Millikan'sOilDropExperimentAtthestartofthe20thcentury,scientistsdiscoveredtheexistenceoftheelectron.Theyknewitpossessedamassandanelectricalcharge,andtheyhaddeterminedthecharge-to-massratio,e/m.In1909,RobertMillikanandHarveyFletcherdevelopedanexperimenttodeterminethefundamentalchargeoftheelectron.Thiswasachievedbymeasuringthechargeofoildropsinaknownelectricfield.Ifallelectronshavethesamecharge,thenthemeasuredchargeontheoildropsmustbemultiplesofthesamefundamentalconstant.Safety Voltage:Allthesignalvoltagesaresmallandharmless.Themains

voltagesarefullycontainedintheapparatusduringnormaluse.

Setupinstructionsforstaff.

InstructionalvideosonhowtosetupthisexperimentareavailableontheLancasterPhysics/Outreach/LabinaBoxwebpage.Unboxing:Theapparatuswillbeprovidedinalargewoodenbox,whichmustbeopenedfromthetoplidfirst.Removetheitemswithin,apartfromtheplatform,thenproceedtolifttherestofthewoodensidesofthebox.Figure1:PhotographofthecablesetupoftheMillikan’sexperimentwithinstructionsonhowtoconnecteachcable.

Millikan'sOilDropExperiment

Figure2:PhotographoftheMillikan’sexperimentsetupwithinstructionsonhowtoconnecttheplatestotheplatform.

Background InMillikan’sexperiment,oildropsaresprayedfromanozzleintoavolumebetweentwoclosely-spacedhorizontalmetalplatesthatareinsulatedfromeachother.Thedropscanbekeptfromfallingbyapplyingapotentialdifferencebetweentheplates.

1. Whentheoildropsaresprayedintothevolumetheyarealreadyelectricallycharged.Whatprocessgivesthemtheircharge?

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...........................................................................................................2.Whatforcesactontheoildropswhilsttheyareinthevolume?LabeltheforcesinFigure2below.

Figure2:Representationofastationaryoildropwithinthevolume,onceavoltageisappliedacrosstheplates.

Millikan'sOilDropExperiment

Thedownwardforceactingontheoildropisduetothegravitationalpullonthedrop:

𝐹! = 𝑚𝑔.Theupwardforceistheelectrostaticforce:

𝐹! = 𝐸𝑞.Theelectricfieldcanbeexpressedasafunctionofthevoltage𝑉acrosstheplatesandthespacing𝑑betweenthem:

𝐸 =𝑉𝑑 .

Writedownthevalueof𝑑 forthisapparatus:............................................................................................................................Inthecorrectconditions,theelectrostaticforceandtheforceofgravitycanbebalanced,suchthattheoildropisbroughttorest.Inthiscase,wecanexpressthechargeontheoildropasfollows:

𝐹! = 𝐹! ,

𝑞 =𝑚𝑔𝑉𝑑 .

Asitisverychallengingtomeasurethemassofindividualoildrops,itishelpfultore-expressthemassoftheoildropintermsofthedensityoftheoilasfollows:

𝜌!"# = 𝑚/𝑉!"!!"! Assumingthattheoildropisapproximatelyspherical,itsvolumeisgivenbythefollowingformula:

𝑉!"!!"! =43𝜋𝑟

! ,whereristheradiusoftheoildrop.Thus,thegravitationalpullontheoildropcanbeexpressedas:

𝐹! =43𝑔𝜋𝑟

!𝜌!"#

Millikan'sOilDropExperiment

Thereisathirdforceactingontheoildrops.Thebuoyancyduetothesurroundingairbetweentheplatesintroducesanupwardthrust:

𝐹! =43𝑔𝜋𝑟

!𝜌!"# .Thereforetheequationforthechargeoftheoildropcanbere-expressedas

𝑞 =𝑑𝑉

43 𝜋𝑟!(𝜌!"# − 𝜌!"#) .

Oftheparametersinthisequation,theradius𝑟isdifficulttomeasure.Thefollowingsectionfocusesonhowitcanbedetermined.

Background(continued)

Theradiusofanoildropisverysmallanddifficulttomeasure.However,Millikanfoundthat𝑟couldbemeasuredindirectlybyswitchingoffthevoltageandlettingthedropfall.

Asthedropfallsdownthroughtheviscousair,itexperiencesanupwarddragforcegivenbyStokes’slawasfollows:

𝐹!"#$ = 6𝜋𝜂𝑟𝑣

Inthisequation,𝜂istheviscosityofairand𝑣 isthespeedoftheoildrop.Whentheupwardbuoyantandviscousforcesbalancethedownwardgravitationalforce,thenthedropnolongeraccelerates,andmovesatitsterminalvelocity𝑣!.Thedropsreachterminalvelocityveryquickly,andthiscanbemeasuredbytiminghowlongittakesadroplettofallameasureddistance.

Byequatingtheforcesactingonanoildroptravellingatitsterminalvelocity,wecanderiveanexpressionfortheradiusoftheoildrop.Labeltheforcesonthediagrambelow:

Millikan'sOilDropExperiment

Equatingtheseforces:

𝐹! = 𝐹! + 𝐹!"#$

Theforcesactingonthedropare:

𝐹!"#$ = 6𝜋𝜂𝑟𝑣,

𝐹! =!!𝑔𝜋𝑟!𝜌!"# ,

𝐹! =!!𝑔𝜋𝑟!𝜌!"# .

Therefore,

6𝜋𝜂𝑟𝑣! =!!𝜋𝑟!(𝜌!"# − 𝜌!"#),

𝑟 = 3𝜂𝑣!

2𝑔(𝜌!"# − 𝜌!"#) .

Substitutingthisbackintotheexpressionfor𝑞weget:

𝑞 =9𝜋𝑑𝑉

2𝜂!𝑣!!

𝑔(𝜌!"# − 𝜌!"#).

Therefore,theelectricchargeontheoildropcanbefoundintermsofitsterminalvelocityandtheappliedvoltageacrosstheplates.

Task1

Useadesklampangledtowardstheoildropchamber,ifpossible,togiveanevenandbrightilluminationonthemicroscope.

Canyoureadthegraticulescaleonthemicroscope?Ifnot,askamemberofstaffforassistance.Notethateachlargescaledivisioncorrespondsto1[mm].

Makesurethattheoilexitpipeisinlinewiththesmallholesonthechamberandgiveoneortwosharpsquirtsofthebulbtosprayoilintothegapbetweenthemetalplates.

1. Usethecontrolonthemicroscopetofocusontheoildropsandwaitforthesidewaysmotiontocease.Youmayneedtoplaceanobjectnexttothesmallholestominimiseaircurrents.Adjusttheintensityofthelightonthemicroscopeifnecessary,toincreasethecontrast.

Millikan'sOilDropExperiment

2. Switchonthevoltagesupplytothemetalplates,andwatchthe

chargeddropsmoveupanddownasyouvarythepotentialdifference.Chooseabright,easy-to-seeoildropwhichisneartothegraticule,andusethevoltagecontroltomoveitdownnearthetopofthescreen.Holdittherebyfine-tuningthevoltage.

3. Recordthisvalueof𝑉.

4. Next,turnoffthevoltageandstartthestopclock.Watchthe

dropasitfalls.

5. Youwillneedtokeepfocusingasitdriftsdownwards.Itisbesttoletitfallasfaraspossibletominimisetheuncertaintiesintiminganddistance.

6. Onceithasfallenthrough6largegraduationsonthescreen,

stoptheclockandrecordthetimeanddistanceithasfallen.

7. Theterminalvelocity𝑣!isgivenbytheequation𝑆 = 𝑣! 𝑡,whereSisthedistancetravelledandtisthetimetaken.

Notethatthedistanceyoumeasurehastobedividedby2totakeaccountofthemagnificationofthemicroscope.

Question1

Carryoutmeasurementsonanumberofoildrops.Foreachoildrop,usethemeasurementstocalculateavalueofitselectriccharge,q,usingthefollowingequationfromtheabovederivation:

𝑞 =9𝜋𝑑𝑉

2𝜂!𝑣!!

𝑔(𝜌!"# − 𝜌!"#).

Atableisprovidedbelowforyoutorecordyourmeasurements.Don’tforgettoincludetheuncertainitiesinyourmeasurements.

Someusefulconstantsforthisexperimentarealsogivenbelow:

Airviscosity 𝜂 = 1.83 ± 0.04× 10!! 𝑁𝑠 𝑚!!

Oildensity 𝜌!"# = 874 ± 2 𝑘𝑔 𝑚!!,at20°C

Airdensity 𝜌!"# = 1.30 ± 0.05 𝑘𝑔 𝑚!!,at20°C

PlateSpacing 𝑑 = 6.00 ± 0.05× 10!! 𝑚

Electroniccharge:𝑒 = 1.60 × 10!!"C

Millikan'sOilDropExperiment

VolatgeV

[V]Distances

[m]

Timet

[s]Term

inalVelocity𝑣

! [m/s]

Chargeq[C]

№ofelectrons(q/e)

Millikan'sOilDropExperiment

1. Wereyourresultsonthenumberofelectronsalwaysintegers?

Ifnot,canyouexplainwhy?

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2. Wastheexperimentsuccessful?Whatwasthelargestsourceofuncertainty?Howcouldyouimproveyourresults?

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