HistoricalAside DuringBenjaminFranklinstimesuchastonishingelectricalphenomenahadbeenobservedthatwidelyattendedpublicelectricaldisplayshadbecomepopularApublicpresentationofthiskindsostronglyimpress ID: 262589
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Chapter3Experiment1:ElectrostaticForces3.1IntroductionItwasknownsincethetimeoftheancientGreeksthatamberrubbedwithfurwouldbecomeelectri edandattractsmallobjects,aneectalsoeasilyseenbyrubbingapieceofplasticwithwool.ThewordelectricityinfactcomesfromtheGreeknameforamber. HistoricalAside DuringBenjaminFranklin'stime,suchastonishingelectricalphenomenahadbeenobservedthatwidely-attendedpublicelectricaldisplayshadbecomepopular.ApublicpresentationofthiskindsostronglyimpressedFranklinthatheboughtthelecturer'sequipment,andbeganinvestigatingelectricalphenomenaonhisown,inparallelwithothereortsalreadyunderwayinEurope. Othershadalreadyfoundthatthereweretwokindsofelectricalcharge,andthatchargesofthesamekindrepelwhilechargesofoppositekindattracteachother.Franklindesignatedthetwokindsofchargeaspositiveandnegative.Butwhilequalitativeunderstandingwasdevelopingrapidly,aquantitativeunderstandingoftheforcesbetweenelectricallychargedobjectswasstilllacking.ItwasCharlesAugustinCoulomb,aFrenchscientist,who rstquantitativelymeasuredtheelectricalattractionandrepulsionbetweenchargedobjectsandestablishedthattheforcewasproportionaltotheproductofthechargesandinverselyproportionaltothesquareofthedistancebetweenthem.Inmksunits,theelectrostaticforce,Fe,thatchargesq1andq2adistancerapartexertoneachotheris Fe=1 4"0q1q2 r2^r:(3.1)25 CHAPTER3:EXPERIMENT1 Theforceactsinadirectionalongthestraightlineconnectingthetwocharges(^r),andtheforceisrepulsivewhenq1andq2arebothpositiveorbothnegative,correspondingtoapositivevalueofq1q2.Theforceisattractivewhenthechargeshaveoppositesignsothatq1q2isnegative.Thequantity"0,calledthepermittivityconstant,isequalto "0=8:85410�12Coulomb2=(Newton-meter2);andassuresthattheforcewillbeinNewtons(N)whenthechargeisexpressedinCoulombs(C)andthedistanceisinmeters(m). HistoricalAside Thegravitationalforcesimilarlyexhibitsaninversesquaredependenceondistancebetweentwopointmasses.Gravitationalforcesdier,however,bybeingalwaysattractive,neverrepulsive,andbybeinginherentlyweaker,withtheelectrostaticrepulsionbetweentwoprotonsbeing1036timesgreaterthantheirgravitationalattraction. Thismightseempuzzling.GravitationalforcesinvolvingmassiveobjectscanbeliterallystrongenoughtomovetheEarth,constantlyactingtokeepitinanearlycircularorbitaroundtheSun,andcertainlyweexperiencegravitationalforcesonourselvesverydirectly.Buttheelectrostaticforcesbetweenpairsofobjectsinthislaboratoryarebarelystrongenoughtoliftsmallbitsoflintorpaper.Evenconsideringthelargemassoftheearth,thismightseeminconsistentwiththestatementaboutelectrostaticforcesbeing1036timesstrongerthangravitationalforces.Theweaknessofelectrostaticforcesbetweendierenteverydayobjectsre ectsthefactthatmatterconsistsofalmostexactlyequalnumbersofpositivelychargedprotonsandnegativelychargedelectronsthoroughlyintermingledwithoneanother,mainlyintheformofatomswhoseelectronsmovearoundpositivelychargednucleiconsistingofprotonsandneu-trons.Theelectronandprotonhaveequalbutoppositecharge( q=e=1:60210�19C,inmksunits),andtheneutronhaszerocharge.Theforcesofelectrostaticattractionandrepulsionactingbetweenparticleswithinthisintimatemixtureofelectronsandprotonsareindeedsubstantial.Butsoclosetoperfectisthebalancebetweenthenumberofelectronsandprotonsinordinarymatter,andsoclosetozeroisthenetcharge,thattwoseparateobjectsneareachotherhardlyexertanyelectrostaticforceatall.Yetifyouwerestandingatarm'slengthfromsomeoneandeachofyouhadonepercentmoreelectronsthanprotons,theforceofelectrostaticrepulsionwouldbesu cienttoliftaweightequaltothatoftheentireearth. Checkpoint StateCoulomb'sLaw.Howmanytypesofelectricchargeexist? 26 CHAPTER3:EXPERIMENT1 HistoricalAside SinceErnestRutherforddiscoveredthatatomsarecomposedofelectronsmovingaroundmuchheavierpositivelychargednuclei,amajorquestionwaswhythetremendouselectricforcesbetweentheprotonsandtheelectronsdonotcausethemtofusetogether. Similarly,tremendouselectricalrepulsiveforcesactbetweenthepositivelychargedpro-tonscon nedwithintheatomicnucleus,andyettheseforcesdonotusuallysucceedinpushingthenucleusapart.Thisisbecauseoftheadditionalstrongshort-rangeattractivenuclearforcesthatholdtheprotonsandneutronstogetherdespitetheelectricalrepulsion. GeneralInformation Thuselectricalforcesandquantummechanicaleectsactingtogetherdeterminetheprecisepropertiesofamaterial.Withsuchenormousforcesactinginbalancewithinthisintimatemixture,itisnothardtounderstandthatmatter,tendingtokeepitspositiveandnegativechargesinthe nestbalance,canhavegreatstinessandstrength. Alsoasaconsequenceoftheelectricandquantummechanicaleectsthatdeterminethepropertiesofmatter,whenatomscombinetoformsolidsitoftenhappensthatoneormoreelectronsnormallyboundtoeachatomareabletowanderaroundmore-or-lessfreelyinthematerial.Someoftheelectronsarenotneededtoformchemicalbondsandcaneasilymovefromatomtoatom.Thesearetheconductionelectronsinmetals.AconsequenceofCoulomb'slawisthatifametallicobjecthasanetcharge,theexcesselectronswillrepeleachotherandbeattractedtoanyregionwithnetpositivecharge,sothatthenetchargetendstoredistributeitselfovertheobject'ssurface.Asaresult,excesschargeonanisolatedsphericalmetalconductorwilldistributeitselfuniformlyontheoutsideoftheconductingsphere.Forthesamereason,ifawireconnectedtoapipeburiedintheground(forexample,totheplumbinginthebuilding)touchestheelectricallychargedconductingsphere,itsexcesschargewill owintothegroundunderthein uenceoftheCoulombforcebetweencharges.PracticalapplicationsofCoulomb'slawinvolveunbalanced(net)chargedistributedoveranextendedregion,suchasanapproximatelysphericalconductorinthepresentexperiment,andnotactuallychargeconcentratedatapoint. GeneralInformation Thenetchargeoneachobjectarisesfromindividualdiscreteelectronsandprotons,butthesmallsizeoftheelectrons'andprotons'chargeandtheirlargenumbermakethedistributiononeachobjectappearsmoothandcontinuous. 27 CHAPTER3:EXPERIMENT1 Coulomb'slaw,Equation(3.1),thenappliesbyregardingeachchargedobjecttobedividedintosmallsub-regions,andbyusingEquation(3.1)tocalculatetheforcethateachsuchsub-regionofthe rstobjectexertsoneachsmallsub-regionofthesecondobject.Wecouldthenevaluateanappropriatevectorsumto ndthenetforceandtorqueofeachobjectontheother.Applyingamathematicalprocedureequivalenttothatdescribedinthepreviouspara-graphshowsthatexcesschargedistributeduniformlyoverthesurfaceofasphereexertsaforceonasmalltestchargeadistanceawayasifallthechargeonthespherewereconcentratedatitscenter.ForthisspecialcaseEquation(3.1)endsupapplyingintheformgivenprovidedthatthedistance,r,totheexcesschargeonthemetalsphereistakenasthedistancetothecenterofthesphere. Checkpoint Inanucleusthereareseveralprotons,allofwhichhavepositivecharge.Whydoestheelectrostaticrepulsionfailtopushthenucleusapart? 3.1.1ConservationofChargeAmberrubbedwithfuracquiresanetnegativechargebecausesomeofthenegativelychargedelectronsarepulledfromthefurontotheamber,leavingthefurpositivelycharged.Sinceelectrifyingobjectsbyfrictioninvolvesmerelymovingthechargesfromoneplacetoanother,thetotalchargestaysthesame. Checkpoint Whatdoesitmeantosaythatchargeisconserved?Anelectronwithachargeof�1:60210�19Ccancombinewithapositronhavingcharge+1:60210�19Ctoyieldonlyunchargedproducts.Ischargeconservedinthisprocess? GeneralInformation Thisprincipleisfarmorefundamentalandgeneralhowever.Newparticlescanbeproducedinhigh-energyreactionssuchasthoseattheFermi-LabNationalAccelerator.Chargedparticlesarenotmerelymovedfromoneplacetoanotherbutarecreated.Yetineachreactionthenumberofnewlycreatedpositively-chargedparticlesalwaysequalsthenumberofnewparticlesthatarenegativelycharged.Sincethenetchargeinallknownphysicalprocessesstaysthesame,chargeissaidtobeconserved. 28 CHAPTER3:EXPERIMENT1 3.2TheExperiment WARNING ThepithballsarefragileandinshortsupplysodoNOTtouchthepithballsunderanycircumstance...handleonlythestrings. ThemeasurementswedowilltestthedistancedependenceinCoulomb'slaw.Theequip-mentusedtoproducenetchargeneededconsistsoftheelectrostaticgeneratorillustratedinFigure3.1.Wealsouseasmallpithballhangingbyanylonline.Thepithballisasmallinsulatingspherewithaconductingoutersurfacethatallowsexcesschargetodistributeitselfevenly. Figure3.1:AsketchoftheVandeGraagenera-torshowinghowaninsulatingbelttransportselectronstothedome.Aknownchargeqis rstplacedontheball,andthenthegeneratorisusedtoplaceachargeQonthesphericalgeneratordome.Theelectrostaticforceactstomovetheballawayfromthegeneratordome,butisbalancedbythegravitationalforce(andthestring'stension)tendingtopullthepithballbacktoitslowestposition.Themeasureddisplacementoftheballintheformoftheanglebetweenthestringandaverticallinedeterminesthecomponentofgravitationalforceinthedirectiontendingtorotatethepithballandstringbackvertical;therefore,theangleofde ectionmeasurestheelectrostaticforceactingonthepithball.Wequicklymeasurethestringangleforseveralvaluesofthepositionx1;x2;:::;xNanddistance,R,tothecenterofthedome,andimmediatelyrepeatthemeasurementforthe rstvalueofx1.TheRdependenceofthemeasuredelectrostaticforcecanthenbecomparedwiththatpredictedbyCoulomb'slaw.Figure3.1showstheelectrostaticgenerator.Thebottomrollerisdrivenbyamotor,causingthecontinuousrubberbelttoturn.Frictionbetweenthebeltandthewoolonthebottomrollertransferschargedionsbetweenthewoolandthebeltsothatthewoolbecomespositivelychargedandthebeltbecomesnegativelycharged.Thebeltcarriesthenegativechargetothetoprollerwherethechargeistransferredtothetopcollector.Then,frictioncauseschargedionstobeexchangedbetweenthetoprollerandthebeltandleavestherollernegativelychargedandthebeltpositivelycharged.Thematerialsinthebeltandrollersarecarefullychosentomakethechargetransfershavethecorrectsign.Theregionofpositivechargeonthebeltiscarriedbacktothebottomroller.29 CHAPTER3:EXPERIMENT1 Figure3.2:AphotographoftheapparatusweusetotestCoulomb'slaw.xisthedome'slocationduringtheexperimentandisthepithball'sde ectionfromvertical.ReferalsotoFigures3.4and3.6.Acontactplacedclosetothebottomrollerisgrounded(meaningthatitisattachedtoanearbypipeorwirethatultimatelyisconnectedtotheearthusingagroundrod).The owofchargetoorfromthegroundcancelsouttheexcesschargeonthepartofthebeltnearthecontact.Bythismechanism,thebeltandrollersystemactasapump,doingworkonthenegativeelectricchargeagainsttherepulsiveforceofthechargealreadyonthedome,anddepositingitonthedomewithahighpotentialenergyperunitcharge,correspondingtoahighvoltage. Checkpoint HowdoestheVandeGraageneratoroperate? Thechargesgeneratedinthisexperimentarenotdangerous,butyoumightexperiencesomeunpleasant,disconcertingshocksbynotfollowingtheinstructionspreciselyasgiven.Youmayalsopickupchargethatcansubjectyoutoaminorelectricalshock,aboutasstrongasacarpetshock,whenyoutouchagroundedobject.30 CHAPTER3:EXPERIMENT1 Themeasurementstobemadeandlateranalyzedusingtheset-upinFigure3.2requirethreebasicsteps:1)Aligningthedomewithrespecttothepithball'spivot,2)placingacharge,q,ontheballanddeterminingitsvalue,and3)measuringtheposition,x,ofthechargeQonthegeneratorandtheangle,,betweenthestringandaverticalline.3.2.1AligningtheApparatusAsshowninFigure3.4,theelectrostaticgeneratorisarrangedsoitcanbedisplacedalongameterstickfastenedtoyourlaboratorybench.Inaddition,eachset-upincludesadullaluminumspheregroundedtoearthandmountedonaninsulatingrodofLucite;thealuminumsphereisforuseingroundingtheconductinggeneratordomeand/orthepithballwhennecessarytoremovetheexcesscharge. HelpfulTip Ifthepithballstringmovestowardorawayfromtheprotractor,theanglemeasurementwillbedistorted.Inthatcasemovetheprotractorbyrotatingthesupportrodonthetabletogetthestringtomoveparalleltotheprotractor'sface. Figure3.4:Thetopofthedomeisremovablesothatwecanaccuratelymatchthespheres'centersandmeasuretheVandeGraa'sref-erenceposition,x0.Firstyouneedtodeterminetheposition,x0,ofthevoltagegeneratoratthepointwherethepithball,hangingvertically,isatitscenter.Todothis,groundthegeneratorspherebytouchingitwiththedullaluminumsphere.Removethetophalfofthedome,relocateoneofthepithballstotheplasticscrew,placethesmallplasticruleracrossthedome'sdiameter,accuratelylocatethecenter,andshiftthegenerator'sbaseuntilthecenterofthepith HelpfulTip Fromtheendofthelabbench,seethatthemotionofthelargedomealongthemeterstickisalignedwiththeprotractor.Ifnecessary,askyourTAtohelpyouoptimizeyourapparatus. 31 CHAPTER3:EXPERIMENT1 Figure3.3:Thephysicsofchargingpithballs.Anobjectwithnegativechargeisbroughtnearthedielectricballs.Positivechargeintheballsareattractedcloserandnegativechargeisrepelledfarther.Sincethepositivechargesarecloser,thenetforceisattractive.Theballstouchtherod,pickupanetnegativecharge,andarerepelledfromtherodandeachother.ballliesatthecenterofthesphere.(SeeFigure3.4.)NotethattheVandeGraageneratorissomewhat imsysothatcarelessnesscancausethedometomovewithoutthebasemovingcommensurately.Takecaretohandlethegeneratorfromthebaseonly.Ifnecessary,askyourteachingassistanttoadjusttheheightofthehorizontalbarsupportingthepithballssotheirmidpointcoincideswiththecenterofthegeneratorsphere.Recordthepositionofthesideofthegeneratoralongthetableontheclampedmeterstick.Thismeasuredlocationalongtheruleriscalledx0intheequationswewillbeusinglater.(SeeFigure3.6.)Whenthebaseisatx0,thelargesphere'scenterisdirectlyunderthepithballpendulum'spivot.Recordtheprotractorreadingfortheun-de ectedstringas0.AnglestowardthesupportrodarepositiveandanglestowardtheVandeGraageneratorarenegative.Recordyourerrorandunits.Iwouldsuggestasmalltableinyournotebook(andlaterinyourreport)toholdx0,L,0,r0,andm1m2. Checkpoint AftertheVandeGraageneratorhasbeenrunningandisturnedowithitsdomestillcharged,howwouldthechargedistributioninthegroundedaluminumspherebeaectedbybringingitnearthedomewithoutmakingcontact?Explainthiseectintermsoftheelectrostaticforcesactingandthepropertiesofthemetallicsphere. 32 CHAPTER3:EXPERIMENT1 Nowmovethegeneratoralongtherulerinthedirectionawayfromtheverticalpoleuntilthepithballclearsthedomebymorethan10cmandreplacethetopofthedome.IftheexperimentweredonewiththeVandeGraageneratortooclosetotheverticalpole,thechargeinducedintheconductingpolewouldinturnexertCoulombforcestoredistributethechargeinthedomemakingitsphericallyasymmetric. Checkpoint Whyisitpossibletousetheformulafortheforcebetweentwopointcharges,Equa-tion(3.1),fortheforcebetweenthechargedpithballandthedomeoftheVandeGraageneratorwhentheelectri eddomeisnotevenapproximatelyapointcharge? 3.2.2ChargingthePithBallsItiswisetoconstructatableinyournotebooktokeepyourxvs.data.Whileyou'reatit,youmightconsideranothertabletokeep0,x0,L,m1,m2,andr0.Iwouldsuggestonepartnerstandstillandreadtheprotractorwhiletheotherplacesthefunctiongenerator,readsx,andwritesdownthedata. HelpfulTip Staticchargesdissipateveryquicklyespeciallywhenhumidityishigh.Becauseofthisitisnecessarytominimizethetimespenttakingdata. Besureyouknowexactlywhattodoand,onceyoustart,continuetakingmeasurementsuntilyou nish.Savethecalculationsanddelayablemeasurementsforlater.Repeatingthe Figure3.5:Illustratesthestrategyformeas-uringthepithballs'charges.measurementsonlytakesalittletime,sothereisnoreasonforstressifthe rstorsecondtrydoesnotgosmoothly.First,groundbothpithballsbytouchingthemwiththedullaluminumsphere.Thenwrapthefuraroundthepointedendoftherubberrodandbrisklyrubtherodwiththefurtoproduceanetnegativechargeontherod.Bringthechargedrodclosetothepithballs.Apositivechargewill rstbeinducedonthesidesofthepithballsclosesttotherod,asshowninFigure3.3,causingtheballstobeattractedtotherod.Aftertherodandpithballsmakecontact,anegativecharge(consistingofelectrons)willpassfromthe33 CHAPTER3:EXPERIMENT1 rodtotheballs.Whenenoughnegativechargehasbeentransferred,theballswill yawayfromtherodandwillrepeleachother.Thesimilarballscollectingchargefromthesamesourcewillacquiresimilarcharges;wewillassumethesechargesareidentical.Iftheballsdonot yawayfromtherodwithin10seconds,theyaretoodryandmustbemoistenedbybreathinggentlyonthem.Theseparationbetweenthetwoballsoncechargedshouldbebetween2-6cm.Donottouchthepithballsortheymaybepartiallydischarged.Untanglethepithballstrings.Placetherodonthetableandadjusttheplasticcaliper'swidthtobethedistancebetweenthecentersoftheballs;gentlytouchingtheballswiththeplasticisok.Carefullylaythecalipersasideuntilyouaredonewiththestaticcharges.3.2.3TakingtheDataLiftoneofthechargedballsbyitslineanddrapeitovertheinsulatedplasticpegmountedonthemeterstickwithouttouchingordischargingtheotherball.Ifyoudisturbtheremainingpithball'schargebeforeyoucompleteyourdatatable,youwillneedtobeginagain...Slidetheelectrostaticgeneratoralongthetableuntilthechargedballisabout5-10cmfromthesurfaceofthegenerator'ssphere.Usethepowercordswitchtorunthegeneratorforseveralshortburstsuntiltheballisde ectedabout5to10asmeasuredbytheprotractoratthelinesupportpoint.YoucannowslidetheVandeGraageneratorclosertoseelargerde ectionangles.Setthegeneratortoexactcentimeter(cm)positionstomakereadingandrecordingthepositionsquicker.Recordthede ectionangle,1,andthehorizontalposition,x1,ofthegenerator.Nowshiftthegeneratorsidewayssoastodecreasethede ectionangleandagainrecordx2and2.Makesixseparatemeasurementstoobtainvaluesofforx=x�x0between10cmand60cm(seeFigure3.6).Youwillnotgetverygoodresultsunlessyouutilizeatleasthalfofthemeterstick.ThedataobtainedwillbeusedtodeterminethetotalchargeQonthegeneratorsphereandtoverifytheinversesquaredistancedependenceinCoulomb'slaw. WARNING KeepthegroundedspherefarawayfromthepithballandtheVandeGraasphere.Itwilldistortthechargedistribution(s);canyouexplainwhy? Checkpoint Ishumidityintheroomaconcerninthisexperiment?Whyorwhynot?Astimepasses,thepithballslosetheirexcesscharge.Wheredoesitgo? Immediatelyafterthelastmeasurement,returnthegeneratortothepositionofthe rstmeasurement,x1,andrecordthevalueof01.34 CHAPTER3:EXPERIMENT1 HelpfulTip Now,youcanslowdownalittleandre ectuponwhatyouhavedone. NowyoucanplacebothpithballsandtheVandeGraadomeinelectricalcontactwiththegroundedspheretodischargeeverything;leavethemconnectedsothatchargedaircurrentsfromadjacentexperimentsiscontinuallydissipated.Howclosely01agreeswith1providesinformationabouthowmuchthedissipationofthechargeaectedyourresults;onemightexpectthatthesameanglewouldbemeasuredforx1bothtimes.Don'tforgettomeasurethedistancer0betweenthepithballsstoredbyyourplasticdividers.Howaccuratelycanyouestimatethepositionsofthepithballs?Howaccuratelycanyoumeasuretheseparationbetweenthecaliper'stips.3.3ComposingandPresentingtheData WARNING Staticelectricitykillselectroniccircuitry.Keepawayfromthecomputersuntileveryoneinthelabhasdischargedtheirapparatus. Onceyou nishcollectingyourdata,youcanuseEquation3.2tocomputethechargeonyourtwopithballs.Youcanmeasurethestring'slengthandthepithballs'masses.Youcannoteobservationsthatmightaectyourdata.Youcanevenrepeattheexperimentonceortwicetoprovideyouwithachoiceortwo.Youcanconstructyourreport'sskeleton.Thereisnoreasontositidlewhileyourclassmatescatchup.YouprobablywillhavenoticedbythispointthattheVandeGraageneratorcanproduceimpressiveelectricaldischarges.Figure3.5(b)showsthevectordiagramforthethreeforcesactingoneachballinequilibrium.Theyare:thetensionTinthenylonline,thepithballs'weightmg,andtheCoulombforceFe.Themassmiswrittenoneachballorontheprotractorwherethestringattachesinunitsofmilligrams(mg)andthelengthofthenylonlinemustbemeasured.1)Beforeyoucometolab,showthatifeachpithballhasmassm,hangsfromastringoflengthL,andthetwoareseparatedbyadistancer0becauseoftheircharge,thechargeqoneachisq=vuuuut 2"0mgr30 r L2�r0 22:(3.2)2)Addthisprooftoyournotebook.35 CHAPTER3:EXPERIMENT1 3)Calculatethechargeq(inCoulombs)onthepithballs.4)OneoptioninyourAnalysisistodiscussthenumberofelectronsneededtomakeupthischarge.. Figure3.6:Thegeometryneededtomeasurethedistancebetweenthecharges(reddiagonal).ReferalsotoFigures3.4and3.2. Checkpoint Whydoestheexperimentre-quireusingtwopithballsratherthanone?ProveEquation(3.2)ofthislabwrite-upoutsideofclass. ConsiderFigure3.6andnotethatthespheremoveshorizontallyasxchanges.jx�x0jisthehori-zontaldistancebetweenthepivotandthesphere'scenter.Whenthepithballisde ected,thehor-izontaldistancefromthepivotisoppositetoandRx=jx�x0j+Lsin.Theheightsofthepithballandspherearedierentbecausethestringrotatedaboutthepivot.TheywereatthesameheightwhenthepithballwasLbelowthepivotandwhenthepithballisde ecteditisLcosbelowthepivotasshowninFigure3.6.Thedierenceinheightbetweenthesphereandthede ectedpithballisRy=L�Lcos.TheredlineinFigure3.6showsthedistance,R,betweenthecenterofthegeneratorsphereandthecenteroftheballandisgivenbyR=q R2x+R2y=q (jx�x0j+Lsin)2+(L�Lcos)2:(3.3)Theangle betweenthelinejoiningthetwocentersandthehorizontalis =tan�1 Ry Rx!=tan�1 L�Lcos(�0) jx�x0j+Lsin(�0)!:(3.4)Byresolvingtheforcesperpendiculartothethreadsupportingtheball,itcanbeshownthattheelectricalforceFeactingontheballisgivenintermsoftheobservedbyFe=mgsin(�0) cos(�0� )(3.5)36 CHAPTER3:EXPERIMENT1 Ifwemaketheassumptionthat issmall,thisforcecanbewrittenasFe=mgtan(�0)(3.6)Equation(3.6)canbeusedtocalculatetheforceFeforeachmeasuredvalueofandthereforeateachRinEquation(3.3).WeseektocomparetheobserveddependenceofFeonRwiththatinCoulomb'slaw,Fe=qQ 4"0R2:ItispossibletousetheprogramGa3tohavethecomputertomakethesecalculations,toplottheresults,andthento tthedatatoapowerlawequationbychoosingoptimumvaluesforR'sexponentialpowerandtheconstantA=qQ 4"0.3.3.1VerifyingtheInverseSquareLawWewilluseVernierSoftware'sGraphicalAnalysis3.4(Ga3)programtoanalyzeourdata.Asuitablesetup leforGa3canbedownloadedfromthelab'swebsiteathttp://groups.physics.northwestern.edu/lab/electrostatic.htmlFirstentertherawdata,xand,inthe rsttwocolumnsPositionandAngle.YoumightwanttoverifythatthecomputerisusingthecorrectformulastocalculatethedistanceRbetweenthechargessqrt((abs(Position-x0)+Length*sin(Angle-Angle0))^2+(Length*(1-cos(Angle-Angle0)))^2)/100: Checkpoint Whydidwedivideeverythingby100? HelpfulTip Parametersx0,Length,Angle0,andMasscanbeadjustedatthebottomlefttomatchyourmeasurements. Inasimilarwayverifythecolumnforelectrostaticforce,Fe,usesEquation(3.6).InthiscasetheformulashouldbeMass*9.807*tan(Angle-Angle0)/10^637 CHAPTER3:EXPERIMENT1 Checkpoint Whydidwedivideeverythingby106? Onceyouhavethecorrectnumberseverywhereinyourtable,verifythattheelectrostaticforceisontheverticalaxisandthatthedistancebetweenthechargesisonthehorizontalaxis.YoushouldseeaninverserelationontheplotwhereasRgetslargerFebecomessmaller.Youwouldliketoverifythatthisisaninversesquarerelation.Usethemousetodragaboxaroundthedatapointssothatallrowsinthetableturngrey.Ifsomerowsinyourtabledonotturngrey,Data/SortfromthemenuandsortinincreasingR.Selectthedataagain.Analyze/CurveFit...fromthemenu.Whentheboxopens,selectthe`Power'formulafromthesetof`Stockfunctions'andTryFit.ThevariableBshouldtellyoutowhatpowerthedatadependsonR.Coulomb'slawpredictsittobe�2.Besurethatthecontinuousmodellinepassesamongyourdatapoints.Sometimesthe tmightneedalittlehelp.InthiscasetypeinvaluesofAandBtoseehowthey tyourdata.LargerAmakesthelinemoveupandlargerBmakesthelinefalloquickerwithdistance.Oncethe tisprettygood,writedownthe tparameters(AandB),selectAutomaticattop-right,andTryFitagain.Trytogetautomatic ttoworksothecomputerwillgenerateuncertainties.Thecomputerwilltryto ndAandBsuchthatFe=ARB(3.7)modelsyourdataascloselyaspossible.Computersarestupid;youmustdecidewhetherthemodelcurverepresentsyourdatapoints.UsingthevalueAandyourpithballcharge,calculatethechargeontheVandeGraageneratordome.Alsoindicatethesignofthechargeifitismadeofelectrons. Checkpoint Howmanyexcesselectronscomposethegenerator'scharge?Doyourdataindicatewhetherthepithballchargeanddomechargehavethesamesign?Dothedataindicatewhichis+andwhichis�? 3.4AnalysisHowdidthe rstvaluedierwhenremeasuredattheend,andwhatdoesthistellyouaboutanyexperimentalerrorcausedbychargeleakingothepithballsandthegenerator?IsyourexponentBtoolargeortoosmall?Isthisconsistentwithchargedissipatingtoground?(Youmightwanttoconsiderthechronologicalorderingofyourdatapointswhile38 CHAPTER3:EXPERIMENT1 answeringthis.)Doesthecomputer'sestimateduncertainty(B)inBbracketCoulomb'spredictionof-2?SeeSection2.9.1.Whataresomesubtlesourcesoferrorthatwehavenotrecorded?Mightwindfromthetemperaturecontrolbesigni cant?Whataboutyourclassmates'charges?Arethechargedistributionssphericallysymmetricordoesthepresenceoftheotherchargedisturbthis?IfalloftheR'swerealittlelarger,wouldthisimproveyouragreement?WhatiftheVandeGraageneratortiltedonitsbasewhiletheexperimentwasinprogress?Whatifthelinebetweenthechargecenterswasnotveryhorizontal?Istheratioofyourcalculatedchargesapproximatelythesameastheratioofsurfaceareas?DoyourdatasatisfyCoulomb'slawtowithinreasonableexperimentalerror?Explainthepossiblesourcesofanydisagreement.Howdothevaluesofthechargesyoumeasuredcomparewithyourexpectations?Howhardwoulditbetoplaceacoulombortwoofchargeonthedomeofthegenerator?Asinthiscase,manytimeswemusttestatheoryonepieceatatimewhenwedon'thavethetechnologytomeasureallofthevariables.WecandeterminewhetherFe/R�2isconsistentwithourobservations.WecanuseCoulomb'slawtocalculatethepithballs'charge.WecanuseCoulomb'slawandourdata tparameterAtodeterminethedome'scharge.WecanuseourexperiencetodecidewhetherusingCoulomb'slawyieldsreasonablevaluesforthesecharges.However,wedidnotmeasurethestaticchargesorthedissipatedchargesindependentlyofCoulomb'slaw,sothiscircumstantialevidenceislessthancomplete.IfallofthesepiecessupportCoulomb'slaw,thisisstillprettycompelling;ifsomedonotsupportCoulomb'slawitislessandlessso.Thedi cultyinperformingtheexperimentwellmakesitdi culttoconcludethatonlyCoulomb'slaw'sfailurecanexplainthediscrepancies.3.5ConclusionsWhatvaluesandunitsofchargedidyoumeasure?Doesyourdatasupport,contradict,orsaynothingaboutCoulomb'slaw?Whatconstantexponentdidyoumeasure?Alwayscommunicatewithcompletesentencesandde neallsymbols.YourreadershouldbeabletounderstandyourConclusionswithoutreadinganythingelseinyourreport.Whatchangesmightimprovetheexperiment?Canyouthinkofapplicationsforanythingyouhaveusedorobserved? HelpfulTip ReviewAppendixEfrequentlywhileassemblingyourreports. 39