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Measuringthee/mratio
In1897,thephysicistJ.J.Thomsonsetouttoprovethatthecathoderaysoriginatingfromaheatedcathodewereactuallyastreamofsmallnegativelychargedparticles,nowknownaselectrons.Bystudyingthecentripetalaccelerationofelectronsinamagneticfield,Thomsonwasabletosuccessfullydeterminetheircharge-to-massratio.Thomson’sworkprovidedthefirstempiricalevidencethatatomsarenotthemselvesfundamentalbuildingblocksofmatter,pavingthewayforthefieldofsubatomicphysics.Thomson’sdiscoveryoftheelectronearnthimtheNobelPrizeforPhysicsin1906.Safety Electricalsafety:Allthesignalvoltagesaresmallandharmless.
Thevoltagesinthemains-poweredequipmentisdangerousbutarescreenedinnormaluse.
HighVoltage:Thebeamtuberequiresdangerouscontactvoltagesofupto300V.Pleaseobservethefollowingsafetyprecautions:
• Onlyusethesuppliedconnectingleads.• Switchoffallpowersuppliesbeforealteringthesetup.• Donotpoweruptheexperimentuntiltheapparatusisfully
assembledandcabledup.• Donothandletheapparatus,particularlytheHelmholtzcoils,
duringoperation.
Fragileequipment:Thefinebeamtubeisanevacuatedglassvesselandpresentsadangerofimplosion.
• Donotsubjectthefinebeamtubetomechanicalstresses.• Donotremovethetubefromitsholder.• Takecarewiththeplugattheglassbase.
Background Themassmoftheelectronishardtomeasureexperimentally.Itis
easiertodeterminethechargetomassratio 𝑒/𝑚,fromwhichthemass𝑚canbecalculatediftheelementarycharge𝑒isknown.Thee/mratiocanbedeterminedexperimentallybymeasuringthetrajectoryofelectronsinanappliedmagneticfield.
1. Howwillelectronschangetheirmotioninthepresenceofamagneticfield?Whatpathwilltheelectronsfollowinthefield?Figure1belowisprovidedtoassistthisquestion.
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Measuringthee/mratio
Figure1:Thetrajectoryofanelectroninanappliedmagneticfield.Anelectronmovingatvelocity𝑣perpendicularlytoahomogeneousmagneticfield𝐵,issubjecttotheLorentzforce:
𝐹!"#$%& = 𝑒𝑣𝐵Tomakeanelectronmoveonacircularpath,itneedsacontinuouscentripetalforcepullingittowardthecenterofthecircle:
𝐹!"#$%&'"$() = 𝑚 !!
!
TheLorentzforceprovidesthiscontinuouspullontheelectron.
2. ByequatingthecentripetalforcetotheLorentzforce,derivean
expressionforthecharge-to-massratioe/m.
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Answer: !!= !
!"
Measuringthee/mratio
Inthisexperiment,theelectronsareinitiallyacceleratedinafinebeamtubebyapplyingapotential𝑉.Theresultingkineticenergyis
𝑒 𝑉 = !!𝑚 𝑣!.
Bysubstitutingthevelocity𝑣intotheaboveexpressionforthechargetomassratiooftheelectron,weobtain:
𝑒𝑚 =
2𝑉(𝐵𝑟)!
Themagneticfield𝐵isgeneratedbyapairofHelmholtzcoilsandisproportionaltothecurrent𝐼inthecoilsasfollows:
𝐵 = 𝑘𝐼Theproportionalityfactor𝑘hasalreadybeendeterminedforourapparatusandisindicatedontheapparatus.Thereforetheexpressionforthee/mratiocanbeexpressedas:
𝑒𝑚 =
2𝑉(𝐼𝑘𝑟)!
Thefinebeamtubecontainshydrogenmoleculesatlowpressure,whichemitlightthroughcollisionswithelectrons.Thismakestheorbitalmotionoftheelectronsindirectlyvisible,andtheradius𝑟ofthismotioncanbemeasuredwitharuler.Thisinturnenablesthee/mratiotobecalculated.
3.Whatuncertaintiesdoyouthinkaffectthemeasurementoftheorbitalradiusoftheelectron?Whatcouldyoudotoreducetheseuncertainties?
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Measuringthee/mratio
Task1 • Theapparatusshouldalreadybeconnected.Ifnot,aska
memberofstafftoconnecttheleads.
• Turnallthedialstozeroandswitchonthepower.
• Setthebeamtubevoltageto𝑉 = 250[V](donotgohigherthanthis).Adjustthe50[V]powersupplytogiveanarrow,well-definedelectronbeamwithclearedgedefinition.
• AdjusttheDCpowersupplyoftheHelmholtzcoilsandlookforthecurrent𝐼atwhichtheelectronbeamisdeflectedintoaclosedorbit.
Question1
Usetheslidersandmirrorstomeasuretheorbitalradiusoftheelectronbeam.Caremustbetakentoreduceparallaxuncertainty.Movetheleftslidesothatitsinneredge,mirrorimage,andtheescapeapertureoftheelectronbeamalllayonthesamelineofsight.Nowmovetherightslidesothatitsinneredge,mirrorimage,andtheright-handsideofthecircularelectronbeamorbitalllayonthesamelineofsight.Notethatyoucanmeasurethediameteroftheorbittodeterminetheradius.
Comparethecircuitdiagramontheapparatuswithwhatisconnected,anddeterminehowyoucanvaryandmeasurethebeampotential𝑉,andtheHelmholtzcoilcurrent𝐼.ExperimentwithdifferentvaluesofthepotentialVandcurrentIandstudyhowtheradiusrchangeswiththeseparameters.
1. Whateffectsdoyouobserve?Howcanyouadjusttheseparameterstominimisetheuncertaintyintheradiusr?
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Writedownthevalueof𝑘 forthisapparatus:...............................................................................................................................
Measuringthee/mratio
Task2
Measuretheradius𝑟foranumberofdifferentvaluesof𝑉and𝐼.Fromthesemeasurements,usetheformulafor𝑉tocalculatee/m.Rememberthattherewillbeanuncertaintyon𝑉,𝐼,𝑘and𝑟.Whichhasthehighestuncertainty?Estimatetheuncertaintyine/m.
Theacceptedvalueforthecharge-to-massratiooftheelectronise/m= 𝟏.𝟕𝟔 × 𝟏𝟎𝟏𝟏C/kg
1. Howdoyourresultscomparetotheacceptedvalue?Whatisthepercentagedifferencebetweenyourmeasuredvaluesofe/manditsacceptedvalue?
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V[V]
I[mA]
r[m]
e/m[C/kg]
Measuringthee/mratio
Question2 Agraphicalmethodcanbeemployedtomeasurethee/mratiowithan
improvedprecision.Thedependenceoftheacceleratingpotential𝑉andthecurrent𝐼canbefoundbyrearrangingtheequationfore/mandsubstitutingfor𝐵:
𝑉 = 0.5 𝑘! 𝑟! 𝑒/𝑚 ×𝐼!.
1. Howcanyougraphicallyinfere/musingtheformulaabove?
(Hint:rememberthat𝑘and𝑟arefixedvaluesinthistaskandthatthetheorypredictsthat𝑉isproportionalto𝐼!)
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Task3
Positiontheleftslidetoalignwiththeescapeaperture,asbefore.Nowsettherightslidesotheinsideedgesareseparatedby80mm,aligntheinsideedgeoftherightslidewithitsmirrorimage,andadjustthecoilcurrent𝐼untiltheelectronbeamrunstangentiallyalongtheslideedgecoveringthemirrorimage.
Reducethepotential𝑉instepsof250[V]downto150[V],andadjustthecoilcurrent𝐼sothattheorbitoftheelectronbeamhasadiameterof80[mm].Recordyourfindingsinthetablebelow.
𝑉[V]
𝐼[A]
𝐼![𝐴!]
250
225
200
175
150
Measuringthee/mratio
Toassessyourplotthe𝑉(𝐼!)functionforthechosenrangeofvalues.YoumayusethegridbeloworacomputerprogramsuchasExcel.
Measuringthee/mratio
Question4
1. DoesyourgraphofVvsI2showalineartrend?
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2. Calculatethee/mratiofromtheslopeofthegraph,includinganestimateoftheuncertainty.Compareyourresultwiththeacceptedvalue.
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3. Wastheexperimentsuccessful?Howdoyouthinkyoucouldimproveyourresults?
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