeltotheraymodel,weshouldexpecttogetthemostaccurate resultswhenthewavelengthismuchshorterthanthedistancesinvolved|forthesakeofthisnumericalexample,let'ssaythata wavelengthis1/10oftheshortestreectedpathfromAtoB.The table,2,showsthedistancestraveledbythe25rays. Notehowsimilararethedistancestraveledbythegroupof7 rays,indicatedwithabracket,thatcomeclosesttoobeyingthe principleofleasttime.Ifwethinkofeachoneasawave,then all7areagainnearlyinphaseatpointB.However,theraysthat arefartherfromsatisfyingtheprincipleofleasttimeshowmore rapidlychangingdistances;onreunitingatpointB,theirphases arearandomjumble,andtheywillverynearlycanceleachother out.Thus,almostnoneofthewaveenergydeliveredtopointB Section5.8 R ? ThePrincipleofLeastTime 89

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goesbytheselongerpaths.Physicallywend,forinstance,that awavepulseemittedatAisobservedatBafteratimeinterval correspondingverynearlytotheshortestpossiblepath,andthe pulseisnotverysmearedout"whenitgetsthere.Theshorter thewavelengthcomparedtothedimensionsofthegure,themore accuratetheseapproximatestatementsbecome. Insteadofdrawinganitenumberofrays,such25,whathappensifwethinkoftheangle, ,ofemissionoftherayasacontinuouslyvaryingvariable?Minimizingthedistance L requires d L d =0. Because L ischangingslowlyinthevicinityoftheanglethat satisestheprincipleofleasttime,alltheraysthatcomeoutclose tothisanglehaveverynearlythesame L ,andremainverynearlyin phasewhentheyreachB.Thisisthebasicreasonwhythediscrete table,aa/2,turnedouttohaveagroupofraysthatalltraveled nearlythesamedistance. Asdiscussedinsection1.5,theprincipleofleasttimeisreallya principleofleast orgreatest time.Thismakesperfectsense,since d L= d =0caningeneraldescribeeitheraminimumoramaximum Theprincipleofleasttimeisverygeneral.Itdoesnotapplyjust torefractionandreection|itcanevenbeusedtoprovethatlight raystravelinastraightlinethroughemptyspace,withouttaking detours!ThisgeneralapproachtowavemotionwasusedbyRichard Feynman,oneofthepioneerswhointhe1950'sreconciledquantum mechanicswithrelativitybook6inthisseries.Averyreadable explanationisgiveninabookFeynmanwroteforlaypeople,QED: TheStrangeTheoryofLightandMatter. 90 Chapter5WaveOptics

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Summary SelectedVocabulary diraction....thebehaviorofawavewhenitencountersan obstacleoranonuniformityinitsmedium; ingeneral,diractioncausesawavetobend aroundobstaclesandmakepatternsofstrong andweakwavesradiatingoutbeyondtheobstacle. coherent.....alightwavewhosepartsareallinphasewith eachother OtherTerminologyandNotation wavelets.....theripplesinHuygens'principle Summary Waveopticsisamoregeneraltheoryoflightthanrayoptics. Whenlightinteractswithmaterialobjectsthataremuchlargerthen onewavelengthofthelight,theraymodeloflightisapproximately correct,butinothercasesthewavemodelisrequired. Huygens'principlestatesthat,givenawavefrontatonemoment intime,thefuturebehaviorofthewavecanbefoundbybreaking thewavefrontupintoalargenumberofsmall,side-by-sidewave peaks,eachofwhichthencreatesapatternofcircularorspherical ripples.Asthesesetsofripplesaddtogether,thewaveevolvesand movesthroughspace.SinceHuygens'principleisapurelygeometricalconstruction,diractioneectsobeyasimplescalingrule:the behaviorisunchangedifthewavelengthandthedimensionsofthe diractingobjectsarebothscaledupordownbythesamefactor.If wewishtopredicttheanglesatwhichvariousfeaturesofthediractionpatternradiateout,scalingrequiresthattheseanglesdepend onlyontheunitlessratio /d,where d isthesizeofsomefeatureof thediractingobject. Double-slitdiractioniseasilyanalyzedusingHuygens'principleiftheslitsarenarrowerthanonewavelength.Weneedonly constructtwosetsofripples,onespreadingoutfromeachslit.The anglesofthemaximabrightestpointsinthebrightfringesand minimadarkestpointsinthedarkfringesaregivenbytheequation d = sin m where d isthecenter-to-centerspacingoftheslits,and m isan integeratamaximumoranintegerplus1/2ataminimum. Ifsomefeatureofadiractingobjectisrepeated,thediraction fringesremaininthesameplaces,butbecomenarrowerwitheach repetition.Byrepeatingadouble-slitpatternhundredsorthousandsoftimes,weobtainadiractiongrating. Asingleslitcanproducediractionfringesifitislargerthan Summary 91

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onewavelength.Manypracticalinstancesofdiractioncanbeinterpretedassingle-slitdiraction,e.g.,diractionintelescopes.The mainthingtorealizeaboutsingle-slitdiractionisthatitexhibits thesamekindofrelationshipbetween d ,andanglesoffringesas inanyothertypeofdiraction. 92 Chapter5WaveOptics

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Whywouldblueorvioletlightbethebestformicroscopy? 2 MatchgratingsA-Cwiththediractionpatterns1-3thatthey produce.Explain. 3 Thebeamofalaserpassesthroughadiractiongrating,fans out,andilluminatesawallthatisperpendiculartotheoriginal beam,lyingatadistanceof2.0mfromthegrating.Thebeam isproducedbyahelium-neonlaser,andhasawavelengthof694.3 nm.Thegratinghas2000linespercentimeter.aWhatisthe distanceonthewallbetweenthecentralmaximumandthemaxima immediatelytoitsrightandleft?bHowmuchdoesyouranswer changewhenyouusethesmall-angleapproximations sin tan ? p 4 Whenwhitelightpassesthroughadiractiongrating,what isthesmallestvalueof m forwhichthevisiblespectrumoforder m overlapsthenextone,oforder m +1?Thevisiblespectrumruns fromabout400nmtoabout700nm. Problems 93

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5 Ultrasound,i.e.,soundwaveswithfrequenciestoohightobe audible,canbeusedforimagingfetusesinthewomborforbreakingupkidneystonessothattheycanbeeliminatedbythebody. Considerthelatterapplication.Lensescanbebuilttofocussound waves,butbecausethewavelengthofthesoundisnotallthatsmall comparedtothediameterofthelens,thesoundwillnotbeconcentratedexactlyatthegeometricalfocalpoint.Instead,adiraction patternwillbecreatedwithanintensecentralspotsurroundedby fainterrings.About85%ofthepowerisconcentratedwithinthe centralspot.Theangleoftherstminimumsurroundingthecentralspotisgivenbysin = =b ,where b isthediameterofthelens. Thisissimilartothecorrespondingequationforasingleslit,but withafactorof1.22infrontwhicharisesfromthecircularshapeof theaperture.Letthedistancefromthelenstothepatient'skidney stonebe L =20cm.Youwillwant f> 20kHz,sothatthesound isinaudible.Findvaluesof b and f thatwouldresultinausable design,wherethecentralspotissmallenoughtoliewithinakidney stone1cmindiameter. 6 Forstarimagessuchastheonesingurex,estimatethe angularwidthofthediractionspotduetodiractionatthemouth ofthetelescope.Assumeatelescopewithadiameterof10meters thelargestcurrentlyinexistence,andlightwithawavelengthin themiddleofthevisiblerange.Comparewiththeactualangular sizeofastarofdiameter10 9 mseenfromadistanceof10 17 m. Whatdoesthistellyou? 7 Underwhatcircumstancescouldonegetamathematically undenedresultbysolvingthedouble-slitdiractionequationfor ? Giveaphysicalinterpretationofwhatwouldactuallybeobserved. 8 Whenultrasoundisusedformedicalimaging,thefrequency maybeashighas5-20MHz.Anothermedicalapplicationofultrasoundisfortherapeuticheatingoftissuesinsidethebody;here,the frequencyistypically1-3MHz.Whatfundamentalphysicalreasons couldyousuggestfortheuseofhigherfrequenciesforimaging? 94 Chapter5WaveOptics

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9 Thegurebelowshowstwodiractionpatterns,bothmade withthesamewavelengthofredlight.aWhattypeofslitsmade thepatterns?Isitasingleslit,doubleslits,orsomethingelse? Explain.bComparethedimensionsoftheslitsusedtomakethe topandbottompattern.Giveanumericalratio,andstatewhich waytheratiois,i.e.,whichslitpatternwasthelargerone.Explain. 10 Thegurebelowshowstwodiractionpatterns.Thetopone wasmadewithyellowlight,andthebottomonewithred.Could theslitsusedtomakethetwopatternshavebeenthesame? Problems 95

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Problems12and13. 11 Thegurebelowshowsthreediractionpatterns.Allwere madeunderidenticalconditions,exceptthatadierentsetofdouble slitswasusedforeachone.Theslitsusedtomakethetoppattern hadacenter-to-centerseparation d =0.50mm,andeachslitwas w =0.04mmwide.aDetermine d and w fortheslitsusedto makethepatterninthemiddle.bDothesamefortheslitsused tomakethebottompattern. 12 Thegureshowsadiractionpatternmadebyadoubleslit, alongwithanimageofametersticktoshowthescale.Theslits were146cmawayfromthescreenonwhichthediractionpattern wasprojected.Thespacingoftheslitswas0.050mm.Whatwas thewavelengthofthelight? 13 Thegureshowsadiractionpatternmadebyadouble slit,alongwithanimageofametersticktoshowthescale.Sketch thediractionpatternfromthegureonyourpaper.Nowconsider thefourvariablesintheequation =d =sin =m .Whichofthese arethesameforallvefringes,andwhicharedierentforeach fringe?Whichvariablewouldyounaturallyuseinordertolabel whichfringewaswhich?Labelthefringesonyoursketchusingthe valuesofthatvariable. 96 Chapter5WaveOptics

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Appendix1:Exercises Exercise2A:ExploringImagesWithaCurvedMirror Equipment: concavemirrorswithdeepcurvature concavemirrorswithgentlecurvature convexmirrors 1.Obtainacurvedmirrorfromyourinstructor.Ifitissilveredonbothsides,makesureyou're workingwiththeconcaveside,whichbendslightraysinward.Lookatyourownfaceinthe mirror.Nowchangethedistancebetweenyourfaceandthemirror,andseewhathappens. Explorethefullrangeofpossibledistancesbetweenyourfaceandthemirror. Intheseobservationsyou'vebeenchangingtwovariablesatonce:thedistancebetweenthe objectyourfaceandthemirror,andthedistancefromthemirrortoyoureye.Ingeneral, scienticexperimentsbecomeeasiertointerpretifwepracticeisolationofvariables,i.e.,only changeonevariablewhilekeepingalltheothersconstant.Inparts2and3you'llformanimage ofanobjectthat'snotyourface,sothatyoucanhaveindependentcontroloftheobjectdistance andthepointofview. 2.Withthemirrorheldfarawayfromyou,observetheimageofsomethingbehindyou,over yourshoulder.Nowbringyoureyecloserandclosertothemirror.Canyouseetheimagewith youreyeveryclosetothemirror?Seeifyoucanexplainyourobservationbydrawingaray diagram. ||||||{ > turnpage

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3.Nowimaginethefollowingnewsituation,but don'tactuallydoityet .Supposeyoulaythe mirrorface-uponapieceoftissuepaper,putyourngerafewcmabovethemirror,andlook attheimageofyournger.Asinpart2,youcanbringyoureyecloserandclosertothemirror. Willyoubeabletoseetheimagewithyoureyeveryclosetothemirror?Drawaraydiagram tohelpyoupredictwhatyouwillobserve. Prediction: Nowtestyourprediction.Ifyourpredictionwasincorrect,seeifyoucangureoutwhatwent wrong,oraskyourinstructorforhelp. 4.Forparts4and5,it'smoreconvenienttouseconcavemirrorsthataremoregentlycurved; obtainonefromyourinstructor.Laythemirroronthetissuepaper,anduseittocreatean imageoftheoverheadlightsonapieceofpaperaboveitandalittleototheside.Whatdo youhavetodoinordertomaketheimageclear?Canyouexplainthisobservationusingaray diagram? ||||||{ > turnpage 98 Appendix1:Exercises

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5.Nowimaginethefollowingexperiment,but don'tdoityet .Whatwillhappentotheimage onthepaperifyoucoverhalfofthemirrorwithyourhand? Prediction: Testyourprediction.Ifyourpredictionwasincorrect,canyouexplainwhathappened? 6.Nowimagineforminganimagewithaconvexmirroronethatbulgesoutward,andthat thereforebendslightraysawayfromthecentralaxisi.e.,isdiverging.Drawatypicalray diagram. Istheimagereal,orvirtual?Willtherebemorethanonetypeofimage? Prediction: Testyourprediction. 99

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Exercise3A:ObjectandImageDistances Equipment: opticalbenches convergingmirrors illuminatedobjects 1.Setuptheopticalbenchwiththemirroratzeroonthecentimeterscale.Setupthe illuminatedobjectonthebenchaswell. 2.Eachgroupwilllocatetheimagefortheirownvalueoftheobjectdistance,byndingwhere apieceofpaperhastobeplacedinordertoseetheimageonit.Theinstructorwilldoone pointaswell.Notethatyouwillhavetotiltthemirroralittlesothatthepaperonwhichyou projecttheimagedoesn'tblockthelightfromtheilluminatedobject. Istheimagerealorvirtual?Howdoyouknow?Isitinverted,oruninverted? Drawaraydiagram. 3.Measuretheimagedistanceandwriteyourresultinthetableontheboard.Dothesamefor themagnication. 4.Whatdoyounoticeaboutthetrendofthedataontheboard?Drawasecondraydiagram withadierentobjectdistance,andshowwhythismakessense.Sometipsfordoingthis correctly:Forsimplicity,usethepointontheobjectthatisonthemirror'saxis.You needtotracetworaystolocatetheimage.Tosavework,don'tjustdotworaysatrandom angles.Youcaneitherusetheon-axisrayasoneray,ordotworaysthatcomeoatthesame angle,oneaboveandonebelowtheaxis.Whereeachrayhitsthemirror,drawthenormal line,andmakesuretherayisatequalanglesonbothsidesofthenormal. 5.Wewillndthemirror'sfocallengthfromtheinstructor'sdata-point.Then,usingthisfocal length,calculateatheoreticalpredictionoftheimagedistance,andwriteitontheboardnext totheexperimentallydeterminedimagedistance. 100 Appendix1:Exercises

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Exercise4A:Howstrongareyourglasses? ThisexercisewascreatedbyDanMacIsaac. Equipment: eyeglasses outbendinglensesforstudentswhodon'twearglasses,orwhouseinbendingglasses rulersandmetersticks scratchpaper markingpens Mostpeoplewhowearglasseshaveglasseswhoselensesareoutbending,whichallowsthemto focusonobjectsfaraway.Suchalenscannotformarealimage,soitsfocallengthcannotbe measuredaseasilyasthatofaninbendinglens.Inthisexerciseyouwilldeterminethefocal lengthofyourownglassesbytakingthemo,holdingthematadistancefromyourface,and lookingthroughthematasetofparallellinesonapieceofpaper.Thelineswillbereduced thelens'smagnicationislessthanone,andbyadjustingthedistancebetweenthelensand thepaper,youcanmakethemagnicationequal1/2exactly,sothattwospacesbetweenlines asseenthroughthelenstintoonespaceasseensimultaneouslytothesideofthelens.This objectdistancecanbeusedinordertondthefocallengthofthelens. 1.Useamarkertodrawthreeevenlyspacedparallellinesonthepaper.Aspacingofafew cmworkswell. 2.Doesthistechniquereallymeasuremagnicationordoesitmeasureangularmagnication? Whatcanyoudoinyourexperimentinordertomakethesetwoquantitiesnearlythesame,so themathissimpler? 3.Beforetakinganynumericaldata,usealgebratondthefocallengthofthelensintermsof d o ,theobjectdistancethatresultsinamagnicationof1/2. 4.Measuretheobjectdistancethatresultsinamagnicationof1/2,anddeterminethefocal lengthofyourlens. 101

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Exercise5A:Double-SourceInterference 1.Twosourcesseparatedbyadistance d =2cmmakecircularrippleswithawavelengthof =1cm.Onapieceofpaper,makealife-sizedrawingofthetwosourcesinthedefaultsetup, andlocatethefollowingpoints: A.Thepointthatis10wavelengthsfromsource#1and10wavelengthsfromsource#2. B.Thepointthatis10.5wavelengthsfrom#1and10.5from#2. C.Thepointthatis11wavelengthsfrom#1and11from#2. D.Thepointthatis10wavelengthsfrom#1and10.5from#2. E.Thepointthatis11wavelengthsfrom#1and11.5from#2. F.Thepointthatis10wavelengthsfrom#1and11from#2. G.Thepointthatis11wavelengthsfrom#1and12from#2. Youcandothiseitherusingacompassorbyputtingthenextpageunderyourpaperand tracing.Itisnotnecessarytotraceallthearcscompletely,anddoingsoisunnecessarilytimeconsuming;youcanfairlyeasilyestimatewherethesepointswouldlie,andjusttracearcslong enoughtondtherelevantintersections. Whatdothesepointscorrespondtointherealwavepattern? 2.Makeafreshcopyofyourdrawing,showingonlypointFandthetwosources,whichforma long,skinnytriangle.Nowsupposeyouweretochangethesetupbydoubling d ,whileleaving thesame.It'seasiesttounderstandwhat'shappeningonthedrawingifyoumovebothsources outward,keepingthecenterxed.Basedonyourdrawing,whatwillhappentothepositionof pointFwhenyoudouble d ?Measureitsanglewithaprotractor. 3.Inpart2,yousawtheeectofdoubling d whileleaving thesame.Nowwhatdoyouthink wouldhappentoyouranglesif,startingfromthestandardsetup,youdoubled whileleaving d thesame? 4.Suppose wasamillionthofacentimeter,while d wasstillasinthestandardsetup.What wouldhappentotheangles?Whatdoesthistellyouaboutobservingdiractionoflight? 102 Appendix1:Exercises

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Exercise5B:Single-slitdiffraction Equipment: rulers computerwithwebbrowser Thefollowingpageisadiagramofasingleslitandascreenontowhichitsdiractionpattern isprojected.Theclasswillmakeanumericalpredictionoftheintensityofthepatternatthe dierentpointsonthescreen.Eachgroupwillberesponsibleforcalculatingtheintensityat oneofthepoints.Either11groupsorsixwillworknicely{inthelattercase,onlypointsa, c,e,g,i,andkareused.Theideaistobreakupthewavefrontinthemouthoftheslitinto nineparts,eachofwhichisassumedtoradiatesemicircularripplesasinHuygens'principle. Thewavelengthofthewaveis1cm,andweassumeforsimplicitythateachsetofrippleshas anamplitudeof1unitwhenitreachesthescreen. 1.Forsimplicity,let'simaginethatwewereonlytousetwosetsofripplesratherthannine. Youcouldmeasurethedistancefromeachofthetwopointsinsidetheslittoyourpointon thescreen.Supposethedistanceswereboth25.0cm.Whatwouldbetheamplitudeofthe superimposedwavesatthispointonthescreen? Supposeonedistancewas24.0cmandtheotherwas25.0cm.Whatwouldhappen? Whatifonewas24.0cmandtheotherwas26.0cm? Whatifonewas24.5cmandtheotherwas25.0cm? Ingeneral,whatcombinationsofdistanceswillleadtocompletelydestructiveandcompletely constructiveinterference? Canyouestimatetheanswerinthecasewherethedistancesare24.7and25.0cm? 2.Althoughitispossibletocalculatemathematicallytheamplitudeofthesinewavethatresults fromsuperimposingtwosinewaveswithanarbitraryphasedierencebetweenthem,thealgebra isratherlaborious,anditbecomeevenmoretediouswhenwehavemorethantwowavestosuperimpose.Instead,onecansimplyuseacomputerspreadsheetorsomeothercomputerprogramto addupthesinewavesnumericallyataseriesofpointscoveringonecompletecycle.Thisiswhat wewillactuallydo.Youjustneedtoentertherelevantdataintothecomputer,thenexaminethe resultsandpickotheamplitudefromtheresultinglistofnumbers.Youcanrunthesoftware throughawebinterfaceat http://lightandmatter.com/cgi-bin/diffraction1.cgi 3.Measureallninedistancestoyourgroup'spointonthescreen,andwritethemontheboard -thatwayeveryonecanseeeveryoneelse'sdata,andtheclasscantrytomakesenseofwhythe resultscameoutthewaytheydid.Determinetheamplitudeofthecombinedwave,andwrite itontheboardaswell. Theclasswilldiscusswhytheresultscameoutthewaytheydid. 104 Appendix1:Exercises

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Exercise5C:DiffractionofLight Equipment: slitpatterns,lasers,straight-lamentbulbs station1 Youhaveamaskwithabunchofdierentdoubleslitscutoutofit.Thevaluesofwanddare asfollows: patternAw=0.04mmd=.250mm patternBw=0.04mmd=.500mm patternCw=0.08mmd=.250mm patternDw=0.08mmd=.500mm Predicthowthepatternswilllookdierent,andtestyourprediction.Theeasiestwaytoget thelasertopointatdierentsetsofslitsistostickfoldeduppiecesofpaperinonesideorthe otheroftheholders. station2 Thisisjustlikestation1,butwithsingleslits: patternAw=0.02mm patternBw=0.04mm patternCw=0.08mm patternDw=0.16mm Predictwhatwillhappen,andtestyourpredictions.Ifyouhavetime,checktheactualnumerical ratiosofthewvaluesagainsttheratiosofthesizesofthediractionpatterns station3 Thisislikestation1,buttheonlydierenceamongthesetsofslitsishowmanyslitsthereare: patternAdoubleslit patternB3slits patternC4slits patternD5slits station4 Holdthediractiongratinguptoyoureye,andlookthroughitatthestraight-lamentlight bulb.Ifyouorientthegratingcorrectly,youshouldbeabletoseethe m =1and m = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 diractionpatternsotheleftandright.Ifyouhaveitorientedthewrongway,they'llbeabove andbelowthebulbinstead,whichisinconvenientbecausethebulb'slamentisvertical.Where isthe m =0fringe?Canyousee m =2,etc.? Station5hasthesameequipmentasstation4.Ifyou'reassignedtostation5rst,youshould actuallydoactivity4rst,becauseit'seasier. station5 Usethetransformertoincreaseanddecreasethevoltageacrossthebulb.Thisallowsyouto controlthelament'stemperature.Sketchgraphsofintensityasafunctionofwavelengthfor varioustemperatures.Theinabilityofthewavemodeloflighttoexplainthemathematical shapesofthesecurveswashistoricallyoneofthereasonsforcreatinganewmodel,inwhich lightisbothaparticleandawave. 106 Appendix1:Exercises

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Appendix2:PhotoCredits Exceptasspecicallynotedbeloworinaparentheticalcreditinthecaptionofagure,alltheillustrationsin thisbookareundermyowncopyright,andarecopyleftlicensedunderthesamelicenseastherestofthebook. Insomecasesit'sclearfromthedatethatthegureispublicdomain,butIdon'tknowthenameoftheartist orphotographer;Iwouldbegratefultoanyonewhocouldhelpmetogivepropercredit.Ihaveassumedthat imagesthatcomefromU.S.governmentwebpagesarecopyright-free,sinceproductsoffederalagenciesfallinto thepublicdomain.I'veincludedsomepublic-domainpaintings;photographicreproductionsofthemarenot copyrightableintheU.S.BridgemanArtLibrary,Ltd.v.CorelCorp.,36F.Supp.2d191,S.D.N.Y.1999. WhenPSSCPhysics"isgivenasacredit,itindicatesthatthegureisfromthersteditionofthetextbook entitledPhysics,bythePhysicalScienceStudyCommittee.Theearlyeditionsofthesebooksneverhadtheir copyrightsrenewed,andarenowthereforeinthepublicdomain.Thereisalsoablanketpermissiongivenin thelaterPSSCCollegePhysicsedition,whichstatesonthecopyrightpagethatThematerialstakenfromthe originalandsecondeditionsandtheAdvancedTopicsofPSSCPHYSICSincludedinthistextwillbeavailable toallpublishersforuseinEnglishafterDecember31,1970,andintranslationsafterDecember31,1975." CreditstoMillikanandGalerefertothetextbooksPracticalPhysicsandElementsofPhysics. Botharepublicdomain.The1927versiondidnothaveitscopyrightrenewed.Sinceispossiblethatsomeof theillustrationsinthe1927versionhadtheircopyrightsrenewedandarestillundercopyright,Ihaveonlyused themwhenitwasclearthattheywereoriginallytakenfrompublicdomainsources. Inafewcases,Ihavemadeuseofimagesunderthefairusedoctrine.However,Iamnotalawyer,andthelaws onfairusearevague,soyoushouldnotassumethatit'slegalforyoutousetheseimages.Inparticular,fairuse lawmaygiveyoulessleewaythanitgivesme,becauseI'musingtheimagesforeducationalpurposes,andgiving thebookawayforfree.Likewise,ifthephotocreditsayscourtesyof...,"thatmeansthecopyrightownergave mepermissiontouseit,butthatdoesn'tmeanyouhavepermissiontouseit. PhotocreditstoNEIrefertophotosfromtheNationalEyeInstitute,partoftheNationalInstitutesofHealth, http://www.nei.nih.gov/photo/.Itemsherearenotcopyrighted.However,wedoaskthatyoucreditasfollows: NationalEyeInstitute,NationalInstitutesofHealthexceptwhereindicatedotherwise." Cover Photocollage: Thephotooftheroseisbytheauthor.Thecross-sectionofthehumaneyeisfromNEI.The photocollageisbytheauthor. Contents X-rayofhand: PabloAlbertoSalgueroQuiles,WikimediaCommons, GFDL1.2. Contents Insect'seye: WikimediaCommons,GFDL1.2,userReytan. Contents Man'seye: NEI. Contents Mirrorball: Photobytheauthor. Contents Soapbubble: WikimediaCommons,GFDL/CC-BY-SA, userTagishsimon. Contents Radiotelescopes: WikimediaCommons,GFDL1.2,userHajor. 11 Raysofsunlight: WikipediauserPiccoloNamek,GFDL1.2. 14 JupiterandIo: NASA/JPL/UniversityofArizona. 29 Narcissus: Caravaggio,ca.1598. 31 Praxinoscope: ThomasB.Greenslade,Jr.. 36 Flower: Basedonaphoto byWikimediaCommonsuserFir0002,GFDL1.2. 23 Ray-tracedimage: GillesTran,WikimediaCommons, publicdomain. 36 Moon: Wikimediacommonsimage. 53 Fish-eyelens: MartinDurrschnabel,CC-BY-SA. 54 Hubblespacetelescope: NASA,publicdomain. 59 Flatworm: CC-BY-SA,AlejandroSanchezAlvarado, Planaria.neuro.utah.edu. 59 Nautilus: CC-BY-SA,WikimediaCommonsuserOpencage,opencage.info. 59 Humaneye: JoaoEstevaoA.deFreitas,Therearenousagerestrictionsforthisphoto". 60 Cross-section ofeye: NEI. 60 Eye'sanatomy: Afterapublic-domaindrawingfromNEI. 66 Ulcer: WikipediauserAspersions,GFDL1.2. 64 Waterwaverefracting: OriginalphotofromPSSC. 74 Binoculars: Wikimedia commons,GFDL. 74 Porroprisms: RedrawnfromagurebyWikipediauserDrBob,GFDL. 77 Pleiades: NASA/ESA/AURA/Caltech,publicdomain. 78 Diractionofwaterwaves: AssembledfromphotosinPSSC. 80 Huygens: Contemporarypainting?. 78 Counterfactuallackofdiractionofwaterwaves: Assembledfrom photosinPSSC. 79 Scalingofdiraction: AssembledfromphotosinPSSC. 81 Diractionofwaterwaves: AssembledfromphotosinPSSC. 82 Young: WikimediaCommons,AfteraportraitbySirThomasLawrence, From:ArthurShuster&ArthurE.Shipley:Britain'sHeritageofScience.London,1917". 78 Diractionof waterwaves: AssembledfromphotosinPSSC. 87 Single-slitdiractionofwaterwaves: PSSC. 87 Simulationofasingleslitusingthreesources: PSSC. 88 Pleiades: NASA/ESA/AURA/Caltech,publicdomain. 88 Radiotelescope: WikipediauserHajor,GFDLandCC-BY-SA.

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Appendix3:HintsandSolutions AnswerstoSelf-Checks AnswerstoSelf-ChecksforChapter1 Page22,self-checkA: Only1iscorrect.Ifyoudrawthenormalthatbisectsthesolidray,it alsobisectsthedashedray. AnswerstoSelf-ChecksforChapter2 Page30,self-checkA: Youshouldhavefoundfromyourraydiagramthatanimageisstill formed,andithassimplymoveddownthesamedistanceastherealface.However,thisnew imagewouldonlybevisiblefromhighup,andthepersoncannolongerseehisownimage. Page35,self-checkB: Increasingthedistancefromthefacetothemirrorhasdecreasedthe distancefromtheimagetothemirror.Thisistheoppositeofwhathappenedwiththevirtual image. AnswerstoSelf-ChecksforChapter3 Page48,self-checkA: Atthetopofthegraph, d i approachesinnitywhen d o approaches f Interpretation:theraysjustbarelyconvergetotherightofthemirror. Onthefarright, d i approaches f as d o approachesinnity;thisisthedenitionofthefocal length. Atthebottom, d i approachesnegativeinnitywhen d o approaches f fromtheotherside. Interpretation:theraysdon'tquiteconvergeontherightsideofthemirror,sotheyappearto havecomefromavirtualimagepointveryfartotheleftofthemirror. AnswerstoSelf-ChecksforChapter4 Page63,self-checkA: If n 1 and n 2 areequal,Snell'slawbecomessin 1 =sin 2 ,which implies 1 = 2 ,sincebothanglesarebetween0and90 .Thegraphwouldbeastraightline alongthediagonalofthegraph.Thegraphisfarthestfromthediagonalwhentheangles arelarge,i.e.,whentheraystrikestheinterfaceatagrazingangle. Page67,self-checkB: In1,therayscrosstheimage,soit'sreal.In2,theraysonly appeartohavecomefromtheimagepoint,sotheimageisvirtual.Araysisalwayscloser tothenormalinthemediumwiththehigherindexofrefraction.Therstleftturnmakesthe rayclosertothenormal,whichiswhatshouldhappeninglass.Thesecondleftturnmakesthe rayfartherfromthenormal,andthat'swhatshouldhappeninair.Takethetopmostrayas anexample.Itwillstilltaketworightturns,butsinceit'senteringthelensatasteeperangle, itwillalsoleaveatasteeperangle.Tracingbackwardtotheimage,thesteeperlineswillmeet closertothelens. AnswerstoSelf-ChecksforChapter5 Page81,self-checkA: Itwouldhavetohaveawavelengthontheorderofcentimetersor

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meters,thesamedistancescaleasthatofyourbody.Thesewouldbemicrowavesorradio waves.ThiseectcaneasilybenoticedwhenapersonaectsaTV'sreceptionbystanding neartheantenna.Noneofthiscontradictsthecorrespondenceprinciple,whichonlystatesthat thewavemodelmustagreewiththeraymodelwhentheraymodelisapplicable.Theraymodel isnotapplicableherebecause =d isontheorderof1. Page83,self-checkB: Atthispoint,bothwaveswouldhavetravelednineandahalfwavelengths.Theywouldbothbeatanegativeextreme,sotherewouldbeconstructiveinterference. Page87,self-checkC: Judgingbythedistancefromonebrightwavecresttothenext,the wavelengthappearstobeabout2/3or3/4asgreatasthewidthoftheslit. Page88,self-checkD: Sincethewavelengthsofradiowavesarethousandsoftimeslonger, diractioncausestheresolutionofaradiotelescopetobethousandsoftimesworse,allother thingsbeingequal.Tocompensateforthewavelength,it'sdesirabletomakethetelescopevery large,asingureyonpage88. 109

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SolutionstoSelectedHomeworkProblems SolutionsforChapter3 Page57,problem2: Seetheraydiagrambelow.Decreasing o decreases i ,sotheequation f = i + o musthaveoppositesignsontheright.Since o isbiggerthan i ,theonlyway togetapositive f isifthesignsare f = )]TJ/F20 10.9091 Tf 8.485 0 Td [( i + o .Thisgives1 =f = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 =d i +1 =d o Page58,problem10: aTheobjectdistanceislessthanthefocallength,sotheimageis virtual:becausetheobjectissoclose,theconeofraysisdivergingtoostronglyforthemirrorto bringitbacktoafocus.bAtanobjectdistanceof30cm,it'sclearlygoingtobereal.With theobjectdistanceof20cm,we'rerightatthecrossing-pointbetweenrealandvirtual.For thisobjectposition,thereectedrayswillbeparallel.Wecouldconsiderthistobeanimage atinnity.c,dAdivergingmirrorcanonlymakevirtualimages. SolutionsforChapter4 Page73,problem13: Since d o ismuchgreaterthan d i ,thelens-lmdistance d i isessentially thesameas f .aSplittingthetriangleinsidethecameraintotworighttriangles,straightforwardtrigonometrygives =2tan )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 w 2 f fortheeldofview.Thiscomesouttobe39 and64 forthetwolenses.bForsmallangles, thetangentisapproximatelythesameastheangleitself,providedwemeasureeverythingin radians.Theequationabovethensimpliesto = w f Theresultsforthetwolensesare.70rad=40 ,and1.25rad=72 .Thisisadecent approximation. cWiththe28-mmlens,whichisclosertothelm,theentireeldofviewwehadwiththe 50-mmlensisnowconnedtoasmallpartofthelm.Usingoursmall-angleapproximation = w=f ,theamountoflightcontainedwithinthesameangularwidth isnowstrikingapiece ofthelmwhoselineardimensionsaresmallerbytheratio28/50.Areadependsonthesquare ofthelineardimensions,soallotherthingsbeingequal,thelmwouldnowbeoverexposedby afactorof = 28 2 =3.2.Tocompensate,weneedtoshortentheexposurebyafactorof3.2. 110 Appendix3:HintsandSolutions

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Index aberration,51 absorption,15 angularmagnication,36 Bohr Niels,80 brightnessoflight,17 Bush,George,59 color,65 concave dened,39 converging,33 convex dened,39 correspondenceprinciple,80 diraction dened,78 double-slit,82 fringe,79 scalingof,79 single-slit,87 diractiongrating,87 diusereection,16 diopter,47 diverging,39 double-slitdiraction,82 EmpedoclesofAcragas,12 endoscope,66 evolution,59 eye evolutionof,59 human,61 Fermat'sprinciple,24 atworm,60 focalangle,45 focallength,46 focalpoint,46 fringe diraction,79 Galileo,13 Hertz,Heinrich Heinrich,82 Huygens'principle,81 images formedbycurvedmirrors,33 formedbyplanemirrors,30 locationof,44 ofimages,35 real,34 virtual,30 incoherentlight,79 indexofrefraction dened,62 relatedtospeedoflight,63 Io,14 Jupiter,14 lens,67 lensmaker'sequation,68 light absorptionof,15 brightnessof,17 particlemodelof,17 raymodelof,17 speedof,13 wavemodelof,17 magnication angular,36 byaconvergingmirror,33 negative,55 Maxwell,JamesClerk,82 mirror converging,44 mollusc,60 Moses,59 nautilus,60 Newton,Isaac,35,81 particlemodeloflight,17,81 Porroprism,75 praxinoscope,31

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prism Porro,75 Pythagoras,12 raydiagrams,19 raymodeloflight,17,81 reection diuse,16 specular,20 refraction andcolor,65 dened,60 repetitionofdiractingobjects,86 retina,35 reversibility,22 Roemer,14 single-slit diraction,87 Snell'slaw,62 derivationof,64 mechanicalmodelof,64 Squid,60 telescope,35,88 timereversal,22 totalinternalreection,66 vision,12 wavemodeloflight,17,82 Wigner,Eugene,43 Young,Thomas,82 112 Index

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Index 113

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UsefulData MetricPrexes M-mega-10 6 k-kilo-10 3 m-milli-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(3 -Greekmumicro-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(6 n-nano-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(9 p-pico-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(12 f-femto-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(15 Centi-,10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(2 ,isusedonlyinthecentimeter. NotationandUnits quantityunitsymbol distancemeter,m x x timesecond,s t t masskilogram,kg m densitykg = m 3 velocitym/s v accelerationm = s 2 a forceN=kg m = s 2 F pressurePa=1N = m 2 P energyJ=kg m 2 = s 2 E powerW=1J = s P momentumkg m = s p periods T wavelengthm frequencys )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 orHz f focallengthm f magnicationunitless M indexofrefractionunitless n FundamentalConstants gravitationalconstant G =6.67 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(11 N m 2 = kg 2 Coulombconstant k =8.99 10 9 N m 2 = C 2 quantumofcharge e =1.60 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(19 C speedoflight c =3.00 10 8 m/s Conversions Nonmetricunitsintermsofmetricones: 1inch=25.4mmbydenition 1pound-force=4.5newtonsofforce kg g =2.2pounds-force 1scienticcalorie=4.18J 1kcal=4.18 10 3 J 1gallon=3.78 10 3 cm 3 1horsepower=746W Whenspeakingoffoodenergy,thewordCalorie"isused tomean1kcal,i.e.,1000calories.Inwriting,thecapitalC maybeusedtoindicate1Calorie=1000calories. RelationshipsamongU.S.units: 1footft=12inches 1yardyd=3feet 1milemi=5280feet SomeIndicesof Refraction substanceindexofrefraction vacuum1bydenition air1.0003 water1.3 glass1.5to1.9 diamond2.4 Notethatindicesofrefraction,exceptinvacuum,dependonwavelength.Thesevaluesareaboutrightfor themiddleofthevisiblespectrumyellow. SubatomicParticles particlemasskgradiusfm electron9.109 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(31 0.01 proton1.673 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(27 1.1 neutron1.675 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(27 1.1 Theradiiofprotonsandneutronscanonlybegivenapproximately,sincetheyhavefuzzysurfaces.Forcomparison,a typicalatomisaboutamillionfminradius. 114 Index