Transcript
EXPERIMENT#34 LENS THINCONVERGING MEASUREMENT OF THE FOCALI.ENGTH Theory,Definitions twice- firstwhere througha glasslensmaybe refracted A rayof lightpassing of the it enterstheglass,andthenwhereit leavestheglass.Thereis a netdeviation Usually eachsurfaceof thelensis partof a sphere. rayfromitsoriginaldirection. closetogether thatthedistance sufficiently surfaces Lenseswithtwospherical arecalledthin lenses.lf their of a lens)canbeneglected them(thethickness between theyarecalledthick lenses. ' cannotbe neglected, thickness a beamof parallellightintoa small A converginglens is capableof focusing regionat a focal point.A converging fensis anylensthickerin thecenterthanat the (convex surfaces). edges A diverginglensdeviateraysawayfromtheaxisinsucha waythattheyappear lensis anylensthicker is,thena virtualfocus.A diverging to comefroma focusF,which, surfaces). at theedgesthaninthecenter(concave parallel point point bya converging raysareconverged which is the to Focal lens, froma diverging lensor fromwhichtheparallelbeamsappearto diverge point of the lens and center focal the the between Focallengthis thedistance (seeFig.34-1'1. incomingrays
outgoing rays
mage space x' Fig. 34-1.A principalray diagramfor a convergingthin lens The followingsymbols are introducedin Fig. 34-1
V,Y' f,f
F, F'
objectand imagedistances,respectively objectand imageheights,respectively first(near)and second(fa0 focallengths,respectively first(near),second(far)focalpoints,respectively.
r22
(1) (2) (3)
Threeprincipalrays are usuallydrawnto findan image{seeFig.34-1) A ray parallelto the opticalaxis,afterrefractionby the lens,passesthrough the secondfocalpoint(appearsto comefromthe secondfocalpointof a lens). diverging A raythrougha centerof the thin lensis not deviated. A raythroughthe firstfocalpointemergesparalleltothe opticalaxis. Magnificationm is definedby the formula
'm =Y'
(34.1)
v
lf the magnification is negative,the imageis inverted,and if the magnification lf the imageis createdby extendingthe is positive,the image is erect (uptighf),, the imagedistanceis negative outgoingraysbaclorards,outgoingraysare divergent, and in sucha,casethe lenggivesa virlqal image. lenseswhenthe objectdistance Virtual imagesareformedby thinconverging lensare erect is lessthanthe focallength.Virtualimagesformedby a thinconverging (orientation of imageand objectarethe same). Real images are formedby thin converginglenswhenthe objectdistanceis greaterthanthe focallength.In this case,realimagesare inverted(oppositeorientation of imagewith respectto that of the object). The power of a lene in dioptersis the reciprocalof the focallengthin meters. Note lens(negativediopters). Nearsightedeye (myopiceye)needsconverging Farcightedeye (hyperopic eye)needsdiverginglens(positivediopters). I/re Sl unit of the focal lengthis[fl = *. = 1. IfieS/unitofthemagniftcationis[ml
I '' ' ' : :
:'
: I ,,,..,,,, to the opticalaxis ln Fig g4-1,wehavean objectof a finitesize,perpendicular (parallelto the axis of the lens). Usingthe sign conventionwhere all distances measuredin the objectspace(onthe leftfromthe centerof a lens)are supposedas the rightfromthe center negative,and all distancesmeasuredin the imagespaee-(on positive. distances measuredabovethe all Analogously, of the lens)are taken as belowthe oplicalaxisare opticalaxisare supposedto be positive'andthose:measured give in Fig. 34-1 the followingequations negative.Then,,similarrighttrianglesshown
Y = . - , \ '. Y ' = t ' -Y' x'-f' x' :v'
(34.2)
to obtain above,dividebyx' andrearrange We nowequatebothequations 't'11
x'xf'
=
t23
(34.3)
or since
-t
f'=
we can write
,:'
1 1 xx'f
,
= ". *1
Eq, (34.3)is knownas the thin lens equation' , :: )r. ,.
PrinciPleof the Method lens,we can usethe thinlens To deletmin"i'n"io*l lengthof a thinconverging distances
onlyoneasilymeasured ln" tJ*u ladepending eque'qh"(bi.ii''t;a"'J";u" X,X''
(34.1),we cancombineitwithEq.(34'3)and of magnification Usinqthe defin1ion for the focallengthcalculation obtainanotfierformula -' f'=
(34.41
xt
1-m
',.,;i.
Objectives'ofttre Measurement lens'Gonsider Findthe focallengthof a thinconverging i.' (34'3) equation expret._ng.fjl"lformulafor f frpmEq' a) ,,,,the,!h1n-bris of the lens- useEq' (3a'a)' magnification bi U)anOnnAaveiages'ColPa.re'bothresults' Calculatef accordingto a) Z. "nJ so the errorwas minimized' Try to arrangeyourmeasuremeni 3. Procedureof the Measurement of distancesx, (at least5 measurements) performa seriesof measurements x', and ready'for variouslocationsof the converging . AccuracYof the Measurement in Eqs'(3a'3)and of allthe quantities in rneasurement Estimatethe uncertainties 1. " ':i': (34.4). the accuracyof methods1a)and 1b)' wfrighdetermine Try to findthe quantities 2. Analyzethe reasons specify *ni"n on" shouldbe more,preciseand why? accordingto the theoryof errors' oi i tot both methodsas the error of repeated g. Calculatethe uncertainties measurements'
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EXPERIMENT #35 THINCONVERGING LENS - THE BESSELNfrETHOD OF THEFOCALLENGTHMEASUREMENT ':" TheorynDefinitions - see Experiment#gl. Theoreticalbackground and definitions Principle of the method The Besselmethodis basedon the factthat - for a givendistanceof an object and a screen- we can find two differentpositionof a converginglens givingsharp imageson the screen- see Fig. 35-1.This fact followsfrom possibleexchangeof objectand imagespaces.' Of cource,eachsuchpositionof the lensgivesa differentmagnification. The first positiongivingthe largerimageis shownin the upperpartof Fig.3S-t,andthe second, givinga smaltimage,is shownbelowit. Let us denotethe distancebetweenthe objectandscreenas 4 AEdthe distance betweenthe two posltionsof the lensas d. Objectand imagedistances"x,,fconespond respectively to indexesto the firstor secondpositionof the tens. Fig.35-1showsthat ., : Xt-Xt ::: ,
=
=
'X2
Xz-Xz
=
d
,
(35.1)
andthat Xj
i
Xz
=
-X1
(35.2)
Moreover,we can write
d=x,+lx.l;d=*,-lt.l
(35.3)
Substituting the aboverelations (35.1)and(35.2)into(35.3),andusingthethin lensequation(34.2),we canexpressthefonnulafor focallengthcalculation in the next form (d' - 6') i' (35.41 4d Objectivesof the Measurement 1. Usingthe Besselmethod,determinethe focallengthof a thin concave lens. 2. Usingthe findingsof the theoryof errors,try to findthe bestarrangement of yourmeasurement. Procedureof the Measurement Performa seriesof measurement (at least5), measuredistancesd and d. Estimatethe maximumerrorsof yourmeasurements. t26
larger image
Pi 2. lens position 'ffialbr P,image
xi
Fig.35-1.TheBesselmethodof focallengthmeasurement Accuracy of the Measurement Makea roughestimateof the errorsof eachmeasuredquantity,usethe theoryof Eliminatethe calculation. of focallengthmeasurement errorsfor the uncertainty as an arithmetic focalrlength the calculate errsrs, then with too large measurements . , , , , , ' : . 'i ' ' : . a v e r a g e o f y o u r m e a s u r e maenndtdse, t e r m i n e i t s e r r o r . , ' Glossary- see Experiment#34 Student's Notes
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