AbstractTheseclassnotesaredesignedforuseoftheinstructorandstudentsofthecoursePHYS2020GeneralPhysicsIItaughtbyDrDonaldLuttermoseratEastTennesseeStateUniversityThesenotesmakereferencetotheCollegePhy ID: 206582
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PHYS-2020:GeneralPhysicsIICourseLectureNotesSectionXIIDr.DonaldG.LuttermoserEastTennesseeStateUniversityEdition4.0 AbstractTheseclassnotesaredesignedforuseoftheinstructorandstudentsofthecoursePHYS-2020:GeneralPhysicsIItaughtbyDr.DonaldLuttermoseratEastTennesseeStateUniversity.ThesenotesmakereferencetotheCollegePhysics,10thHybridEdition(2015)textbookbySerwayandVuille. XII.MirrorsandLensesA.PlaneMirrors.1.Imagesformedbyplane(i.e.,\rat)mirrorshavethefollowingproperties:a)Theimageisasfarbehindthemirrorastheobjectisinfront.b)Theimageisunmagni ed,virtual,anderect.2.Imageorientation:a)Erect:Imageisorientedthesameastheobject.b)Inverted:Imageis\ripped180withrespecttotheob-ject.3.Imageclassi cation:a)Real:Imageisonthesamesideofmirrorastheobject=)lightraysactuallypassthroughtheimagepoint.b)Virtual:Imageisontheoppositesideofmirrorfromobject=)lightraysappeartodivergefromimagepoint.4.Imagesizeisdeterminedbythemagni cationofanobjectwhichisgivenbyMimageheightobjectheight=h0h.(XII-1)XII{1 XII{2PHYS-2020:GeneralPhysicsIIjMj�1=)Imageisbiggerthanobject(magni ed).jMj=1=)Imageisunmagni ed(likeaplanemirror).jMj1=)Imageissmallerthanobject(demagni ed).M&#x-7.9;硢0=)Imageiserect.M0=)Imageisinverted.M=0=)Noimageisformed.5.RayTracingRules:a)Imagesformatthepointwhereraysoflightactuallyin-tersect(forrealimages)orfromwhichtheyappeartooriginate(forvirtualimages).b)Forplanemirrors,p(theobjectdistancefromthemirror)=q(theimagedistancefromthemirror)andh=h0.c)Thefollowingdiagramshowshowimagesareconstructedforaplanemirror.i)Onerayrunsparalleltotheopticalaxis(?linetothemirrorsurfaceatthecenterofthemirror)fromtheheadoftheobject(e.g.,Ray1inthe gure).ii)Oneraytravelsfromtheheadtothemirroratthepointwheretheopticalaxisintersectsthemirror(e.g.,Ray2inthe gure). DonaldG.Luttermoser,ETSUXII{3B.SphericalMirrors1.Sphericalmirrorshavetheshapeofasegmentofasphere.a)Concavemirror:Re\rectingsurfaceisonthe\inside"ofthecurvedsurface.b)Convexmirror:Re\rectingsurfaceisonthe\outside"ofthecurvedsurface.2.Constructingtheimage.Considerthefollowingconcavemirror: XII{4PHYS-2020:GeneralPhysicsIIa)Thelinethatisnormaltothemirrorsurfaceattheexactcenteriscalledtheopticalaxisofthemirror.b)Thepointwheretheopticalaxisintersectsthemirrorsur-faceiscalledthevertex(labeled`V'intheprecedingdiagram).c)Point`C'indicatesthepositionofthecenterofcurva-tureofthemirror=)lineCVisequaltotheradiusofcurvature,R,ofthemirror.d)Notethatthelengthsp,q,andRareallmeasuredwithrespecttothevertexposition.Alsonotethattheobjectpositionislabeledwith`O'andtheimagepositionwith`I'intheprecedingdiagram.e)Constructtheimageusingthelawofre\rection:i= r;(XII-2)whereismeasuredwithrespecttothenormalofthemirrorsurface.The`negative'signisintroducedheretonotethatthere\rectedanglesweepsawayfromtheopticalaxisintheopposite`sense'oftheincidentangle.i)AllnormallinesonsphericalconcavemirrorsgothroughcenterofcurvaturepointC(i=r=0)!ii)Nowre\rectarayo thevertexofthemirrorV.f)Usingtrigonometry,weseethath0q=tanr&hp=tani:i)Sincei= r,wegettani= tanr,andhenceh0q= tani= hpor DonaldG.Luttermoser,ETSUXII{5M=h0h= qp.(XII-3)ii)Themagni cationalsocanbedeterminedbytheratiooftheimagetotheobjectdistance.g)Usingthe\ "trianglesintheprecedingdiagram,wecanwritetan =hp R&tan = h0R q;orhp R= h0R q;orh0h= R qp R:Finally,usingEq.(XII-3)givesqp=R qp R:SolvingthisaboveexpressiongivesR qq=p RpRq 1=1 RpRq+Rp=1+1=2:Finally,1p+1q=2R,(XII-4)whichisthemirrorequation.i)Ifp R,then1=p 2=R,sowesaythatasp!1,1=p!0and1q=2Rorq=R2 XII{6PHYS-2020:GeneralPhysicsII=)theimageisformed(i.e.,comestoafocus)halfwayouttothecenterofcurvature.ii)Sowhenp R,thefocallengthofthemirrorisf=R2:(XII-5)iii)Whenp R,themirrorappears\thin"tothedistantobject,thereforeEq.(XII-5)iscalledthethinmirrorapproximationandwerewriteEq.(XII-4)as1p+1q=1f.(XII-6)3.BothconvexandconcavemirrorsuseEq.(XII-6),exceptthereisa\change"insignfortheradiusandfocallengthofthemirror.TableXII-1showsthesignconventionsusedforthegeometricopticsparametersforcurvedmirrors.4.ImagelocationcaneitherbedeterminedalgebraicallyfromEqs.(XII-3)&(XII-6)orbydrawingraydiagrams.Therearethreeprincipleraysthatde netheimagelocation(see guresonpageXII-8):a)Concavemirror:Ray1isdrawnparalleltotheopticalaxisandisre\rectedbackthroughthefocalpoint,F.b)Concavemirror:Ray2isdrawnthroughthefocalpoint,F,andre\rectedparalleltotheopticalaxis.c)Concavemirror:Ray3isdrawnthroughthecenterofcurvature,C,andre\rectedbackonitself. DonaldG.Luttermoser,ETSUXII{7TableXII{1:SignConventionsforCurvedMirrors+SIGNS{pobjectleftofmirrorobjectrightofmirror(realobject)(virtualobject)imagesamesideofimageoppositesideofqmirrorasobjectmirrorasobject(realimage)(virtualimage)hobjectiserectobjectisinvertedh0imageiserectimageisinvertedMimageisinsameimageisinvertedorientationasobjectwithrespecttoobjectRconcavemirrorconvexmirrorfconcavemirrorconvexmirrorsymbol XII{8PHYS-2020:GeneralPhysicsIId)Convexmirror:Ray4isdrawnparalleltotheopticalaxisandisre\rectedbackawayfromthefocalpoint,F,onthebacksideofthemirror.e)Convexmirror:Ray5isdrawntowardthefocalpoint,F,onthebacksideofthemirrorandre\rectedback,paralleltotheopticalaxis.f)Convexmirror:Ray6isdrawntowardthecenterofcurva-tureonthebacksideofthemirror,C,andre\rectedbackonitself.CFVaxisConcave Mirror Special RaysRay 1Ray 2Ray 3Radius Rf = R/2to objectFC DonaldG.Luttermoser,ETSUXII{9ExampleXII{1.Aconvexsphericalmirrorwithradiusofcur-vatureof10.0cmproducesavirtualimageone-thirdthesizeoftherealobject.Whereistheobject?Solution:Aconvexmirror(R0)willonlyproduceanerect,virtualimagesofrealobjects,thusM&#x-7.9;硢0.ThenEq.(XII-3)givesM= qp=+13orq= p=3.Usingthisinthemirrorequation(Eq.XII-4),weonlyneedtosolveforp,thelocationoftheobject:1p+1q=2R1p 3p=2R 2p=2 10:0cmp=+10:0cmortheobjectis10.0cminfrontofthemirror.C.ImagesFormedbyRefraction.1.UsingSnell'slawandalittletrigonometry,itcanbeshownthatn1p+n2q=n2 n1R(XII-7)forasphericalsurfacethattransmitslight.a)n1=indexofrefractionofmediumcontainingtheobject.b)n2=indexofrefractionofmediumcontainingtheimage.c)Theothervariableshavethesamemeaningastheyhadformirrors. XII{10PHYS-2020:GeneralPhysicsII2.Furthermore,itcanbeshownthatthemagni cationgoingthroughsucha\transmitting"medium(i.e.,alens)isM=h0h= n1qn2p.(XII-8)Thesignconventionsarethesameasmirrorsexceptforq,whereq�0whentheimageisontheoppositesideofthelensandq0whentheimageisonthesamesideofthelensastheobject(seeTableXII-2).3.Planerefractingsurfacesdonotmagnifyimages:a)R!1,so(n2 n1)=R!0=)n1=p+n2=q=0orq= n2n1p.(XII-9)b)Ascanbeseenfromthisequation,sinceqisnegative(assumingpispositive),theimageformedbyaplanerefractingsurfaceisonthesamesideofthesurfaceastheobject.D.ThinLenses1.Ifanobjectisplacedadistance,p,thatismuchfartherthanthefocallengthofalens,f(i.e.,p f),thenthethicknessofthelenscanbeconsiderednegligiblewithrespecttoq(theimagedistance),p,andf=)thinlensapproximation.2.Theimageformsatthefocallengthwhentheobjectis\in- nitely"faraway:p!1;1p!0or1p+1q=1f=)1q!1fasp!1: DonaldG.Luttermoser,ETSUXII{113.Thereare2basictypesoflenses:a)Converginglens:i)Lensthickeratcenterthanedges.ii)Lightraysarerefractedtowardsthefocalpoint,F,ontheothersideofthelens.FFb)Diverginglens:i)Lensthinneratcenterthanedges.ii)Lightraysarerefractedinadirectionawayfromthefocalpoint,F,ontheinnersideofthelens.FF4.Justaswehadformirrors,thethinlensequationis1p+1q=1f:(XII-10)withmagni cationbeinggivenbyMimageheightobjectheight=h0h= qp:(XII-11) XII{12PHYS-2020:GeneralPhysicsII5.Thefocallengthforalensinairisrelatedtothecurvaturesofitsfrontandbacksurfacesviathelensmaker'sequation:1f=(n 1) 1R1 1R2!;(XII-12)wherenindexofrefractionofthelens,ffocallength(i.e.,distancefromthelenstothefocalpointF),R1radiusofcurvatureoffrontsurface,andR2radiusofcurvatureofbacksurface.6.ThevariablesinEqs.(XII-10,11,12)takeonpositiveornegativevaluesbasedupontheirrelativepositionswithrespecttothegivenlens.TableXII-2givesthesignconventionsforthinlenses.7.RayDiagramsforThinLenses(see guresonpageXII-14).a)The rstray(i.e.,Ray1)isdrawnparalleltotheopticalaxisfromthetopoftheobject.Afterbeingrefractedbythelens,thisrayeitherpassesthroughthefocalpoint,F,ontheothersideofthelens(foraconverginglens),orappearstocomefromthenearsidefocalpoint,F,infrontofthelens(foradiverginglens).b)Thesecondray(i.e.,Ray2)isdrawnfromthetopoftheobjectandthroughthecenterofthelens.Thisraycontinuesontheothersideofthelensasastraightline.c)Thethirdray(i.e.,Ray3)isdrawnthroughthefocalpoint,F,onthenearsideandemergesfromthelensontheoppositeside,paralleltotheopticalaxis. DonaldG.Luttermoser,ETSUXII{13TableXII{2:SignConventionsforThinLenses+SIGNS{pobjectinfrontoflensobjectinbackoflens(realobject)(virtualobject)qimageinbackoflensimageinfrontoflens(realimage)(virtualimage)hobjectiserectobjectisinvertedh0imageiserectimageisinvertedMimageisinsameimageisinvertedorientationasobjectwithrespecttoobjectR1;R2centerofcurvatureincenterofcurvatureinbackoflensfrontoflensfconverginglensdiverginglenssymbol XII{14PHYS-2020:GeneralPhysicsIIFFORay 1Ray 2Ray 3IFRONTBACKreal &invertedFFORay 1Ray 2Ray 3IFRONTBACKvirtual& erectFF DonaldG.Luttermoser,ETSUXII{158.Refractingtelescopesandmicroscopesuse2ormorelensesinunisontomagnifyimages.ExampleXII{2.Aprojectionlensinacertainslideprojectorisasinglethinlens.Aslide24.0mmhighistobeprojectedsothatitsimage llsascreen1.80mhigh.Theslide-to-screendistanceis3.00m.(a)Determinethefocallengthoftheprojectionlens.(b)Howfarfromtheslideshouldthelensoftheprojectorbeplacedinordertoformtheimageonthescreen?Solution(b):Forthisproblem,itiseasiertodothequestioninPart(b) rst,thenuseitsresultsinPart(a).Thequestiondoesn'ttelluswhattypeoflenstouse.However,sincetheimageisreal(i.e.,ontheothersideofthelensfromtheslide)andmagni ed,thelensmustbeaconverginglens.Also,toproduceareal,magni edimage,theimagewillalwaysbeinverted(i.e.,h00)goingthroughaconverginglens.Therefore,themagni cationequationgivesforthissetupthefollowingM=h0h= 1:80m24:010 3m= 75:0= qp;orq=75:0p:Also,weknowthatp+q=3:00mp+75:0p=3:00m76:0p=3:00mp=3:9510 2m=39:5mm. XII{16PHYS-2020:GeneralPhysicsIISolution(a):NowwecanusethesolutionfromPart(b)todeterminetheanswerforPart(a).Thethinlensequation(Eq.XII-10)thengives1p+1q=1f1p+175:0p=1f75:075:0p+175:0p=1f76:075:0p=1ff=75:0p76:0=75:076:0(39:5mm)=39:0mm.