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'
124
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
127