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ACompoundMicroscopeThisisanexampleforimageformationinacompoundmicroscope,similartotheonedoneinthelecture.Thefollowinggivesastep-by-stepdescriptionhowtoanalyzetheimage formationprocess.Eachof the14solutionsteps is illustratedbyadrawing(figure),showninaseparatepdf-andppt-file.Youshoulddownloadthisfigurefilebefore startingon theproblemand solutionbelow.Look through the sequenceoffiguresinthisfiletogetanoverviewofthesolution,whenworkingthroughthe14solutionstepsbelow.Youmustalwaysbeginthesolutiontoproblemslikethisbymakingalarge,cleandrawing.Startwiththeinformationgivenintheproblemstatement.Thenfillinmoredetails,asyouworkyourway,stepbystep,throughthesolution.The(pdf/ppt)figurefile accompanying this solution outline illustrates how such a drawing should bemade,asyouproceedwitheachstepofyourcalculation.ProblemStatement:Acompoundmicroscope isbuiltwithanobjective lens (=Lens1) of focal lengthf1=0.65cm,andaneyepiecelens(=Lens2)offocallengthf2=2.50cmwithLens2totherightofLens1.Asmallobject(Obj1)ofsize0.032cmisplaced0.70cmtotheleftofLens1.Thedistancebetweenthetwolensesisadjustedsothatthefinalimage(Img2),asseen by the eye, looking through the eyepiece, is located at the eye’s near pointdistance,20cmtotheleftofLens2.Stateorcalculateallimageparameters:d1,d1’,d2,d2’,h1,h1’,h2,h2’.Calculatethelens-to-lensdistanceL.CalculatetheangleQesubtendedattheEyebythefinalimage,Img2,whenviewedthroughtheeyepieceLens2.CalculatetheangleQrefsubtendedattheEyebytheoriginalobject,placedatthenearpoint,20cminfrontoftheeyeandviewedwithoutmicroscope.FromQeandQref,calculatetheangularmagnificationMQ=Qe/Qref.SolutionSteps:
1. Drawlenses,originalsourceobject(Obj1)andreceiveroffinalimage(Eye)onopticalaxis,withObj1drawnasanarrowpointingupward.
2. Tracethegeneralraydirectionthroughlenses:fromthesource(Obj1)tothereceiver(Eye).
3. Mark“in”,“out”,“notin”and“notout”foreachlens,usingraytracedthroughbothlensestofigureoutwhichsideofeachlensis“in”,whichsideis“out”.
4. MarkfocalpointF1forLens1onopticalaxis,relativetoObj1andLens1.Sincef1>0,F1mustbeonthe“in”sideofLens1.So,F1istotheleftofLens1.Also:|d1|>|f1|,soObj1isfurtherawaythanF1fromLens1.
5. Drawandmarkobjectdistanced1asanarrowfromLens1toObj1.SinceObj1is, by definition, on the “in” side of Lens1, Obj1 is a real object andd1>0: d1=+0.70cm
6. Calculateimagedistanced1’ofImg1fromLens1: d1’=(1/f1-1/d1)-1=(1/0.65–1/0.70)-1cm=+9.10cm>0CalculatethelateralmagnificationofLens1: m1=-d1’/d1=-(9.10/0.70)=-13.0<0,sincebothd1>0,d1’>0.Sinced1’>0,Img1isreal,i.e.,onthe“out”sideofLens1.Sincem1’<0,Img1isinverted relative to Obj1. Draw Img1 on the optical axis accordingly: as adownwardarrowtotherightofLens1.Drawandmarkimagedistanced1’asandarrowfromLens1toImg1.Note:atthispoint,youmaynotknowyetwhetherImg1isplacedtotheleftortotherightofLens2!?Pickoneofthetwochoices!Ifitlaterturnsouttobewrongyoucancorrectthedrawingthen.Thepdf-FigureforStep6showsImg1to the left, i.e., on the “in” sideofLens2,which turnsout tobe thecorrectchoiceofplacement(seeStep8).
7. DrawImg2,theimageofLens2.SinceImg2isthefinalimage,tobeseenbytheEye,itmustbeplacedinfrontoftheEye,i.e.,atleastanear-pointdistanceaway,totheleftoftheEyeandLens2.SincethisplacesImg2onthe“notout”sideofLens2,Img2isavirtualimageanditsimagedistanced2’mustbenegative: d2’=-|d2’|=-dnear=-20.0cm<0Drawandmarkimagedistanced2’asanarrowfromLens2toImg2.Note: at this point, youmay not know yet whether Img2 has the same oropposite(invertedorerect?)orientationasObj2ºImg1!?Pickoneofthetwochoices!Ifitlaterturnsouttobewrongyoucancorrectthedrawingthen.Thepdf-Figure for Step 7 shows Img2 pointing down, i.e., having the sameorientation as Obj2ºImg1, which turns out to be the correct choice oforientation(seeStep8).
8. Calculateobjectdistanced2ofObj2ºImg1fromLens1:
d2=(1/f2-1/d2)-1=(1/2.50–1/(-20.0))-1cm=+2.22cm>0CalculatethelateralmagnificationofLens2: m2=-d2’/d2=-((-20)/2.22)=+9.00>0,sinced2>0,d2’<0.Sinced2>0,Obj2isreal,i.e.,locatedonthe“in”sideofLens2.SoObj2istotheleftofLens2.IfyoudidplaceImg1ºObj2incorrectlyrelativetoLens2(totherightofLens2),inStep6,correctitnow:Img1ºObj2andLens2mustbedrawnsuchthatObj2istotheleftofLens2.Drawandmarkobjectdistanced2asanarrowfromLens2toImg2.
Sincem2’>0,Img2isnotinvertedrelativetoObj2:Img2musthavethesameorientationasObj2ºImg1.So,ifyoudiddrawImg2incorrectly(asanupward-pointingarrow),inStep7,correctitnow:Img2mustbedrawnasanarrowpointingdownward.
9. CalculatetheLens1-Lens2distanceL: L=d1’+d2=+9.10cm+2.22cm=11.32cmDrawLbetweenLens1andLens2.
10. Statetheoriginalobjectheighth1forObj1andcalculateimageheighth1’forImg1: h1=0.032cm h1’=m1h1=(-d1’/d1)h1=-(9.10/0.70)(0.032cm)=-0.416cmMarkh1andh1’inthedrawing.
11. Statetheobjectheighth2forObj2.SinceImg1fromLens1servesastheobjectforLens2,Obj2,seedrawing: h2=h1’=-0.416cmThencalculateimageheighth2’forImg2: h2’=m2h2=(-d2’/d2)h2=-((-20.)/(+2.22)(-0.416cm)=-3.74cmMarkh2andh2’inthedrawing.
12. CalculatetheangleQe subtendedat theEyebythe final image, Img2,when
viewedthroughtheeyepieceLens2.Theheight,he,andapproximatedistancefromEye(piece),de,forthatfinalimagearegiven he=|h2’|=3.74cm and de=|d2’|=20.0cmThus: Qe»tan(Qe)=he/de=|h2’|/|d2’|=3.74/20=0.187radMark he, de,Qe in the microscope drawing. Draw he, de,Qe in a separatedrawing,ifneededforyourownclarification.
13. CalculatetheangleQrefsubtendedattheEyebytheoriginalobject,placedattheeye’snearpoint,infrontoftheeyeandviewedwithoutmicroscope.Theheight,href,anddistancefromEye(piece),dref,fortheoriginalobjectaregiven href=h1=0.032cmand dref=dnear=20.0cmThus: Qref»tan(Qref)=href/dref=h1/dnear=0.032/20=0.0016radMakeaseparatedrawing,toshowonlyhref=h1,dref=dnearandQref
14. Calculatetheangularmagnification:
MQ=Qe/Qref=0.187/0.0016=117.Exercises:
1. Repeattheforegoingcalculationusingthesameinputparameters,exceptthatthefinalimage,Img2,isnowplacedveryfarawayfromtheEye(piece),atadistance de=|d2‘|=¥ sothat 1/d2‘=0.
2. Repeat the foregoingcalculations,both forde=|d2‘|=20cm andde=|d2‘|=¥,usingthesameinputparameters,exceptthatLens2isreplacedbyadivergentlenswithnegativefocallength:f2=-2.50cm.Whathappenstod2andm2now?DoyouneedtomodifytheplacementofObj2relativetoLens2inyourdrawing?IsObj2ºImg1totherightortotheleftofLens2now?Note:Obj2maybeavirtualobjectnow,eventhoughImg1isstillarealimage.Do you need to modify the orientation of the final image, Img2, in yourdrawing?IsImg2anupwardoradownwardarrownow?HowdoesMQchangewhencomparedtothecasef2=+2.50cm?
3. Use the approximate angular magnification formula given in the textbookwhich,inournotationandignoringthesign,readsasfollows:
MQ=dneard1‘/|f1f2|[ThetranslationOurNotationßàTextbookNotationisasfollows:OurNotation: dnear d1‘ f1 f2TextbookNotation: N di fobjective feyepiece.]
Howdotheresultsfromthetextbookformulacomparetotheexactresults,calculatedfromMQ=Qe/Qref?
Eye
Compound Microscope: Step1
Object 1
Lens 1 Lens 2
Eye
Compound Microscope: Step2
General Direction of Light RaysPassing thru LensesObject 1
Lens 1 Lens 2
1: in 1: not out
1: not in 1: out
2: in 2: not out
2: not in 2: out
Eye
Compound Microscope: Step3
General Direction of Light RaysPassing thru Lenses
1: in 1: not out
1: not in 1: out
2: in 2: not out
2: not in 2: out
F1Eye
Compound Microscope: Step4
1: in 1: not out
1: not in 1: out
2: in 2: not out
2: not in 2: out
F1
d1
Eye
Compound Microscope: Step5
1: in 1: not out
1: not in 1: out
2: in 2: not out
2: not in 2: out
F1
d1 d1’
Eye
Compound Microscope: Step6
1: in 1: not out
1: not in 1: out
2: in 2: not out
2: not in 2: out
F1
d2’
d1 d1’
Eye
Compound Microscope: Step7
1: in 1: not out
1: not in 1: out
2: in 2: not out
2: not in 2: out
F2
F1
d2
d2’
d1 d1’
Eye
Compound Microscope: Step8
1: in 1: not out
1: not in 1: out
2: in 2: not out
2: not in 2: out
F2
F1
d2
d2’
d1 d1’
L
Eye
Compound Microscope: Step9
1: in 1: not out
1: not in 1: out
2: in 2: not out
2: not in 2: out
h1’
h1 F2
F1
d2
d2’
d1 d1’
L
Eye
Compound Microscope: Step10
1: in 1: not out
1: not in 1: out
2: in 2: not out
2: not in 2: out
h2’
h1’=h2
h1 F2
F1
d2
d2’
d1 d1’
L
Eye
Compound Microscope: Step11
θe
|h2’|=he
|d2’|=de
Eye
Compound Microscope: Step12
θrefhref = h1
Eye
Compound Microscope: Step13
dref = dnear
Compound Microscope: Step14
θe
|h2’|=he
|d2’|=de
θrefhref=h1
MΘ = θe / θref
dref = dnear
1: in 1: not out
1: not in 1: out
2: in 2: not out
2: not in 2: out
θe
h2’
h1’=h2
h1 F2
F1
d2
d2’
d1 d1’
L
Eye
Compound Microscope
1: in 1: not out
1: not in 1: out
2: in 2: not out
2: not in 2: out
θe
|h2’|=he
h1’=h2
h1 F2
F1
d2
|d2’|=de
d1 d1’
L
Eye
Compound Microscope
