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Electricity and Magnetism

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Electricity and Magnetism
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Science, Physics, Electricity, Atoms, Nucleus, Circuits, Fields of Force, Electromagnetism, Capacitance, Inductance, atomic force, electrons, alpha decay, nuclear fission, ionizing radiation, electrical current, voltage, resistance, Atomic number, Parallel resistances, magnetic fields, gravitational field, electric field, Capacitors, Inductors
Energy, Physics, Scientific Concepts
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This is a text on electricity and magnetism for an introductory college physics class. The treatment is algebra-based, with applications of calculus discussed in optional sections. Contents: 1) Electricity and the Atom. 2) The Nucleus. 3) Circuits, Part 1. 4) Circuits, Part 2. 5) Fields of Force. 6) Electromagnetism. Appendix: Capacitance and Inductance. This is book 4 in the Light and Matter series of free introductory physics textbooks.
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Community College, Higher Education
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Adobe PDF Reader.
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Benjamin Crowell, Fullerton College, CA
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Textbook
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www.lightandmatter.com
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http://florida.theorangegrove.org/og/file/c04f56d6-847d-84ba-ead6-0e13d5257220/1/Electricity.pdf

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Copyright 1999-2008 Benjamin Crowell. This book is licensed under the Creative Commons Attribution-ShareAlike license, version 1.0, http://creativecommons.org/licenses/by-sa/1.0/ except for those photographs and drawings of which I am not the author, as listed in the photo credits. If you agree to the license, it grants you certain privileges that you …
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Book4intheLightandMatterseriesoffreeintroductoryphysicstextbooks www.lightandmatter.com

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The LightandMatter seriesof introductoryphysicstextbooks: 1NewtonianPhysics 2ConservationLaws 3VibrationsandWaves 4ElectricityandMagnetism 5Optics 6TheModernRevolutioninPhysics

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BenjaminCrowell www.lightandmatter.com

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Fullerton,California www.lightandmatter.com copyright1999-2008BenjaminCrowell rev.October7,2008 ThisbookislicensedundertheCreativeCommonsAttribution-ShareAlikelicense,version1.0, http://creativecommons.org/licenses/by-sa/1.0/,except forthosephotographsanddrawingsofwhichIamnot theauthor,aslistedinthephotocredits.Ifyouagree tothelicense,itgrantsyoucertainprivilegesthatyou wouldnototherwisehave,suchastherighttocopythe book,ordownloadthedigitalversionfreeofchargefrom www.lightandmatter.com.Atyouroption,youmayalso copythisbookundertheGNUFreeDocumentation Licenseversion1.2,http://www.gnu.org/licenses/fdl.txt, withnoinvariantsections,nofront-covertexts,andno back-covertexts. ISBN0-9704670-4-4

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ToArnoldArons.

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BriefContents 1ElectricityandtheAtom13 2TheNucleus41 3Circuits,Part177 4Circuits,Part2105 5FieldsofForce123 6Electromagnetism143 ACapacitanceandInductance169

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Contents 1ElectricityandtheAtom 1.1Thequestfortheatomicforce...14 1.2Charge,electricityandmagnetism.15 Charge,15.|Conservationofcharge, 17.|Electricalforcesinvolvingneutral objects,18.|Thepathahead,18.| Magneticforces,18. 1.3Atoms.............20 Atomism,20.|Atoms,light,andeverythingelse,22.|Thechemicalelements, 23.|Makingsenseoftheelements,24.| Directproofthatatomsexisted,25. 1.4Quantizationofcharge......26 1.5Theelectron..........30 Cathoderays,30.|Werecathoderays aformoflight,orofmatter?,30.| Thomson'sexperiments,31.|Thecathode rayasasubatomicparticle:theelectron, 33. 1.6Theraisincookiemodeloftheatom34 Summary.............37 Problems.............39 2TheNucleus 2.1Radioactivity..........41 Becquerel'sdiscoveryofradioactivity, 41.|Threekindsofradiations",43.| Radium:amoreintensesourceof radioactivity,43.|Trackingdownthenatureofalphas,betas,andgammas,43. 2.2Theplanetarymodeloftheatom..45 Somephenomenaexplainedwiththeplanetarymodel,48. 2.3Atomicnumber.........48 2.4Thestructureofnuclei......53 Theproton,53.|Theneutron,54.| Isotopes,55.|Sizesandshapesofnuclei, 56. 2.5Thestrongnuclearforce,alphadecay andssion.............56 Randomnessinphysics,59. 2.6Theweaknuclearforce;betadecay59 Thesolarneutrinoproblem,61. 2.7Fusion.............63 2.8Nuclearenergyandbindingenergies64 2.9Biologicaleffectsofionizingradiation67 2.10 ? Thecreationoftheelements..70 CreationofhydrogenandheliumintheBig bang,70.|Wearestardust,70.|Articial synthesisofheavyelements,71. Summary.............73 Problems.............75 3Circuits,Part1 3.1Current............78 Unityofalltypesofelectricity,78.| Electriccurrent,79. 3.2Circuits............81 3.3Voltage............82 Thevoltunit,82.|Thevoltageconceptin general,83. 3.4Resistance...........87 Resistance,87.|Superconductors,89.| Constantvoltagethroughoutaconductor, 90.|Shortcircuits,91.|Resistors,91.| Lightbulb,92.|Polygraph,92.|Fuse, 92.|Voltmeter,93. 3.5Current-conductingpropertiesof materials.............94 Solids,94.|Gases,95.|Liquids,95.| Speedofcurrentsandelectricalsignals,96. 3.6 R ApplicationsofCalculus....97 Summary.............99 Problems.............101 10

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4Circuits,Part2 4.1Schematics..........106 4.2Parallelresistancesandthejunction rule................107 4.3Seriesresistances........112 Summary.............118 Problems.............119 5FieldsofForce 5.1Whyelds?..........123 Timedelaysinforcesexertedatadistance, 123.|Moreevidencethateldsofforceare real:theycarryenergy.,124. 5.2Thegravitationaleld......125 Sourcesandsinks,127.|Superpositionof elds,127.|Gravitationalwaves,128. 5.3Theelectriceld........129 Denition,129.|Dipoles,130.| Alternativedenitionoftheelectriceld, 132.|Voltagerelatedtoelectriceld,132. 5.4 R VoltageforNonuniformFields..134 5.5TwoorThreeDimensions.....136 5.6 R ? ElectricFieldofaContinuous ChargeDistribution.........138 Summary.............140 Problems.............141 6Electromagnetism 6.1Themagneticeld........144 Nomagneticmonopoles,144.|Denition ofthemagneticeld,145. 6.2Calculatingmagneticeldsand forces...............146 Magnetostatics,146.|Forceonacharge movingthroughamagneticeld,148. 6.3Induction............149 Electromagnetismandrelativemotion, 149.|Theprincipleofinduction,151. 6.4Electromagneticwaves......154 Polarization,154.|Lightisanelectromagneticwave,155.|Theelectromagnetic spectrum,155. 6.5Calculatingenergyinelds....156 6.6 ? Symmetryandhandedness...160 Summary.............162 Problems.............163 ACapacitanceandInductance A.1Capacitanceandinductance...169 Capacitors,170.|Inductors,170. A.2Oscillations..........172 A.3VoltageandCurrent.......175 A.4Decay.............179 Therccircuit,179.|Therlcircuit,180. A.5Impedance...........182 Problems.............185 Appendix1:Exercises 186 Appendix2:PhotoCredits 201 Appendix3:HintsandSolutions 203 11

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Chapter1 ElectricityandtheAtom Wherethetelescopeends,themicroscopebegins.Whichofthetwo hasthegranderview? VictorHugo Hisfatherdiedduringhismother'spregnancy.Rejectedbyher asaboy,hewaspackedotoboardingschoolwhensheremarried. Hehimselfnevermarried,butinmiddleageheformedanintense relationshipwithamuchyoungerman,arelationshipthatheterminatedwhenheunderwentapsychoticbreak.Followinghisearly scienticsuccesses,hespenttherestofhisprofessionallifemostly infrustrationoverhisinabilitytounlockthesecretsofalchemy. ThemanbeingdescribedisIsaacNewton,butnotthetriumphant Newtonofthestandardtextbookhagiography.Whydwellonthe sadsideofhislife?Tothemodernscienceeducator,Newton'slifelongobsessionwithalchemymayseemanembarrassment,adistractionfromhismainachievement,thecreationthemodernscienceof mechanics.ToNewton,however,hisalchemicalresearcheswerenaturallyrelatedtohisinvestigationsofforceandmotion.Whatwas radicalaboutNewton'sanalysisofmotionwasitsuniversality:it succeededindescribingboththeheavensandtheearthwiththe sameequations,whereaspreviouslyithadbeenassumedthatthe sun,moon,stars,andplanetswerefundamentallydierentfrom earthlyobjects.ButNewtonrealizedthatifsciencewastodescribe allofnatureinauniedway,itwasnotenoughtounitethehuman scalewiththescaleoftheuniverse:hewouldnotbesatiseduntil 13

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hetthemicroscopicuniverseintothepictureaswell. ItshouldnotsurpriseusthatNewtonfailed.Althoughhewasa rmbelieverintheexistenceofatoms,therewasnomoreexperimentalevidencefortheirexistencethantherehadbeenwhentheancient Greeksrstpositedthemonpurelyphilosophicalgrounds.Alchemy laboredunderatraditionofsecrecyandmysticism.Newtonhad alreadyalmostsingle-handedlytransformedthefuzzyheadedeld ofnaturalphilosophy"intosomethingwewouldrecognizeasthe modernscienceofphysics,anditwouldbeunjusttocriticizehim forfailingtochangealchemyintomodernchemistryaswell.The timewasnotripe.Themicroscopewasanewinvention,anditwas cutting-edgesciencewhenNewton'scontemporaryHookediscovered thatlivingthingsweremadeoutofcells. 1.1Thequestfortheatomicforce Newtonwasnottherstoftheageofreason.Hewasthelastof themagicians. JohnMaynardKeynes NeverthelessitwillbeinstructivetopickupNewton'strainof thoughtandseewhereitleadsuswiththebenetofmodernhindsight.Inunitingthehumanandcosmicscalesofexistence,hehad reimaginedbothasstagesonwhichtheactorswereobjectstrees andhouses,planetsandstarsthatinteractedthroughattractions andrepulsions.Hewasalreadyconvincedthattheobjectsinhabitingthemicroworldwereatoms,soitremainedonlytodetermine whatkindsofforcestheyexertedoneachother. Hisnextinsightwasnolessbrilliantforhisinabilitytobringitto fruition.Herealizedthatthemanyhuman-scaleforces|friction, stickyforces,thenormalforcesthatkeepobjectsfromoccupying thesamespace,andsoon|mustallsimplybeexpressionsofa morefundamentalforceactingbetweenatoms.Tapestickstopaper becausetheatomsinthetapeattracttheatomsinthepaper.My housedoesn'tfalltothecenteroftheearthbecauseitsatomsrepel theatomsofthedirtunderit. Herehegotstuck.Itwastemptingtothinkthattheatomicforce wasaformofgravity,whichheknewtobeuniversal,fundamental, andmathematicallysimple.Gravity,however,isalwaysattractive, sohowcouldheuseittoexplaintheexistenceofbothattractive andrepulsiveatomicforces?Thegravitationalforcebetweenobjectsofordinarysizeisalsoextremelysmall,whichiswhywenever noticecarsandhousesattractingusgravitationally.Itwouldbe hardtounderstandhowgravitycouldberesponsibleforanything asvigorousasthebeatingofaheartortheexplosionofgunpowder. Newtonwentontowriteamillionwordsofalchemicalnoteslled withspeculationaboutsomeotherforce,perhapsadivineforce"or vegetativeforce"thatwouldforexamplebecarriedbythesperm 14 Chapter1ElectricityandtheAtom

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a / Fourpiecesoftapeare prepared,1,asdescribedinthe text.Dependingonwhichcombinationistested,theinteraction canbeeitherrepulsive,2,or attractive,3. totheegg. Luckily,wenowknowenoughtoinvestigateadierentsuspect asacandidatefortheatomicforce:electricity.Electricforcesare oftenobservedbetweenobjectsthathavebeenpreparedbyrubbing orothersurfaceinteractions,forinstancewhenclothesrubagainst eachotherinthedryer.Ausefulexampleisshowningurea/1: sticktwopiecesoftapeonatabletop,andthenputtwomorepieces ontopofthem.Lifteachpairfromthetable,andthenseparate them.Thetwotoppieceswillthenrepeleachother,a/2,aswill thetwobottompieces.Abottompiecewillattractatoppiece, however,a/3.Electricalforceslikethesearesimilarincertainways togravity,theotherforcethatwealreadyknowtobefundamental: Electricalforcesare universal .Althoughsomesubstances, suchasfur,rubber,andplastic,respondmorestronglyto electricalpreparationthanothers,allmatterparticipatesin electricalforcestosomedegree.Thereisnosuchthingasa nonelectric"substance.Matterisbothinherentlygravitationalandinherentlyelectrical. Experimentsshowthattheelectricalforce,likethegravitationalforce,isan inversesquare force.Thatis,theelectrical forcebetweentwospheresisproportionalto1 =r 2 ,where r is thecenter-to-centerdistancebetweenthem. Furthermore,electricalforcesmakemoresensethangravityas candidatesforthefundamentalforcebetweenatoms,becausewe haveobservedthattheycanbeeitherattractiveorrepulsive. 1.2Charge,electricityandmagnetism Charge Charge"isthetechnicaltermusedtoindicatethatanobject hasbeenpreparedsoastoparticipateinelectricalforces.Thisis tobedistinguishedfromthecommonusage,inwhichthetermis usedindiscriminatelyforanythingelectrical.Forexample,although wespeakcolloquiallyofcharging"abattery,youmayeasilyverify thatabatteryhasnochargeinthetechnicalsense,e.g.,itdoesnot exertanyelectricalforceonapieceoftapethathasbeenprepared asdescribedintheprevioussection. Twotypesofcharge Wecaneasilycollectreamsofdataonelectricalforcesbetween dierentsubstancesthathavebeenchargedindierentways.We ndforexamplethatcatfurpreparedbyrubbingagainstrabbit furwillattractglassthathasbeenrubbedonsilk.Howcanwe makeanysenseofallthisinformation?Avastsimplicationis achievedbynotingthattherearereallyonlytwotypesofcharge. Section1.2Charge,electricityandmagnetism 15

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Supposewepickcatfurrubbedonrabbitfurasarepresentativeof typeA,andglassrubbedonsilkfortypeB.Wewillnowndthat thereisnotypeC."Anyobjectelectriedbyanymethodiseither A-like,attractingthingsAattractsandrepellingthoseitrepels,or B-like,displayingthesameattractionsandrepulsionsasB.Thetwo types,AandB,alwaysdisplayoppositeinteractions.IfAdisplays anattractionwithsomechargedobject,thenBisguaranteedto undergorepulsionwithit,andvice-versa. Thecoulomb Althoughthereareonlytwotypesofcharge,eachtypecancome indierentamounts.Themetricunitofchargeisthecoulomb rhymeswithdroolon",denedasfollows: OneCoulombCistheamountofchargesuchthataforceof 9.0 10 9 Noccursbetweentwopointlikeobjectswithcharges of1Cseparatedbyadistanceof1m. Thenotationforanamountofchargeis q .Thenumericalfactor inthedenitionishistoricalinorigin,andisnotworthmemorizing.Thedenitionisstatedforpointlike,i.e.,verysmall,objects, becauseotherwisedierentpartsofthemwouldbeatdierentdistancesfromeachother. Amodeloftwotypesofchargedparticles Experimentsshowthatallthemethodsofrubbingorotherwise chargingobjectsinvolvetwoobjects,andbothofthemendupgettingcharged.Ifoneobjectacquiresacertainamountofonetypeof charge,thentheotherendsupwithanequalamountoftheother type.Variousinterpretationsofthisarepossible,butthesimplest isthatthebasicbuildingblocksofmattercomeintwoavors,one witheachtypeofcharge.Rubbingobjectstogetherresultsinthe transferofsomeoftheseparticlesfromoneobjecttotheother.In thismodel,anobjectthathasnotbeenelectricallypreparedmayactuallypossessesagreatdealof both typesofcharge,buttheamounts areequalandtheyaredistributedinthesamewaythroughoutit. SincetypeArepelsanythingthattypeBattracts,andviceversa, theobjectwillmakeatotalforceofzeroonanyotherobject.The restofthischaptereshesoutthismodelanddiscusseshowthese mysteriousparticlescanbeunderstoodasbeinginternalpartsof atoms. Useofpositiveandnegativesignsforcharge Becausethetwotypesofchargetendtocancelouteachother's forces,itmakessensetolabelthemusingpositiveandnegativesigns, andtodiscussthe total chargeofanobject.Itisentirelyarbitrary whichtypeofchargetocallnegativeandwhichtocallpositive. BenjaminFranklindecidedtodescribetheonewe'vebeencalling A"asnegative,butitreallydoesn'tmatteraslongaseveryoneis 16 Chapter1ElectricityandtheAtom

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consistentwitheveryoneelse.Anobjectwithatotalchargeofzero equalamountsofbothtypesisreferredtoaselectrically neutral self-checkA Criticizethefollowingstatement:Therearetwotypesofcharge,attractiveandrepulsive. Answer,p. 203 Coulomb'slaw Alargebodyofexperimentalobservationscanbesummarized asfollows: Coulomb'slaw:Themagnitudeoftheforceactingbetween pointlikechargedobjectsatacenter-to-centerdistance r isgiven bytheequation j F j = k j q 1 jj q 2 j r 2 wheretheconstant k equals9.0 10 9 N m 2 = C 2 .Theforceisattractiveifthechargesareofdierentsigns,andrepulsiveiftheyhave thesamesign. Clevermoderntechniqueshaveallowedthe1 =r 2 formofCoulomb's lawtobetestedtoincredibleaccuracy,showingthattheexponent isintherangefrom1.9999999999999998to2.0000000000000002. NotethatCoulomb'slawiscloselyanalogoustoNewton'slaw ofgravity,wherethemagnitudeoftheforceis Gm 1 m 2 =r 2 ,except thatthereisonlyonetypeofmass,nottwo,andgravitationalforces areneverrepulsive.Becauseofthiscloseanalogybetweenthetwo typesofforces,wecanrecycleagreatdealofourknowledgeof gravitationalforces.Forinstance,thereisanelectricalequivalent oftheshelltheorem:theelectricalforcesexertedexternallybya uniformlychargedsphericalshellarethesameasifallthecharge wasconcentratedatitscenter,andtheforcesexertedinternallyare zero. Conservationofcharge Anevenmorefundamentalreasonforusingpositiveandnegativesignsforelectricalchargeisthatexperimentsshowthatcharge isconservedaccordingtothisdenition:inanyclosedsystem,the totalamountofchargeisaconstant.Thisiswhyweobservethat rubbinginitiallyunchargedsubstancestogetheralwayshastheresultthatonegainsacertainamountofonetypeofcharge,while theotheracquiresanequalamountoftheothertype.Conservation ofchargeseemsnaturalinourmodelinwhichmatterismadeof positiveandnegativeparticles.Ifthechargeoneachparticleisa xedpropertyofthattypeofparticle,andiftheparticlesthemselves canbeneithercreatednordestroyed,thenconservationofchargeis inevitable. Section1.2Charge,electricityandmagnetism 17

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b / Achargedpieceoftape attractsunchargedpiecesof paperfromadistance,andthey leapuptoit. c / Thepaperhaszerototal charge,butitdoeshavecharged particlesinitthatcanmove. Electricalforcesinvolvingneutralobjects Asshowningureb,anelectricallychargedobjectcanattract objectsthatareuncharged.Howisthispossible?Thekeyisthat eventhougheachpieceofpaperhasatotalchargeofzero,ithasat leastsomechargedparticlesinitthathavesomefreedomtomove. Supposethatthetapeispositivelycharged,c.Mobileparticles inthepaperwillrespondtothetape'sforces,causingoneendof thepapertobecomenegativelychargedandtheothertobecome positive.Theattractionisbetweenthepaperandthetapeisnow strongerthantherepulsion,becausethenegativelychargedendis closertothetape. self-checkB Whatwouldhavehappenedifthetapewasnegativelycharged? Answer,p.203 Thepathahead Wehavebeguntoencountercomplexelectricalbehaviorthatwe wouldneverhaverealizedwasoccurringjustfromtheevidenceofour eyes.Unlikethepulleys,blocks,andinclinedplanesofmechanics, theactorsonthestageofelectricityandmagnetismareinvisible phenomenaalientooureverydayexperience.Forthisreason,the avorofthesecondhalfofyourphysicseducationisdramatically dierent,focusingmuchmoreonexperimentsandtechniques.Even thoughyouwillneveractuallyseechargemovingthroughawire, youcanlearntouseanammetertomeasuretheow. Studentsalsotendtogettheimpressionfromtheirrstsemester ofphysicsthatitisadeadscience.Notso!Weareabouttopick upthehistoricaltrailthatleadsdirectlytothecutting-edgephysics researchyoureadaboutinthenewspaper.Theatom-smashingexperimentsthatbeganaround1900,whichwewillbestudyinginthis chapter,werenotthatdierentfromtheonesoftheyear2000| justsmaller,simpler,andmuchcheaper. Magneticforces Adetailedmathematicaltreatmentofmagnetismwon'tcome untilmuchlaterinthisbook,butweneedtodevelopafewsimple ideasaboutmagnetismnowbecausemagneticforcesareusedinthe experimentsandtechniqueswecometonext.Everydaymagnets comeintwogeneraltypes.Permanentmagnets,suchastheones onyourrefrigerator,aremadeofironorsubstanceslikesteelthat containironatoms.Certainothersubstancesalsowork,butiron isthecheapestandmostcommon.Theothertypeofmagnet, anexampleofwhichistheonesthatmakeyourstereospeakers vibrate,consistofcoilsofwirethroughwhichelectricchargeows. Bothtypesofmagnetsareabletoattractironthathasnotbeen 18 Chapter1ElectricityandtheAtom

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magneticallyprepared,forinstancethedooroftherefrigerator. Asingleinsightmakestheseapparentlycomplexphenomena muchsimplertounderstand:magneticforcesareinteractionsbetweenmovingcharges,occurringinadditiontotheelectricforces. Supposeapermanentmagnetisbroughtnearamagnetofthecoiledwiretype.Thecoiledwirehasmovingchargesinitbecauseweforce chargetoow.Thepermanentmagnetalsohasmovingchargesin it,butinthiscasethechargesthatnaturallyswirlaroundinsidethe iron.Whatmakesamagnetizedpieceofirondierentfromablock ofwoodisthatthemotionofthechargeinthewoodisrandom ratherthanorganized.Themovingchargesinthecoiled-wiremagnetexertaforceonthemovingchargesinthepermanentmagnet, andvice-versa. Themathematicsofmagnetismissignicantlymorecomplex thantheCoulombforcelawforelectricity,whichiswhywewill waituntilchapter6beforedelvingdeeplyintoit.Twosimplefacts willsucefornow: Ifachargedparticleismovinginaregionofspacenearwhere otherchargedparticlesarealsomoving,theirmagneticforceonit isdirectlyproportionaltoitsvelocity. Themagneticforceonamovingchargedparticleisalways perpendiculartothedirectiontheparticleismoving. Amagneticcompassexample1 TheEarthismolteninside,andlikeapotofboilingwater,itroils andchurns.Tomakeadrasticoversimplication,electriccharge cangetcarriedalongwiththechurningmotion,sotheEarthcontainsmovingcharge.Theneedleofamagneticcompassisitself asmallpermanentmagnet.Themovingchargeinsidetheearth interactsmagneticallywiththemovingchargeinsidethecompass needle,causingthecompassneedletotwistaroundandpoint north. Atelevisiontubeexample2 ATVpictureispaintedbyastreamofelectronscomingfrom thebackofthetubetothefront.Thebeamscansacrossthe wholesurfaceofthetubelikeareaderscanningapageofabook. Magneticforcesareusedtosteerthebeam.Asthebeamcomes fromthebackofthetubetothefront,up-downandleft-rightforces areneededforsteering.Butmagneticforcescannotbeused togetthebeamuptospeedintherstplace,sincetheycan onlypushperpendiculartotheelectrons'directionofmotion,not forwardalongit. DiscussionQuestions A Iftheelectricalattractionbetweentwopointlikeobjectsatadistance of1mis9 10 9 N,whycan'tweinferthattheirchargesare+1and )]TJ/F39 9.9626 Tf 7.748 0 Td [(1C? Whatfurtherobservationswouldweneedtodoinordertoprovethis? Section1.2Charge,electricityandmagnetism 19

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B Anelectricallychargedpieceoftapewillbeattractedtoyourhand. Doesthatallowustotellwhetherthemobilechargedparticlesinyour handarepositiveornegative,orboth? 1.3Atoms Iwasbroughtuptolookattheatomasanice,hardfellow,redor greyincoloraccordingtotaste. Rutherford Atomism TheGreekshavebeenkickedaroundalotinthelastcoupleof millennia:dominatedbytheRomans,bulliedduringthecrusades bywarlordsgoingtoandfromtheHolyLand,andoccupiedby Turkeyuntilrecently.It'snowondertheyprefertoremembertheir saladdays,whentheirbestthinkerscameupwithconceptslike democracyandatoms.Greeceisdemocraticagainafteraperiod ofmilitarydictatorship,andanatomisproudlypicturedononeof theircoins.That'swhyithurtsmetohavetosaythattheancient Greekhypothesisthatmatterismadeofatomswaspureguesswork.Therewasnorealexperimentalevidenceforatoms,andthe 18th-centuryrevivaloftheatomconceptbyDaltonowedlittleto theGreeksotherthanthename,whichmeansunsplittable."SubtractingevenmorecruellyfromGreekglory,thenamewasshown tobeinappropriatein1899whenphysicistJ.J.Thomsonprovedexperimentallythatatomshadevensmallerthingsinsidethem,which couldbeextracted.Thomsoncalledthemelectrons."Theunsplittable"wassplittableafterall. Butthat'sgettingaheadofourstory.Whathappenedtothe atomconceptintheinterveningtwothousandyears?Educatedpeoplecontinuedtodiscusstheidea,andthosewhowereinfavorofit couldoftenuseittogiveplausibleexplanationsforvariousfactsand phenomena.Onefactthatwasreadilyexplainedwasconservation ofmass.Forexample,ifyoumix1kgofwaterwith1kgofdirt, yougetexactly2kgofmud,nomoreandnoless.Thesameistrue fortheavarietyofprocessessuchasfreezingofwater,fermenting beer,orpulverizingsandstone.Ifyoubelievedinatoms,conservationofmassmadeperfectsense,becausealltheseprocessescould beinterpretedasmixingandrearrangingatoms,withoutchanging thetotalnumberofatoms.Still,thisisnothinglikeaproofthat atomsexist. Ifatomsdidexist,whattypesofatomswerethere,andwhatdistinguishedthedierenttypesfromeachother?Wasittheirsizes, theirshapes,theirweights,orsomeotherquality?Thechasmbetweentheancientandmodernatomismsbecomesevidentwhenwe considerthewildspeculationsthatexistedontheseissuesuntilthe presentcentury.Theancientsdecidedthattherewerefourtypesof atoms,earth,water,airandre;themostpopularviewwasthat 20 Chapter1ElectricityandtheAtom

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theyweredistinguishedbytheirshapes.Wateratomswerespherical,hencewater'sabilitytoowsmoothly.Fireatomshadsharp points,whichwaswhyrehurtwhenittouchedone'sskin.There wasnoconceptoftemperatureuntilthousandsofyearslater.The drasticallydierentmodernunderstandingofthestructureofatoms wasachievedinthecourseoftherevolutionarydecadestretching 1895to1905.Themainpurposeofthischapteristodescribethose momentousexperiments. Areyounoworhaveyoueverbeenanatomist? Youarewhatyoueat.Theglibmodernphrasemoreorlessassumes theatomicexplanationofdigestion.Afterall,digestionwasprettymysteriousinancienttimes,andpremoderncultureswouldtypicallybelieve thateatingallowedyoutoextractsomekindoflifeforcefromthefood. Mythsaboundtotheeffectthatabstractqualitiessuchasbraveryor ritualimpuritycanenteryourbodyviathefoodyoueat.Incontrastto thesesupernaturalpointsofview,theancientatomistshadanentirely naturalisticinterpretationofdigestion.Thefoodwasmadeofatoms, andwhenyoudigestedityouweresimplyextractingsomeatomsfrom itandrearrangingthemintothecombina-tionsrequiredforyourown bodytissues.Themoreprogressivemedievalandrenaissancescientistslovedthiskindofexplanation.Theywereanxioustodriveastake throughtheheartofAristotelianphysicsanditsembellished,Churchfriendlyversion,scholasticism,whichintheirviewascribedtoomany occultpropertiesandpurposestoobjects.Forinstance,theAristotelianexplanationforwhyarockwouldfalltoearthwasthatitwas itsnatureorpurposetocometorestontheground. Theseeminglyinnocentattempttoexplaindigestionnaturalistically, however,endedupgettingtheatomistsinbigtroublewiththeChurch. TheproblemwasthattheChurch'smostimportantsacramentinvolves eatingbreadandwineandtherebyreceivingthesupernaturaleffectof forgivenessofsin.Inconnectionwiththisritual,thedoctrineoftransubstantiationassertsthattheblessingoftheeucharisticbreadandwine literallytransformsitintothebloodandeshofChrist.Atomismwas perceivedascontradictingtransubstantiation,sinceatomismseemed todenythattheblessingcouldchangethenatureoftheatoms.Althoughthehistoricalinformationgiveninmostsciencetextbooksabout Galileorepresentshisrun-inwiththeInquisitionasturningontheissue ofwhethertheearthmoves,somehistoriansbelievehispunishment hadmoretodowiththeperceptionthathisadvocacyofatomismsubvertedtransubstantiation.Otherissuesinthecomplexsituationwere Galileo'sconfrontationalstyle,PopeUrban'smilitaryproblems,andrumorsthatthestupidcharacterinGalileo'sdialogueswasmeanttobe thePope.Foralongtime,beliefinatomismservedasabadgeof nonconformityforscientists,awayofassertingapreferencefornatural ratherthansupernaturalinterpreta-tionsofphenomena.Galileoand Newton'sespousalofatomismwasanactofrebellion,likelatergenerations'adoptionofDarwinismorMarxism. Anotherconictbetweenscholasticismandatomismcamefromthe questionofwhatwasbetweentheatoms.Ifyouaskmodernpeoplethis question,theywillprobablyreplynothingoremptyspace.ButAristotleandhisscholasticsuccessorsbelievedthattherecouldbenosuch thingasemptyspace,i.e.,avacuum.Thatwasnotanunreasonable Section1.3Atoms 21

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pointofview,becauseairtendstorushintoanyspaceyouopenup, anditwasn'tuntiltherenaissancethatpeopleguredouthowtomake avacuum. Atoms,light,andeverythingelse AlthoughItendtoridiculeancientGreekphilosopherslikeAristotle,let'stakeamomenttopraisehimforsomething.Ifyouread Aristotle'swritingsonphysicsorjustskimthem,whichisallI've done,themoststrikingthingishowcarefulheisaboutclassifying phenomenaandanalyzingrelationshipsamongphenomena.Thehumanbrainseemstonaturallymakeadistinctionbetweentwotypes ofphysicalphenomena:objectsandmotionofobjects.Whena phenomenonoccursthatdoesnotimmediatelypresentitselfasone ofthese,thereisastrongtendencytoconceptualizeitasoneor theother,oreventoignoreitsexistencecompletely.Forinstance, physicsteachersshudderatstudents'statementsthatthedynamite exploded,andforcecameoutofitinalldirections."Intheseexamples,thenonmaterialconceptofforceisbeingmentallycategorized asifitwasaphysicalsubstance.Thestatementthatwindingthe clockstoresmotioninthespring"isamiscategorizationofelectrical energyasaformofmotion.Anexampleofignoringtheexistence ofaphenomenonaltogethercanbeelicitedbyaskingpeoplewhy weneedlamps.Thetypicalresponsethatthelampilluminates theroomsowecanseethings,"ignoresthenecessaryroleoflight comingintooureyesfromthethingsbeingilluminated. Ifyouasksomeonetotellyoubrieyaboutatoms,thelikely responseisthateverythingismadeofatoms,"butwe'venowseen thatit'sfarfromobviouswhicheverything"thisstatementwould properlyreferto.Forthescientistsoftheearly1900swhowere tryingtoinvestigateatoms,thiswasnotatrivialissueofdenitions.Therewasanewgizmocalledthevacuumtube,ofwhichthe onlyfamiliarexampletodayisthepicturetubeofaTV.Inshort order,electricaltinkerershaddiscoveredawholeockofnewphenomenathatoccurredinandaroundvacuumtubes,andgiventhem picturesquenameslikex-rays,"cathoderays,"Hertzianwaves," andN-rays."Thesewerethetypesofobservationsthatendedup tellingusthatweknowaboutmatter,butercecontroversiesensued overwhetherthesewerethemselvesformsofmatter. Let'sbringourselvesuptothelevelofclassicationofphenomenaemployedbyphysicistsintheyear1900.Theyrecognizedthree categories: Matter hasmass,canhavekineticenergy,andcantravel throughavacuum,transportingitsmassandkineticenergy withit.Matterisconserved,bothinthesenseofconservation ofmassandconservationofthenumberofatomsofeachelement.Atomscan'toccupythesamespaceasotheratoms,so 22 Chapter1ElectricityandtheAtom

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m He m H =3.97 m Ne m H =20.01 m Sc m H =44.60 d / Examplesofmassesof atomscomparedtothatofhydrogen.Notehowsome,butnotall, areclosetointegers. aconvenientwaytoprovesomethingisnotaformofmatter istoshowthatitcanpassthroughasolidmaterial,inwhich theatomsarepackedtogetherclosely. Light hasnomass,alwayshasenergy,andcantravelthrougha vacuum,transportingitsenergywithit.Twolightbeamscan penetratethrougheachotherandemergefromthecollision withoutbeingweakened,deected,oraectedinanyother way.Lightcanpenetratecertainkindsofmatter,e.g.,glass. Thethirdcategoryiseverythingthatdoesn'ttthedenitionoflightormatter.Thiscatch-allcategoryincludes,for example,time,velocity,heat,andforce. Thechemicalelements Howwouldonendoutwhattypesofatomstherewere?Today,itdoesn'tseemlikeitshouldhavebeenverydiculttowork outanexperimentalprogramtoclassifythetypesofatoms.For eachtypeofatom,thereshouldbeacorrespondingelement,i.e.,a puresubstancemadeoutofnothingbutthattypeofatom.Atoms aresupposedtobeunsplittable,soasubstancelikemilkcouldnot possiblybeelemental,sincechurningitvigorouslycausesittosplit upintotwoseparatesubstances:butterandwhey.Similarly,rust couldnotbeanelement,becauseitcanbemadebycombiningtwo substances:ironandoxygen.Despiteitsapparentreasonableness, nosuchprogramwascarriedoutuntiltheeighteenthcentury.The ancientspresumablydidnotdoitbecauseobservationwasnotuniversallyagreedonastherightwaytoanswerquestionsaboutnature, andalsobecausetheylackedthenecessarytechniquesorthetechniquesweretheprovinceoflaborerswithlowsocialstatus,suchas smithsandminers.Alchemistswerehinderedbyatomism'sreputationforsubversiveness,andbyatendencytowardmysticismand secrecy.Themostcelebratedchallengefacingthealchemists,that ofconvertingleadintogold,isonewenowknowtobeimpossible, sinceleadandgoldarebothelements. By1900,however,chemistshaddoneareasonablygoodjobof ndingoutwhattheelementswere.Theyalsohaddeterminedthe ratiosofthedierentatoms'massesfairlyaccurately.Atypical techniquewouldbetomeasurehowmanygramsofsodiumNa wouldcombinewithonegramofchlorineCltomakesaltNaCl. Thisassumesyou'vealreadydecidedbasedonotherevidencethat saltconsistedofequalnumbersofNaandClatoms.Themassesof individualatoms,asopposedtothemassratios,wereknownonly towithinafewordersofmagnitudebasedonindirectevidence,and plentyofphysicistsandchemistsdeniedthatindividualatomswere anythingmorethanconvenientsymbols. Thefollowingtablegivestheatomicmassesofalltheelements, onastandardscaleinwhichthemassofhydrogenisverycloseto1.0. Section1.3Atoms 23

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Theabsolutecalibrationofthewholescalewasonlyveryroughly knownforalongtime,butwaseventuallytieddown,withthemass ofahydrogenatombeingdeterminedtobeabout1.7 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(27 kg. Ag107.9 Eu152.0 Mo95.9 Sc45.0 Al27.0 F19.0 N14.0 Se79.0 Ar39.9 Fe55.8 Na23.0 Si28.1 As74.9 Ga69.7 Nb92.9 Sn118.7 Au197.0 Gd157.2 Nd144.2 Sr87.6 B10.8 Ge72.6 Ne20.2 Ta180.9 Ba137.3 H1.0 Ni58.7 Tb158.9 Be9.0 He4.0 O16.0 Te127.6 Bi209.0 Hf178.5 Os190.2 Ti47.9 Br79.9 Hg200.6 P31.0 Tl204.4 C12.0 Ho164.9 Pb207.2 Tm168.9 Ca40.1 In114.8 Pd106.4 U238 Ce140.1 Ir192.2 Pt195.1 V50.9 Cl35.5 K39.1 Pr140.9 W183.8 Co58.9 Kr83.8 Rb85.5 Xe131.3 Cr52.0 La138.9 Re186.2 Y88.9 Cs132.9 Li6.9 Rh102.9 Yb173.0 Cu63.5 Lu175.0 Ru101.1 Zn65.4 Dy162.5 Mg24.3 S32.1 Zr91.2 Er167.3 Mn54.9 Sb121.8 Makingsenseoftheelements Astheinformationaccumulated,thechallengewastonda wayofsystematizingit;themodernscientist'saestheticsenserebels againstcomplication.Thishodgepodgeofelementswasanembarrassment.Onecontemporaryobserver,WilliamCrookes,described theelementsasextendingbeforeusasstretchedthewideAtlantic beforethegazeofColumbus,mocking,tauntingandmurmuring strangeriddles,whichnomanhasyetbeenabletosolve."Itwasn't longbeforepeoplestartedrecognizingthatmanyatoms'masseswere nearlyintegermultiplesofthemassofhydrogen,thelightestelement.Afewexcitabletypesbeganspeculatingthathydrogenwas thebasicbuildingblock,andthattheheavierelementsweremade ofclustersofhydrogen.Itwasn'tlong,however,beforetheirparade wasrainedonbymoreaccuratemeasurements,whichshowedthat notalloftheelementshadatomicmassesthatwerenearinteger multiplesofhydrogen,andeventheonesthatwereclosetobeing integermultipleswereobyonepercentorso. ChemistryprofessorDmitriMendeleev,preparinghislecturesin 1869,wantedtondsomewaytoorganizehisknowledgeforhisstudentstomakeitmoreunderstandable.Hewrotethenamesofall theelementsoncardsandbeganarrangingthemindierentways onhisdesk,tryingtondanarrangementthatwouldmakesenseof themuddle.Therow-and-columnschemehecameupwithisessentiallyourmodernperiodictable.Thecolumnsofthemodernversion representgroupsofelementswithsimilarchemicalproperties,and eachrowismoremassivethantheoneaboveit.Goingacrosseach row,thisalmostalwaysresultedinplacingtheatomsinsequence 24 Chapter1ElectricityandtheAtom

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e / Amodernperiodictable.Elementsinthesamecolumnhave similarchemicalproperties.The modernatomicnumbers,discussedinsection2.3,werenot knowninMendeleev'stime,since thetablecouldbeippedinvariousways. byweightaswell.Whatmadethesystemsignicantwasitspredictivevalue.TherewerethreeplaceswhereMendeleevhadtoleave gapsinhischeckerboardtokeepchemicallysimilarelementsinthe samecolumn.Hepredictedthatelementswouldexisttollthese gaps,andextrapolatedorinterpolatedfromotherelementsinthe samecolumntopredicttheirnumericalproperties,suchasmasses, boilingpoints,anddensities.Mendeleev'sprofessionalstockskyrocketedwhenhisthreeelementslaternamedgallium,scandium andgermaniumwerediscoveredandfoundtohaveverynearlythe propertieshehadpredicted. OnethingthatMendeleev'stablemadeclearwasthatmasswas notthebasicpropertythatdistinguishedatomsofdierentelements.Tomakehistablework,hehadtodeviatefromordering theelementsstrictlybymass.Forinstance,iodineatomsarelighter thantellurium,butMendeleevhadtoputiodineaftertelluriumso thatitwouldlieinacolumnwithchemicallysimilarelements. Directproofthatatomsexisted Thesuccessofthekinetictheoryofheatwastakenasstrongevidencethat,inadditiontothemotionofanyobjectasawhole,there isaninvisibletypeofmotionallaroundus:therandommotionof atomswithineachobject.Butmanyconservativeswerenotconvincedthatatomsreallyexisted.Nobodyhadeverseenone,after all.Itwasn'tuntilgenerationsafterthekinetictheoryofheatwas developedthatitwasdemonstratedconclusivelythatatomsreally existedandthattheyparticipatedincontinuousmotionthatnever diedout. Thesmokingguntoproveatomsweremorethanmathematical abstractionscamewhensomeold,obscureobservationswerereexaminedbyanunknownSwisspatentclerknamedAlbertEinstein. AbotanistnamedBrown,usingamicroscopethatwasstateofthe artin1827,observedtinygrainsofpolleninadropofwaterona microscopeslide,andfoundthattheyjumpedaroundrandomlyfor Section1.3Atoms 25

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f / AyoungRobertMillikan. g / Asimplieddiagramof Millikan'sapparatus. noapparentreason.Wonderingatrstifthepollenhe'dassumedto bedeadwasactuallyalive,hetriedlookingatparticlesofsoot,and foundthatthesootparticlesalsomovedaround.Thesameresults wouldoccurwithanysmallgrainorparticlesuspendedinaliquid. ThephenomenoncametobereferredtoasBrownianmotion,and itsexistencewasledawayasaquaintandthoroughlyunimportant fact,reallyjustanuisanceforthemicroscopist. Itwasn'tuntil1906thatEinsteinfoundthecorrectinterpretationforBrown'sobservation:thewatermoleculeswereincontinuous randommotion,andwerecollidingwiththeparticleallthetime, kickingitinrandomdirections.Afterallthemillenniaofspeculation aboutatoms,atlasttherewassolidproof.Einstein'scalculations dispelledalldoubt,sincehewasabletomakeaccuratepredictions ofthingsliketheaveragedistancetraveledbytheparticleinacertainamountoftime.EinsteinreceivedtheNobelPrizenotforhis theoryofrelativitybutforhispapersonBrownianmotionandthe photoelectriceect. DiscussionQuestions A Howcouldknowledgeofthesizeofanindividualaluminumatombe usedtoinferanestimateofitsmass,orviceversa? B HowcouldonetestEinstein'sinterpretationofBrownianmotionby observingitatdifferenttemperatures? 1.4Quantizationofcharge Provingthatatomsactuallyexistedwasabigaccomplishment,but demonstratingtheirexistencewasdierentfromunderstandingtheir properties.NotethattheBrown-Einsteinobservationshadnothing atalltodowithelectricity,andyetweknowthatmatterisinherentlyelectrical,andwehavebeensuccessfulininterpretingcertain electricalphenomenaintermsofmobilepositivelyandnegatively chargedparticles.Aretheseparticlesatoms?Partsofatoms?Particlesthatareentirelyseparatefromatoms?Itisperhapsprematuretoattempttoanswerthesequestionswithoutanyconclusive evidenceinfavorofthecharged-particlemodelofelectricity. Strongsupportforthecharged-particlemodelcamefroma1911 experimentbyphysicistRobertMillikanattheUniversityofChicago. Considerajetofdropletsofperfumeorsomeotherliquidmadeby blowingitthroughatinypinhole.Thedropletsemergingfromthe pinholemustbesmallerthanthepinhole,andinfactmostofthem areevenmoremicroscopicthanthat,sincetheturbulentowofair tendstobreakthemup.Millikanreasonedthatthedropletswould acquirealittlebitofelectricchargeastheyrubbedagainstthechannelthroughwhichtheyemerged,andifthecharged-particlemodel ofelectricitywasright,thechargemightbesplitupamongsomany minusculeliquiddropsthatasingledropmighthaveatotalcharge 26 Chapter1ElectricityandtheAtom

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amountingtoanexcessofonlyafewchargedparticles|perhaps anexcessofonepositiveparticleonacertaindrop,oranexcessof twonegativeonesonanother. Millikan'singeniousapparatus,g,consistedoftwometalplates, whichcouldbeelectricallychargedasneeded.Hesprayedacloudof oildropletsintothespacebetweentheplates,andselectedonedrop throughamicroscopeforstudy.First,withnochargeontheplates, hewoulddeterminethedrop'smassbylettingitfallthroughthe airandmeasuringitsterminalvelocity,i.e.,thevelocityatwhich theforceofairfrictioncanceledouttheforceofgravity.Theforce ofairdragonaslowlymovingspherehadalreadybeenfoundby experimenttobe bvr 2 ,where b wasaconstant.Settingthetotal forceequaltozerowhenthedropisatterminalvelocitygives bvr 2 )]TJ/F20 10.9091 Tf 10.909 0 Td [(mg =0, andsettingtheknowndensityofoilequaltothedrop'smassdivided byitsvolumegivesasecondequation, = m 4 3 r 3 Everythingintheseequationscanbemeasureddirectlyexceptfor m and r ,sothesearetwoequationsintwounknowns,whichcanbe solvedinordertodeterminehowbigthedropis. NextMillikanchargedthemetalplates,adjustingtheamount ofchargesoastoexactlycounteractgravityandlevitatethedrop. If,forinstance,thedropbeingexaminedhappenedtohaveatotal chargethatwasnegative,thenpositivechargeputonthetopplate wouldattractit,pullingitup,andnegativechargeonthebottom platewouldrepelit,pushingitup.Theoreticallyonlyoneplate wouldbenecessary,butinpracticeatwo-platearrangementlikethis gaveelectricalforcesthatweremoreuniforminstrengththroughout thespacewheretheoildropswere.Theamountofchargeonthe platesrequiredtolevitatethechargeddropgaveMillikanahandle ontheamountofchargethedropcarried.Themorechargethe drophad,thestrongertheelectricalforcesonitwouldbe,andthe lesschargewouldhavetobeputontheplatestodothetrick.Unfortunately,expressingthisrelationshipusingCoulomb'slawwould havebeenimpractical,becauseitwouldrequireaperfectknowledge ofhowthechargewasdistributedoneachplate,plustheability toperformvectoradditionofalltheforcesbeingexertedonthe dropbyallthechargesontheplate.Instead,Millikanmadeuseof thefactthattheelectricalforceexperiencedbyapointlikecharged objectatacertainpointinspaceisproportionaltoitscharge, F q =constant. Withagivenamountofchargeontheplates,thisconstantcouldbe determinedforinstancebydiscardingtheoildrop,insertingbetween Section1.4Quantizationofcharge 27

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q = .64 q C 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(19 C )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.970 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(18 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12.02 )]TJ/F15 10.9091 Tf 8.485 0 Td [(0.987 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(18 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6.02 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.773 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(18 )]TJ/F15 10.9091 Tf 8.485 0 Td [(16.93 h / AfewsamplesofMillikan's data. theplatesalargerandmoreeasilyhandledobjectwithaknown chargeonit,andmeasuringtheforcewithconventionalmethods. Millikanactuallyusedaslightlydierentsetoftechniquesfordeterminingtheconstant,buttheconceptisthesame.Theamount offorceontheactualoildrophadtoequal mg ,sinceitwasjust enoughtolevitateit,andoncethecalibrationconstanthadbeen determined,thechargeofthedropcouldthenbefoundbasedonits previouslydeterminedmass. TablehshowsafewoftheresultsfromMillikan's1911paper. Millikantookdataonbothnegativelyandpositivelychargeddrops, butinhispaperhegaveonlyasampleofhisdataonnegatively chargeddrops,sothesenumbersareallnegative.Evenaquick lookatthedataleadstothesuspicionthatthechargesarenot simplyaseriesofrandomnumbers.Forinstance,thesecondcharge isalmostexactlyequaltohalftherstone.Millikanexplainedthe observedchargesasallbeingintegermultiplesofasinglenumber, 1.64 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(19 C.Inthesecondcolumn,dividingbythisconstantgives numbersthatareessentiallyintegers,allowingfortherandomerrors presentintheexperiment.Millikanstatesinhispaperthatthese resultswerea ...directandtangibledemonstration...ofthecorrectnessoftheviewadvancedmanyyearsagoandsupported byevidencefrommanysourcesthatallelectricalcharges, howeverproduced,areexactmultiplesofonedenite, elementaryelectricalcharge,orinotherwords,thatan electricalchargeinsteadofbeingspreaduniformlyover thechargedsurfacehasadenitegranularstructure, consisting,infact,of...specks,oratomsofelectricity,allpreciselyalike,pepperedoverthesurfaceofthe chargedbody. Inotherwords,hehadprovideddirectevidenceforthechargedparticlemodelofelectricityandagainstmodelsinwhichelectricity wasdescribedassomesortofuid.Thebasicchargeisnotated e andthemodernvalueis e =1.60 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(19 C.Theword quantized isusedinphysicstodescribeaquantitythatcanonlyhavecertain numericalvalues,andcannothaveanyofthevaluesbetweenthose. Inthislanguage,wewouldsaythatMillikandiscoveredthatcharge isquantized.Thecharge e isreferredtoasthequantumofcharge. self-checkC Ismoneyquantized?Whatisthequantumofmoney? Answer,p. 203 28 Chapter1ElectricityandtheAtom

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AhistoricalnoteonMillikan'sfraud Veryfewundergraduatephysicstextbooksmentionthewell-documented factthatalthoughMillikan'sconclusionswerecorrect,hewasguiltyof scienticfraud.Histechniquewasdifcultandpainstakingtoperform, andhisoriginalnotebooks,whichhavebeenpreserved,showthatthe datawerefarlessperfectthanheclaimedinhispublishedscientic papers.Inhispublications,hestatedcategoricallythateverysingle oildropobservedhadhadachargethatwasamultipleof e ,withno Exceptionsoromissions.Buthisnotebooksarerepletewithnotations suchasbeautifuldata,keep,andbadrun,throwout.Millikan,then, appearstohaveearnedhisNobelPrizebyadvocatingacorrectposition withdishonestdescriptionsofhisdata. WhydotextbookauthorsfailtomentionMillikan'sfraud?Itmaybe thattheythinkstudentsaretoounsophisticatedtocorrectlyevaluatethe implicationsofthefactthatscienticfraudhassometimesexistedand evenbeenrewardedbythescienticestablishment.Maybetheyare afraidstudentswillreasonthatfudgingdataisOK,sinceMillikangot theNobelPrizeforit.Butfalsifyinghistoryinthenameofencouragingtruthfulnessismorethanalittleironic.Englishteachersdon'tedit Shakespeare'stragediessothatthebadcharactersarealwayspunishedandthegoodonesneversuffer! Anotherpossibleexplanationissimplyalackoforiginality;it'spossiblethatsomeveneratedtextbookwasuncriticalofMillikan'sfraud,and laterauthorssimplyfollowedsuit.BiologistStephenJayGouldhaswrittenanessaytracinganexampleofhowauthorsofbiologytextbooks tendtofollowacertaintraditionaltreatmentofatopic,usingthegiraffe'snecktodiscussthenonheritabilityofacquiredtraits.Yetanother interpretationisthatscientistsderivestatusfromtheirpopularimages asimpartialsearchersafterthetruth,andtheydon'twantthepublicto realizehowhumanandimperfecttheycanbe.Millikanhimselfwasan educationalreformer,andwroteaseriesoftextbooksthatwereofmuch higherqualitythanothersofhisera. NoteaddedSeptember2002 SeveralyearsafterIwrotethishistoricaldigression,Icameacrossan interestingdefenseofMillikanbyDavidGoodsteinAmericanScientist, Jan-Feb2001,pp.54-60.GoodsteinarguesthatalthoughMillikan wroteasentenceinhispaperthatwasalie,Millikanisneverthelessnot guiltyoffraudwhenwetakethatsentenceincontext:Millikanstated thatheneverthrewoutanydata,andhedidthrowoutdata,buthehad good,objectivereasonsforthrowingoutthedata.TheMillikanaffairwill probablyremaincontroversialamonghistorians,butthelessonIwould takeawayisthatalthoughtheepisodemayreduceourcondencein Millikan,itshoulddeepenourfaithinscience.Thecorrectresultwas eventuallyrecognized;itmightnothavebeeninapseudo-scienticeld likepoliticalscience. Section1.4Quantizationofcharge 29

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i / Cathoderaysobservedin avacuumtube. 1.5Theelectron Cathoderays Nineteenth-centuryphysicistsspentalotoftimetryingtocome upwithwild,randomwaystoplaywithelectricity.Thebestexperimentsofthiskindweretheonesthatmadebigsparksorpretty colorsoflight. Onesuchparlortrickwasthecathoderay.Toproduceit,you rsthadtohireagoodglassblowerandndagoodvacuumpump. Theglassblowerwouldcreateahollowtubeandembedtwopiecesof metalinit,calledtheelectrodes,whichwereconnectedtotheoutsideviametalwirespassingthroughtheglass.Beforelettinghim sealupthewholetube,youwouldhookituptoavacuumpump, andspendseveralhourshungandpungawayatthepump's handcranktogetagoodvacuuminside.Then,whileyouwerestill pumpingonthetube,theglassblowerwouldmelttheglassandseal thewholethingshut.Finally,youwouldputalargeamountofpositivechargeononewireandalargeamountofnegativechargeon theother.Metalshavethepropertyoflettingchargemovethrough themeasily,sothechargedepositedononeofthewireswould quicklyspreadoutbecauseoftherepulsionofeachpartofitfor everyotherpart.Thisspreading-outprocesswouldresultinnearly allthechargeendingupintheelectrodes,wherethereismoreroom tospreadoutthanthereisinthewire.Forobscurehistoricalreasonsanegativeelectrodeiscalledacathodeandapositiveoneis ananode. Figureishowsthelight-emittingstreamthatwasobserved.If, asshowninthisgure,aholewasmadeintheanode,thebeam wouldextendonthroughtheholeuntilithittheglass.Drillinga holeinthecathode,howeverwouldnotresultinanybeamcoming outontheleftside,andthisindicatedthatthestu,whateverit was,wascomingfromthecathode.Therayswerethereforechristenedcathoderays."Theterminologyisstillusedtodayinthe termcathoderaytube"orCRT"forthepicturetubeofaTVor computermonitor. Werecathoderaysaformoflight,orofmatter? Werecathoderaysaformoflight,ormatter?Atrstnoonereallycaredwhattheywere,butastheirscienticimportancebecame moreapparent,thelight-versus-matterissueturnedintoacontroversyalongnationalisticlines,withtheGermansadvocatinglight andtheEnglishholdingoutformatter.Thesupportersofthematerialinterpretationimaginedtheraysasconsistingofastreamof atomsrippedfromthesubstanceofthecathode. Oneofourdeningcharacteristicsofmatteristhatmaterial objectscannotpassthrougheachother.Experimentsshowedthat cathoderayscouldpenetrateatleastsomesmallthicknessofmatter, 30 Chapter1ElectricityandtheAtom

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j / J.J.Thomsoninthelab. suchasametalfoilatenthofamillimeterthick,implyingthatthey wereaformoflight. Otherexperiments,however,pointedtothecontraryconclusion. Lightisawavephenomenon,andonedistinguishingpropertyof wavesisdemonstratedbyspeakingintooneendofapapertowel roll.Thesoundwavesdonotemergefromtheotherendofthe tubeasafocusedbeam.Instead,theybeginspreadingoutinall directionsassoonastheyemerge.Thisshowsthatwavesdonot necessarilytravelinstraightlines.Ifapieceofmetalfoilintheshape ofastaroracrosswasplacedinthewayofthecathoderay,then ashadow"ofthesameshapewouldappearontheglass,showing thattheraystraveledinstraightlines.Thisstraight-linemotion suggestedthattheywereastreamofsmallparticlesofmatter. Theseobservationswereinconclusive,sowhatwasreallyneeded wasadeterminationofwhethertherayshadmassandweight.The troublewasthatcathoderayscouldnotsimplybecollectedinacup andputonascale.Whenthecathoderaytubeisinoperation,one doesnotobserveanylossofmaterialfromthecathode,oranycrust beingdepositedontheanode. Nobodycouldthinkofagoodwaytoweighcathoderays,sothe nextmostobviouswayofsettlingthelight/matterdebatewasto checkwhetherthecathoderayspossessedelectricalcharge.Light wasknowntobeuncharged.Ifthecathoderayscarriedcharge, theyweredenitelymatterandnotlight,andtheywerepresumablybeingmadetojumpthegapbythesimultaneousrepulsionof thenegativechargeinthecathodeandattractionofthepositive chargeintheanode.Therayswouldovershoottheanodebecause oftheirmomentum.Althoughelectricallychargedparticlesdonot normallyleapacrossagapofvacuum,verylargeamountsofcharge werebeingused,sotheforceswereunusuallyintense. Thomson'sexperiments PhysicistJ.J.ThomsonatCambridgecarriedoutaseriesof denitiveexperimentsoncathoderaysaroundtheyear1897.By turningthemslightlyocoursewithelectricalforces,k,heshowed thattheywereindeedelectricallycharged,whichwasstrongevidencethattheywerematerial.Notonlythat,butheprovedthat theyhadmass,andmeasuredtheratiooftheirmasstotheircharge, m=q .Sincetheirmasswasnotzero,heconcludedthattheywere aformofmatter,andpresumablymadeupofastreamofmicroscopic,negativelychargedparticles.WhenMillikanpublishedhis resultsfourteenyearslater,itwasreasonabletoassumethatthe chargeofonesuchparticleequaledminusonefundamentalcharge, q = )]TJ/F20 10.9091 Tf 8.485 0 Td [(e ,andfromthecombinationofThomson'sandMillikan'sresultsonecouldthereforedeterminethemassofasinglecathoderay particle. Section1.5Theelectron 31

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k / Thomson'sexperimentproving cathoderayshadelectriccharge redrawnfromhisoriginalpaper. Thecathode,c,andanode,A,are asinanycathoderaytube.The rayspassthroughaslitintheanode,andasecondslit,B,isinterposedinordertomakethebeam thinnerandeliminateraysthat werenotgoingstraight.Charging platesDandEshowsthatcathoderayshavecharge:theyare attractedtowardthepositiveplate Dandrepelledbythenegative plateE. Thebasictechniquefordetermining m=q wassimplytomeasure theanglethroughwhichthechargedplatesbentthebeam.The electricforceactingonacathoderayparticlewhileitwasbetween theplateswouldbeproportionaltoitscharge, F elec =knownconstant q ApplicationofNewton'ssecondlaw, a = F=m ,wouldallow m=q tobedetermined: m q = knownconstant a Therewasjustonecatch.Thomsonneededtoknowthecathode rayparticles'velocityinordertogureouttheiracceleration.At thatpoint,however,nobodyhadevenaneducatedguessastothe speedofthecathoderaysproducedinagivenvacuumtube.The beamappearedtoleapacrossthevacuumtubepracticallyinstantaneously,soitwasnosimplematteroftimingitwithastopwatch! Thomson'scleversolutionwastoobservetheeectofbothelectricandmagneticforcesonthebeam.Themagneticforceexerted byaparticularmagnetwoulddependonboththecathoderay's chargeanditsspeed: F mag =knownconstant#2 qv Thomsonplayedwiththeelectricandmagneticforcesuntileitheronewouldproduceanequaleectonthebeam,allowinghim tosolveforthespeed, v = knownconstant knownconstant#2 Knowingthespeedwhichwasontheorderof10%ofthespeed oflightforhissetup,hewasabletondtheaccelerationandthus themass-to-chargeratio m=q .Thomson'stechniqueswererelatively crudeorperhapsmorecharitablywecouldsaythattheystretched thestateoftheartofthetime,sowithvariousmethodshecame 32 Chapter1ElectricityandtheAtom

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upwith m=q valuesthatrangedoveraboutafactoroftwo,even forcathoderaysextractedfromacathodemadeofasinglematerial.Thebestmodernvalueis m=q =5.69 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(12 kg/C,whichis consistentwiththelowendofThomson'srange. Thecathoderayasasubatomicparticle:theelectron WhatwassignicantaboutThomson'sexperimentwasnotthe actualnumericalvalueof m=q ,however,somuchasthefactthat, combinedwithMillikan'svalueofthefundamentalcharge,itgave amassforthecathoderayparticlesthatwasthousandsoftimes smallerthanthemassofeventhelightestatoms.Evenwithout Millikan'sresults,whichwere14yearsinthefuture,Thomsonrecognizedthatthecathoderays' m=q wasthousandsoftimessmaller thanthe m=q ratiosthathadbeenmeasuredforelectricallycharged atomsinchemicalsolutions.Hecorrectlyinterpretedthisasevidencethatthecathoderaysweresmallerbuildingblocks|hecalled them electrons |outofwhichatomsthemselveswereformed.This wasanextremelyradicalclaim,comingatatimewhenatomshad notyetbeenproventoexist!Eventhosewhousedthewordatom" oftenconsideredthemnomorethanmathematicalabstractions,not literalobjects.Theideaofsearchingforstructureinsideofunsplittable"atomswasseenbysomeaslunacy,butwithintenyears Thomson'sideashadbeenamplyveriedbymanymoredetailed experiments. DiscussionQuestions A Thomsonstartedtobecomeconvincedduringhisexperimentsthat thecathoderaysobservedcomingfromthecathodesofvacuumtubes werebuildingblocksofatomswhatwenowcallelectrons.Hethen carriedoutobservationswithcathodesmadeofavarietyofmetals,and foundthat m = q wasroughlythesameineverycase,consideringhislimitedaccuracy.Givenhissuspicion,whydiditmakesensetotrydifferent metals?Howwouldtheconsistentvaluesof m = q testhishypothesis? B Mystudentshavefrequentlyaskedwhetherthe m = q thatThomson measuredwasthevalueforasingleelectron,orforthewholebeam.Can youanswerthisquestion? C Thomsonfoundthatthe m = q ofanelectronwasthousandsoftimes smallerthanthatofchargedatomsinchemicalsolutions.Wouldthisimply thattheelectronshadmorecharge?Lessmass?Wouldtherebenoway totell?Explain.RememberthatMillikan'sresultswerestillmanyyearsin thefuture,so q wasunknown. D CanyouguessanypracticalreasonwhyThomsoncouldn'tjust letoneelectronyacrossthegapbeforedisconnectingthebatteryand turningoffthebeam,andthenmeasuretheamountofchargedeposited ontheanode,thusallowinghimtomeasurethechargeofasingleelectron directly? E Whyisitnotpossibletodetermine m and q themselves,ratherthan justtheirratio,byobservingelectrons'motioninelectricandmagnetic elds? Section1.5Theelectron 33

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l / Theraisincookiemodelof theatomwithfourunitsof charge,whichwenowknowtobe beryllium. 1.6Theraisincookiemodeloftheatom Basedonhisexperiments,Thomsonproposedapictureoftheatom whichbecameknownastheraisincookiemodel.Intheneutral atom,l,therearefourelectronswithatotalchargeof )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 e ,sitting inaspherethecookie"withachargeof+4 e spreadthroughoutit. Itwasknownthatchemicalreactionscouldnotchangeoneelement intoanother,soinThomson'sscenario,eachelement'scookiesphere hadapermanentlyxedradius,mass,andpositivecharge,dierent fromthoseofotherelements.Theelectrons,however,werenota permanentfeatureoftheatom,andcouldbetackedonorpulledout tomakechargedions.Althoughwenowknow,forinstance,thata neutralatomwithfourelectronsistheelementberyllium,scientists atthetimedidnotknowhowmanyelectronsthevariousneutral atomspossessed. Thismodelisclearlydierentfromtheoneyou'velearnedin gradeschoolorthroughpopularculture,wherethepositivecharge isconcentratedinatinynucleusattheatom'scenter.Anequally importantchangeinideasabouttheatomhasbeentherealization thatatomsandtheirconstituentsubatomicparticlesbehaveentirely dierentlyfromobjectsonthehumanscale.Forinstance,we'llsee laterthatanelectroncanbeinmorethanoneplaceatonetime. Theraisincookiemodelwaspartofalongtraditionofattempts tomakemechanicalmodelsofphenomena,andThomsonandhis contemporariesneverquestionedtheappropriatenessofbuildinga mentalmodelofanatomasamachinewithlittlepartsinside.Today,mechanicalmodelsofatomsarestillusedforinstancethe tinker-toy-stylemolecularmodelingkitsliketheonesusedbyWatsonandCricktogureoutthedoublehelixstructureofDNA,but scientistsrealizethatthephysicalobjectsareonlyaidstohelpour brains'symbolicandvisualprocessesthinkaboutatoms. Althoughtherewasnoclear-cutexperimentalevidenceformany ofthedetailsoftheraisincookiemodel,physicistswentaheadand startedworkingoutitsimplications.Forinstance,supposeyouhad afour-electronatom.Allfourelectronswouldberepellingeach other,buttheywouldalsoallbeattractedtowardthecenterofthe cookie"sphere.Theresultshouldbesomekindofstable,symmetricarrangementinwhichalltheforcescanceledout.People sucientlycleverwithmathsoonshowedthattheelectronsina four-electronatomshouldsettledownattheverticesofapyramid withonelesssidethantheEgyptiankind,i.e.,aregulartetrahedron.Thisdeductionturnsouttobewrongbecauseitwasbased onincorrectfeaturesofthemodel,butthemodelalsohadmany successes,afewofwhichwewillnowdiscuss. Flowofelectricalchargeinwiresexample3 Oneofmyformerstudentswasthesonofanelectrician,and hadbecomeanelectricianhimself.Herelatedtomehowhis 34 Chapter1ElectricityandtheAtom

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fatherhadremainedrefusedtobelieveallhislifethatelectrons reallyowedthroughwires.Iftheyhad,hereasoned,themetal wouldhavegraduallybecomemoreandmoredamaged,eventuallycrumblingtodust. Hisopinionisnotatallunreasonablebasedonthefactthatelectronsarematerialparticles,andthatmattercannotnormallypass throughmatterwithoutmakingaholethroughit.Nineteenthcenturyphysicistswouldhavesharedhisobjectiontoachargedparticlemodeloftheowofelectricalcharge.Intheraisin-cookie model,however,theelectronsareverylowinmass,andthereforepresumablyverysmallinsizeaswell.Itisnotsurprisingthat theycanslipbetweentheatomswithoutdamagingthem. Flowofelectricalchargeacrosscellmembranesexample4 Yournervoussystemisbasedonsignalscarriedbychargemovingfromnervecelltonervecell.Yourbodyisessentiallyallliquid, andatomsinaliquidaremobile.Thismeansthat,unlikethecase ofchargeowinginasolidwire,entirechargedatomscanowin yournervoussystem Emissionofelectronsinacathoderaytubeexample5 Whydoelectronsdetachthemselvesfromthecathodeofavacuumtube?Certainlytheyareencouragedtodosobytherepulsionofthenegativechargeplacedonthecathodeandthe attractionfromthenetpositivechargeoftheanode,buttheseare notstrongenoughtoripelectronsoutofatomsbymainforce iftheywere,thentheentireapparatuswouldhavebeeninstantly vaporizedaseveryatomwassimultaneouslyrippedapart! Theraisincookiemodelleadstoasimpleexplanation.Weknow thatheatistheenergyofrandommotionofatoms.Theatomsin anyobjectarethereforeviolentlyjostlingeachotherallthetime, andafewofthesecollisionsareviolentenoughtoknockelectrons outofatoms.Ifthisoccursnearthesurfaceofasolidobject,the electronmaycancomeloose.Ordinarily,however,thislossof electronsisaself-limitingprocess;thelossofelectronsleaves theobjectwithanetpositivecharge,whichattractsthelostsheep hometothefold.Forobjectsimmersedinairratherthanvacuum, therewillalsobeabalancedexchangeofelectronsbetweenthe airandtheobject. Thisinterpretationexplainsthewarmandfriendlyyellowglowof thevacuumtubesinanantiqueradio.Toencouragetheemission ofelectronsfromthevacuumtubes'cathodes,thecathodesare intentionallywarmedupwithlittleheatercoils. DiscussionQuestions A Todaymanypeoplewoulddeneanionasanatomormolecule withmissingelectronsorextraelectronsaddedon.Howwouldpeople havedenedthewordionbeforethediscoveryoftheelectron? Section1.6Theraisincookiemodeloftheatom 35

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B Sinceelectricallyneutralatomswereknowntoexist,therehadtobe positivelychargedsubatomicstufftocanceloutthenegativelycharged electronsinanatom.Basedonthestateofknowledgeimmediatelyafter theMillikanandThomsonexperiments,wasitpossiblethatthepositively chargedstuffhadanunquantizedamountofcharge?Coulditbequantizedinunitsof+e?Inunitsof+2e?Inunitsof+5/7e? 36 Chapter1ElectricityandtheAtom

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Summary SelectedVocabulary atom.......thebasicunitofoneofthechemicalelements molecule.....agroupofatomsstucktogether electricalforce.oneofthefundamentalforcesofnature;anoncontactforcethatcanbeeitherrepulsiveor attractive charge......anumericalratingofhowstronglyanobject participatesinelectricalforces coulombCthe unitofelectrical charge ion......... anelectricallychargedatomormolecule cathoderay...themysteriousraythatemanatedfromthe cathodeinavacuumtube;shownbyThomson tobeastreamofparticlessmallerthanatoms electron......Thomson'snamefortheparticlesofwhicha cathoderaywasmade quantized....describesquantitysuchasmoneyorelectrical charge,thatcanonlyexistincertainamounts Notation q ..........charge e ..........thequantumofcharge Summary Alltheforcesweencounterineverydaylifeboildowntotwo basictypes:gravitationalforcesandelectricalforces.Aforcesuch asfrictionorastickyforce"arisesfromelectricalforcesbetween individualatoms. Justasweusethewordmass"todescribehowstronglyan objectparticipatesingravitationalforces,weusethewordcharge" fortheintensityofitselectricalforces.Therearetwotypesof charge.Twochargesofthesametyperepeleachother,butobjects whosechargesaredierentattracteachother.Chargeismeasured inunitsofcoulombsC. Mobilechargedparticlemodel:Agreatmanyphenomenaare easilyunderstoodifweimaginematterascontainingtwotypesof chargedparticles,whichareatleastpartiallyabletomovearound. Positiveandnegativecharge:Ordinaryobjectsthathavenot beenspeciallypreparedhavebothtypesofchargespreadevenly throughouttheminequalamounts.Theobjectwillthentendnot toexertelectricalforcesonanyotherobject,sinceanyattraction duetoonetypeofchargewillbebalancedbyanequalrepulsion fromtheother.Wesaytendnotto"becausebringingtheobject nearanobjectwithunbalancedamountsofchargecouldcauseits chargestoseparatefromeachother,andtheforcewouldnolonger Summary 37

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cancelduetotheunequaldistances.Itthereforemakessenseto describethetwotypesofchargeusingpositiveandnegativesigns, sothatanunpreparedobjectwillhavezero total charge. TheCoulombforcelawstatesthatthemagnitudeoftheelectricalforcebetweentwochargedparticlesisgivenby j F j = k j q 1 jj q 2 j =r 2 Conservationofcharge:Anevenmorefundamentalreasonfor usingpositiveandnegativesignsforchargeisthatwiththisdenitionthetotalchargeofaclosedsystemisaconservedquantity. Quantizationofcharge:Millikan'soildropexperimentshowed thatthetotalchargeofanobjectcouldonlybeanintegermultiple ofabasicunitofcharge e .Thissupportedtheideathetheow" ofelectricalchargewasthemotionoftinyparticlesratherthanthe motionofsomesortofmysteriouselectricaluid. Einstein'sanalysisofBrownianmotionwastherstdenitive proofoftheexistenceofatoms.Thomson'sexperimentswithvacuumtubesdemonstratedtheexistenceofanewtypeofmicroscopic particlewithaverysmallratioofmasstocharge.Thomsoncorrectlyinterpretedtheseasbuildingblocksofmatterevensmaller thanatoms:therstdiscoveryofsubatomicparticles.Theseparticlesarecalledelectrons. Theaboveexperimentalevidenceledtotherstusefulmodelof theinteriorstructureofatoms,calledtheraisincookiemodel.In theraisincookiemodel,anatomconsistsofarelativelylarge,massive,positivelychargedspherewithacertainnumberofnegatively chargedelectronsembeddedinit. 38 Chapter1ElectricityandtheAtom

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Problem1.Top:Arealistic pictureofaneuron.Bottom: Asimplieddiagramofone segmentofthetailaxon. Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Thegureshowsaneuron,whichisthetypeofcellyournerves aremadeof.Neuronsservetotransmitsensoryinformationtothe brain,andcommandsfromthebraintothemuscles.Allthisdata istransmittedelectrically,butevenwhenthecellisrestingandnot transmittinganyinformation,thereisalayerofnegativeelectrical chargeontheinsideofthecellmembrane,andalayerofpositive chargejustoutsideit.Thischargeisintheformofvariousions dissolvedintheinteriorandexterioruids.Whywouldthenegative chargeremainplasteredagainsttheinsidesurfaceofthemembrane, andlikewisewhydoesn'tthepositivechargewanderawayfromthe outsidesurface? 2 Usethenutritionalinformationonsomepackagedfoodto makeanorder-of-magnitudeestimateoftheamountofchemical energystoredinoneatomoffood,inunitsofjoules.Assumethat atypicalatomhasamassof10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(26 kg.Thisconstitutesarough estimateoftheamountsofenergythereareontheatomicscale. [Seechapter1ofbook1,NewtonianPhysics,forhelponhowtodo order-of-magnitudeestimates.Notethatanutritionalcalorie"is reallyakilocalorie;seepage210.] p 3 aRecallthatthegravitationalenergyoftwogravitationally interactingspheresisgivenby PE = )]TJ/F20 10.9091 Tf 8.485 0 Td [(Gm 1 m 2 =r ,where r isthe center-to-centerdistance.Whatwouldbetheanalogousequation fortwoelectricallyinteractingspheres?Justifyyourchoiceofa plusorminussignonphysicalgrounds,consideringattractionand repulsion. p bUsethisexpressiontoestimatetheenergyrequiredtopullapart araisin-cookieatomoftheone-electrontype,assumingaradiusof 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(10 m. p cComparethiswiththeresultofproblem2. 4 Aneonlightconsistsofalongglasstubefullofneon,with metalcapsontheends.Positivechargeisplacedononeendofthe tube,andnegativechargeontheother.Theelectricforcesgenerated canbestrongenoughtostripelectronsoofacertainnumberof neonatoms.Assumeforsimplicitythatonlyoneelectronisever strippedoofanyneonatom.Whenanelectronisstrippedoof anatom,boththeelectronandtheneonatomnowanionhave electriccharge,andtheyareacceleratedbytheforcesexertedbythe chargedendsofthetube.Theydonotfeelanysignicantforces fromtheotherionsandelectronswithinthetube,becauseonly atinyminorityofneonatomsevergetsionized.Lightisnally producedwhenionsarereunitedwithelectrons.Giveanumerical Problems 39

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Problem8. Problem6. comparisonofthemagnitudesanddirectionsoftheaccelerationsof theelectronsandions.[Youmayneedsomedatafrompage210.] p 5 Ifyouputtwohydrogenatomsneareachother,theywillfeel anattractiveforce,andtheywillpulltogethertoformamolecule. Moleculesconsistingoftwohydrogenatomsarethenormalformof hydrogengas.Howisthispossible,sinceeachiselectricallyneutral?Shouldn'ttheattractiveandrepulsiveforcesallcancelout exactly?Usetheraisincookiemodel.Studentswhohavetaken chemistryoftentrytousefanciermodelstoexplainthis,butifyou can'texplainitusingasimplemodel,youprobablydon'tunderstand thefancymodelaswellasyouthoughtyoudid!It'snotsoeasy toprovethattheforceshouldactuallybeattractiveratherthanrepulsive,sojustconcentrateonexplainingwhyitdoesn'tnecessarily havetovanishcompletely. 6 Thegureshowsonelayerofthethree-dimensionalstructure ofasaltcrystal.Theatomsextendmuchfartheroinalldirections, butonlyasix-by-sixsquareisshownhere.Thelargercirclesare thechlorineions,whichhavechargesof )]TJ/F20 10.9091 Tf 8.485 0 Td [(e .Thesmallercircles aresodiumions,withchargesof+ e .Thecenter-to-centerdistance betweenneighboringionsisabout0.3nm.Realcrystalsarenever perfect,andthecrystalshownherehastwodefects:amissingatom atonelocation,andanextralithiumatom,shownasagreycircle, insertedinoneofthesmallgaps.Ifthelithiumatomhasacharge of+ e ,whatisthedirectionandmagnitudeofthetotalforceonit? Assumetherearenootherdefectsnearbyinthecrystalbesidesthe twoshownhere.[Hints:Theforceonthelithiumionisthevector sumofalltheforcesofallthequadrillionsofsodiumandchlorine atoms,whichwouldobviouslybetoolaborioustocalculate.Nearly alloftheseforces,however,arecanceledbyaforcefromanionon theoppositesideofthelithium.] p ? 7 TheEarthandMoonareboundtogetherbygravity.If,instead,theforceofattractionweretheresultofeachhavingacharge ofthesamemagnitudebutoppositeinsign,ndthequantityof chargethatwouldhavetobeplacedoneachtoproducetherequiredforce. p 8 Intheseminalsofanelectrostaticcroquettournament, Jessicahitsherpositivelychargedball,sendingitacrosstheplaying eld,rollingtotheleftalongthe x axis.Itisrepelledbytwoother positivecharges.Thesetwoequalchargesarexedonthe y axisat thelocationsshowninthegure.aExpresstheforceontheball intermsoftheball'sposition, x .bAtwhatvalueof x doesthe ballexperiencethegreatestdeceleration?Expressyouanswerin termsof b .[BasedonaproblembyHallidayandResnick.] R 40 Chapter1ElectricityandtheAtom

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a / MarieandPierreCuriewere thersttopurifyradiuminsignicantquantities.Radium'sintense radioactivitymadepossiblethe experimentsthatledtothemodernplanetarymodeloftheatom, inwhichelectronsorbitanucleus madeofprotonsandneutrons. Chapter2 TheNucleus 2.1Radioactivity Becquerel'sdiscoveryofradioactivity Howdidphysicistsgureoutthattheraisincookiemodelwas incorrect,andthattheatom'spositivechargewasconcentratedin atiny,centralnucleus?ThestorybeginswiththediscoveryofradioactivitybytheFrenchchemistBecquerel.Upuntilradioactivity wasdiscovered,alltheprocessesofnaturewerethoughttobebased onchemicalreactions,whichwererearrangementsofcombinations ofatoms.Atomsexertforcesoneachotherwhentheyareclosetogether,sostickingorunstickingthemwouldeitherreleaseorstore electricalenergy.Thatenergycouldbeconvertedtoandfromother forms,aswhenaplantusestheenergyinsunlighttomakesugars andcarbohydrates,orwhenachildeatssugar,releasingtheenergy intheformofkineticenergy. Becquereldiscoveredaprocessthatseemedtoreleaseenergy fromanunknownnewsourcethatwasnotchemical.Becquerel, 41

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b / HenriBecquerel-1908. c / Becquerel'sphotographic plate.Intheexposureatthe bottomoftheimage,hehas foundthathecouldabsorbthe radiations,castingtheshadow ofaMaltesecrossthatwas placedbetweentheplateandthe uraniumsalts. whosefatherandgrandfatherhadalsobeenphysicists,spentthe rsttwentyyearsofhisprofessionallifeasasuccessfulcivilengineer,teachingphysicsonapart-timebasis.Hewasawardedthe chairofphysicsattheMuseed'HistoireNaturelleinParisafterthe deathofhisfather,whohadpreviouslyoccupiedit.Havingnowa signicantamountoftimetodevotetophysics,hebeganstudying theinteractionoflightandmatter.Hebecameinterestedinthephenomenonofphosphorescence,inwhichasubstanceabsorbsenergy fromlight,thenreleasestheenergyviaaglowthatonlygradually goesaway.Oneofthesubstancesheinvestigatedwasauranium compound,thesaltUKSO 5 .Onedayin1896,cloudyweatherinterferedwithhisplantoexposethissubstancetosunlightinorder toobserveitsuorescence.Hestuckitinadrawer,coincidentallyon topofablankphotographicplate|theold-fashionedglass-backed counterpartofthemodernplasticrolloflm.Theplatehadbeen carefullywrapped,butseveraldayslaterwhenBecquerelcheckedit inthedarkroombeforeusingit,hefoundthatitwasruined,asifit hadbeencompletelyexposedtolight. Historyprovidesmanyexamplesofscienticdiscoveriesthat happenedthisway:analertandinquisitiveminddecidestoinvestigateaphenomenonthatmostpeoplewouldnothaveworried aboutexplaining.Becquerelrstdeterminedbyfurtherexperiments thattheeectwasproducedbytheuraniumsalt,despiteathick wrappingofpaperaroundtheplatethatblockedoutalllight.He triedavarietyofcompounds,andfoundthatitwastheuranium thatdidit:theeectwasproducedbyanyuraniumcompound,but notbyanycompoundthatdidn'tincludeuraniumatoms.Theeect couldbeatleastpartiallyblockedbyasucientthicknessofmetal, andhewasabletoproducesilhouettesofcoinsbyinterposingthem betweentheuraniumandtheplate.Thisindicatedthattheeect traveledinastraightline.,sothatitmusthavebeensomekindof rayratherthan,e.g.,theseepageofchemicalsthroughthepaper. Heusedthewordradiations,"sincetheeectradiatedoutfrom theuraniumsalt. AtthispointBecquerelstillbelievedthattheuraniumatoms wereabsorbingenergyfromlightandthengraduallyreleasingthe energyintheformofthemysteriousrays,andthiswashowhe presenteditinhisrstpublishedlecturedescribinghisexperiments. Interesting,butnotearth-shattering.Buthethentriedtodetermine howlongittookfortheuraniumtouseupalltheenergythathad supposedlybeenstoredinitbylight,andhefoundthatitnever seemedtobecomeinactive,nomatterhowlonghewaited.Notonly that,butasamplethathadbeenexposedtointensesunlightfora wholeafternoonwasnomoreorlesseectivethanasamplethat hadalwaysbeenkeptinside.Wasthisaviolationofconservation ofenergy?Iftheenergydidn'tcomefromexposuretolight,where diditcomefrom? 42 Chapter2TheNucleus

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Threekindsofradiations Unabletodeterminethesourceoftheenergydirectly,turn-ofthe-centuryphysicistsinsteadstudiedthebehavioroftheradiations"oncetheyhadbeenemitted.Becquerelhadalreadyshown thattheradioactivitycouldpenetratethroughclothandpaper,so therstobviousthingtodowastoinvestigateinmoredetailwhat thicknessofmaterialtheradioactivitycouldgetthrough.Theysoon learnedthatacertainfractionoftheradioactivity'sintensitywould beeliminatedbyevenafewinchesofair,buttheremainderwas noteliminatedbypassingthroughmoreair.Apparently,then,the radioactivitywasamixtureofmorethanonetype,ofwhichonewas blockedbyair.Theythenfoundthatofthepartthatcouldpenetrateair,afurtherfractioncouldbeeliminatedbyapieceofpaper oraverythinmetalfoil.Whatwasleftafterthat,however,was athird,extremelypenetratingtype,someofwhoseintensitywould stillremainevenafterpassingthroughabrickwall.Theydecided thatthisshowedtherewerethreetypesofradioactivity,andwithouthavingthefaintestideaofwhattheyreallywere,theymadeup namesforthem.Theleastpenetratingtypewasarbitrarilylabeled alpha,therstletteroftheGreekalphabet,andsoonthrough betaandnally gammaforthemostpenetratingtype. Radium:amoreintensesourceofradioactivity Themeasuringdevicesusedtodetectradioactivitywerecrude: photographicplatesorevenhumaneyeballsradioactivitymakes ashesoflightinthejelly-likeuidinsidetheeye,whichcanbe seenbytheeyeball'sownerifitisotherwiseverydark.Becausethe waysofdetectingradioactivityweresocrudeandinsensitive,further progresswashinderedbythefactthattheamountofradioactivity emittedbyuraniumwasnotreallyverygreat.Thevitalcontributionofphysicist/chemistMarieCurieandherhusbandPierrewas todiscovertheelementradium,andtopurifyandisolatesignicant quantitiesit.Radiumemitsaboutamilliontimesmoreradioactivity perunitmassthanuranium,makingitpossibletodotheexperimentsthatwereneededtolearnthetruenatureofradioactivity. Thedangersofradioactivitytohumanhealthwerethenunknown, andMariediedofleukemiathirtyyearslater.Pierrewasrunover andkilledbyahorsecart. Trackingdownthenatureofalphas,betas,andgammas Asradiumwasbecomingavailable,anapprenticescientistnamed ErnestRutherfordarrivedinEnglandfromhisnativeNewZealand andbeganstudyingradioactivityattheCavendishLaboratory.The youngcolonial'srstsuccesswastomeasurethemass-to-chargeratioofbetarays.Thetechniquewasessentiallythesameastheone Thomsonhadusedtomeasurethemass-to-chargeratioofcathode raysbymeasuringtheirdeectionsinelectricandmagneticelds. Theonlydierencewasthatinsteadofthecathodeofavacuum Section2.1Radioactivity 43

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d / Asimpliedversionof Rutherford's1908experiment, showingthatalphaparticleswere doublyionizedheliumatoms. e / Thesepelletsofuranium fuelwillbeinsertedintothemetal fuelrodandusedinanuclear reactor.Thepelletsemitalpha andbetaradiation,whichthe glovesarethickenoughtostop. tube,anuggetofradiumwasusedtosupplythebetarays.Not onlywasthetechniquethesame,butsowastheresult.Betarays hadthesame m=q ratioascathoderays,whichsuggestedtheywere oneandthesame.Nowadays,itwouldmakesensesimplytouse thetermelectron,"andavoidthearchaiccathoderay"andbeta particle,"buttheoldlabelsarestillwidelyused,anditisunfortunatelynecessaryforphysicsstudentstomemorizeallthreenames forthesamething. Atrst,itseemedthatneitheralphasorgammascouldbedeectedinelectricormagneticelds,makingitappearthatneither waselectricallycharged.ButsoonRutherfordobtainedamuchmore powerfulmagnet,andwasabletouseittodeectthealphasbut notthegammas.Thealphashadamuchlargervalueof m=q than thebetasabout4000timesgreater,whichwaswhytheyhadbeen sohardtodeect.Gammasareuncharged,andwerelaterfoundto beaformoflight. The m=q ratioofalphaparticlesturnedouttobethesame asthoseoftwodierenttypesofions,He ++ aheliumatomwith twomissingelectronsandH + 2 twohydrogenatomsbondedintoa molecule,withoneelectronmissing,soitseemedlikelythatthey wereoneortheotherofthose.ThediagramshowsasimpliedversionofRutherford'singeniousexperimentprovingthattheywere He ++ ions.Thegaseouselementradon,analphaemitter,wasintroducedintoonehalfofadoubleglasschamber.Theglasswall dividingthechamberwasmadeextremelythin,sothatsomeofthe rapidlymovingalphaparticleswereabletopenetrateit.Theother chamber,whichwasinitiallyevacuated,graduallybegantoaccumulateapopulationofalphaparticleswhichwouldquicklypickup electronsfromtheirsurroundingsandbecomeelectricallyneutral. Rutherfordthendeterminedthatitwasheliumgasthathadappearedinthesecondchamber.Thusalphaparticleswereprovedto beHe ++ ions.Thenucleuswasyettobediscovered,butinmodern terms,wewoulddescribeaHe ++ ionasthenucleusofaHeatom. Tosummarize,herearethethreetypesofradiationemittedby radioactiveelements,andtheirdescriptionsinmodernterms: particle stoppedbyafewinchesofair Henucleus particle stoppedbyapieceofpaper electron ray penetratesthickshielding atypeoflight DiscussionQuestion A Mostsourcesofradioactivityemitalphas,betas,andgammas,not justoneofthethree.Intheradonexperiment,howdidRutherfordknow thathewasstudyingthealphas? 44 Chapter2TheNucleus

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f / ErnestRutherford1937. g / MarsdenandRutherford's apparatus. 2.2Theplanetarymodeloftheatom Thestagewasnowsetfortheunexpecteddiscoverythatthepositivelychargedpartoftheatomwasatiny,denselumpattheatom's centerratherthanthecookiedough"oftheraisincookiemodel. By1909,Rutherfordwasanestablishedprofessor,andhadstudents workingunderhim.ForarawundergraduatenamedMarsden,he pickedaresearchprojecthethoughtwouldbetediousbutstraightforward. Itwasalreadyknownthatalthoughalphaparticleswouldbe stoppedcompletelybyasheetofpaper,theycouldpassthrougha sucientlythinmetalfoil.Marsdenwastoworkwithagoldfoil only1000atomsthick.Thefoilwasprobablymadebyevaporating alittlegoldinavacuumchambersothatathinlayerwouldbe depositedonaglassmicroscopeslide.Thefoilwouldthenbelifted otheslidebysubmergingtheslideinwater. Rutherfordhadalreadydeterminedinhispreviousexperiments thespeedofthealphaparticlesemittedbyradium,afantastic1.5 10 7 m/s.TheexperimentersinRutherford'sgroupvisualizedthem asverysmall,veryfastcannonballspenetratingthecookiedough" partofthebiggoldatoms.Apieceofpaperhasathicknessofa hundredthousandatomsorso,whichwouldbesucienttostop themcompletely,butcrashingthroughathousandwouldonlyslow themalittleandturnthemslightlyooftheiroriginalpaths. Marsden'ssupposedlyho-humassignmentwastousetheapparatusshowninguregtomeasurehowoftenalphaparticleswere deectedatvariousangles.Atinylumpofradiuminaboxemittedalphaparticles,andathinbeamwascreatedbyblockingall thealphasexceptthosethathappenedtopassoutthroughatube. Typicallydeectedinthegoldbyonlyasmallamount,theywould reachascreenverymuchlikethescreenofaTV'spicturetube, whichwouldmakeaashoflightwhenitwashit.Hereistherst examplewehaveencounteredofanexperimentinwhichabeamof particlesisdetectedoneatatime.Thiswaspossiblebecauseeach alphaparticlecarriedsomuchkineticenergy;theyweremovingat aboutthesamespeedastheelectronsintheThomsonexperiment, buthadtenthousandtimesmoremass. Marsdensatinadarkroom,watchingtheapparatushourafter hourandrecordingthenumberofasheswiththescreenmovedto variousangles.Therateoftheasheswashighestwhenhesetthe screenatanangleclosetothelineofthealphas'originalpath,butif hewatchedanareafartherototheside,hewouldalsooccasionally seeanalphathathadbeendeectedthroughalargerangle.After seeingafewofthese,hegotthecrazyideaofmovingthescreento seeifevenlargerangleseveroccurred,perhapsevenangleslarger than90degrees. Section2.2Theplanetarymodeloftheatom 45

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i / Theplanetarymodelof theatom. h / Alphaparticlesbeingscatteredbyagoldnucleus.Onthisscale, thegoldatomisthesizeofacar,soallthealphaparticlesshownhere areonesthatjusthappenedtocomeunusuallyclosetothenucleus. Fortheseexceptionalalphaparticles,theforcesfromtheelectronsare unimportant,becausetheyaresomuchmoredistantthanthenucleus. Thecrazyideaworked:afewalphaparticlesweredeected throughanglesofupto180degrees,andtheroutineexperiment hadbecomeanepoch-makingone.Rutherfordsaid,Wehavebeen abletogetsomeofthealphaparticlescomingbackwards.Itwas almostasincredibleasifyoureda15-inchshellatapieceoftissue paperanditcamebackandhityou."Explanationswerehardto comebyintheraisincookiemodel.Whatintenseelectricalforces couldhavecausedsomeofthealphaparticles,movingatsuchastronomicalspeeds,tochangedirectionsodrastically?Sinceeachgold atomwaselectricallyneutral,itwouldnotexertmuchforceonan alphaparticleoutsideit.True,ifthealphaparticlewasverynearto orinsideofaparticularatom,thentheforceswouldnotnecessarily canceloutperfectly;ifthealphaparticlehappenedtocomevery closetoaparticularelectron,the1 =r 2 formoftheCoulombforce lawwouldmakeforaverystrongforce.ButMarsdenandRutherfordknewthatanalphaparticlewas8000timesmoremassivethan anelectron,anditissimplynotpossibleforamoremassiveobject toreboundbackwardsfromacollisionwithalessmassiveobject whileconservingmomentumandenergy.Itmightbepossiblein principleforaparticularalphatofollowapaththattookitvery closetooneelectron,andthenveryclosetoanotherelectron,andso on,withthenetresultofalargedeection,butcarefulcalculations showedthatsuchmultiplecloseencounters"withelectronswould bemillionsoftimestooraretoexplainwhatwasactuallyobserved. Atthispoint,RutherfordandMarsdendustedoanunpopularandneglectedmodeloftheatom,inwhichalltheelectrons orbitedaroundasmall,positivelychargedcoreornucleus,"just liketheplanetsorbitingaroundthesun.Allthepositivecharge 46 Chapter2TheNucleus

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andnearlyallthemassoftheatomwouldbeconcentratedinthe nucleus,ratherthanspreadthroughouttheatomasintheraisin cookiemodel.Thepositivelychargedalphaparticleswouldberepelledbythegoldatom'snucleus,butmostofthealphaswouldnot comecloseenoughtoanynucleustohavetheirpathsdrastically altered.Thefewthatdidcomeclosetoanucleus,however,could reboundbackwardsfromasinglesuchencounter,sincethenucleusof aheavygoldatomwouldbeftytimesmoremassivethananalpha particle.Itturnedoutthatitwasnoteventoodiculttoderivea formulagivingtherelativefrequencyofdeectionsthroughvarious angles,andthiscalculationagreedwiththedatawellenoughto within15%,consideringthedicultyingettinggoodexperimental statisticsontherare,verylargeangles. Whathadstartedoutasatediousexercisetogetastudent startedinsciencehadendedasarevolutioninourunderstanding ofnature.Indeed,thewholethingmaysoundalittletoomuch likeamoralisticfableofthescienticmethodwithovertonesof theHoratioAlgergenre.Theskepticalreadermaywonderwhy theplanetarymodelwasignoredsothoroughlyuntilMarsdenand Rutherford'sdiscovery.Issciencereallymoreofasociologicalenterprise,inwhichcertainideasbecomeacceptedbytheestablishment, andother,equallyplausibleexplanationsarearbitrarilydiscarded? Somesocialscientistsarecurrentlyruingalotofscientists'featherswithcritiquesverymuchlikethis,butinthisparticularcase, therewereverysoundreasonsforrejectingtheplanetarymodel.As you'lllearninmoredetaillaterinthiscourse,anychargedparticle thatundergoesanaccelerationdissipateenergyintheformoflight. Intheplanetarymodel,theelectronswereorbitingthenucleusin circlesorellipses,whichmeanttheywereundergoingacceleration, justliketheaccelerationyoufeelinacargoingaroundacurve.They shouldhavedissipatedenergyaslight,andeventuallytheyshould havelostalltheirenergy.Atomsdon'tspontaneouslycollapselike that,whichwaswhytheraisincookiemodel,withitsstationary electrons,wasoriginallypreferred.Therewereotherproblemsas well.Intheplanetarymodel,theone-electronatomwouldhave tobeat,whichwouldbeinconsistentwiththesuccessofmolecularmodelingwithsphericalballsrepresentinghydrogenandatoms. Thesemolecularmodelsalsoseemedtoworkbestifspecicsizes wereusedfordierentatoms,butthereisnoobviousreasoninthe planetarymodelwhytheradiusofanelectron'sorbitshouldbea xednumber.InviewoftheconclusiveMarsden-Rutherfordresults, however,thesebecamefreshpuzzlesinatomicphysics,notreasons fordisbelievingtheplanetarymodel. Section2.2Theplanetarymodeloftheatom 47

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j / Theplanetarymodelappliedtoanonmetal,1,an unmagnetizedmetal,2,anda magnetizedmetal,3.Notethat theseguresareallsimpliedin severalways.Foronething,the electronsofanindividualatomdo notallrevolvearoundthenucleus inthesameplane.Itisalsovery unusualforametaltobecomeso stronglymagnetizedthat100% ofitsatomshavetheirrotations alignedasshowninthisgure. Somephenomenaexplainedwiththeplanetarymodel Theplanetarymodelmaynotbetheultimate,perfectmodelof theatom,butdon'tunderestimateitspower.Italreadyallowsus tovisualizecorrectlyagreatmanyphenomena. Asanexample,let'sconsiderthedistinctionsamongnonmetals, metalsthataremagnetic,andmetalsthatarenonmagnetic.As showningurej,ametaldiersfromanonmetalbecauseitsoutermostelectronsarefreetowanderratherthanowingtheirallegiance toaparticularatom.Ametalthatcanbemagnetizedisonethat iswillingtolineuptherotationsofsomeofitselectronssothat theiraxesareparallel.Recallthatmagneticforcesareforcesmade bymovingcharges;wehavenotyetdiscussedthemathematicsand geometryofmagneticforces,butitiseasytoseehowrandomorientationsoftheatomsinthenonmagneticsubstancewouldleadto cancellationoftheforces. Eveniftheplanetarymodeldoesnotimmediatelyanswersuch questionsaswhyoneelementwouldbeametalandanotheranonmetal,theseideaswouldbedicultorimpossibletoconceptualize intheraisincookiemodel. DiscussionQuestion A Inreality,chargesofthesametyperepeloneanotherandcharges ofdifferenttypesareattracted.Supposetherulesweretheotherway around,givingrepulsionbetweenoppositechargesandattractionbetweensimilarones.Whatwouldtheuniversebelike? 2.3Atomicnumber Asalludedtoinadiscussionquestionintheprevioussection,scientistsofthisperiodhadonlyaveryapproximateideaofhowmany unitsofchargeresidedinthenucleiofthevariouschemicalelements.Althoughwenowassociatethenumberofunitsofnuclear chargewiththeelement'spositionontheperiodictable,andcall ittheatomicnumber,theyhadnoideathatsucharelationship existed.Mendeleev'stablejustseemedlikeanorganizationaltool, notsomethingwithanynecessaryphysicalsignicance.AndeverythingMendeleevhaddoneseemedequallyvalidifyouturnedthe tableupside-downorreverseditsleftandrightsides,soevenifyou wantedtonumbertheelementssequentiallywithintegers,therewas anambiguityastohowtodoit.Mendeleev'soriginaltablewasin factupside-downcomparedtothemodernone. Intheperiodimmediatelyfollowingthediscoveryofthenucleus, physicistsonlyhadroughestimatesofthechargesofthevarious nuclei.Inthecaseoftheverylightestnuclei,theysimplyfound themaximumnumberofelectronstheycouldstripobyvarious methods:chemicalreactions,electricsparks,ultravioletlight,and soon.Forexampletheycouldeasilystripofoneortwoelectrons 48 Chapter2TheNucleus

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k / Amodernperiodictable, labeledwithatomicnumbers. Mendeleev'soriginaltablewas upside-downcomparedtothis one. fromhelium,makingHe + orHe ++ ,butnobodycouldmakeHe +++ presumablybecausethenuclearchargeofheliumwasonly+2 e Unfortunatelyonlyafewofthelightestelementscouldbestripped completely,becausethemoreelectronswerestrippedo,thegreater thepositivenetchargeremaining,andthemorestronglytherestof thenegativelychargedelectronswouldbeheldon.Theheavyelements'atomicnumberscouldonlyberoughlyextrapolatedfromthe lightelements,wheretheatomicnumberwasabouthalftheatom's massexpressedinunitsofthemassofahydrogenatom.Gold,for example,hadamassabout197timesthatofhydrogen,soitsatomic numberwasestimatedtobeabouthalfthat,orsomewherearound 100.Wenowknowittobe79. Howdidwenallyndout?Theriddleofthenuclearcharges wasatlastsuccessfullyattackedusingtwodierenttechniques, whichgaveconsistentresults.Onesetofexperiments,involving x-rays,wasperformedbytheyoungHenryMosely,whosescientic brilliancewassoontobesacricedinabattlebetweenEuropeanimperialistsoverwhowouldowntheDardanelles,duringthatpointless conictthenknownastheWartoEndAllWars,andnowreferred toasWorldWarI. l / Analphaparticlehastocome muchclosertothelow-charged coppernucleusinordertobedeectedthroughthesameangle. Section2.3Atomicnumber 49

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SinceMosely'sanalysisrequiresseveralconceptswithwhichyou arenotyetfamiliar,wewillinsteaddescribethetechniqueused byJamesChadwickataroundthesametime.Anaddedbonusof describingChadwick'sexperimentsisthattheypresagedtheimportantmoderntechniqueofstudying collisions ofsubatomicparticles. Ingradschool,Iworkedwithaprofessorwhosethesisadviser'sthesisadviserwasChadwick,andherelatedsomeinterestingstories abouttheman.Chadwickwasapparentlyalittlenuttyandacompletefanaticaboutscience,totheextentthatwhenhewasheldina GermanprisoncampduringWorldWarII,hemanagedtocajolehis captorsintoallowinghimtoscroungeuppartsfrombrokenradios sothathecouldattempttodophysicsexperiments. Chadwick'sexperimentworkedlikethis.Supposeyouperform twoRutherford-typealphascatteringmeasurements,rstonewitha goldfoilasatargetasinRutherford'soriginalexperiment,andthen onewithacopperfoil.Itispossibletogetlargeanglesofdeection inbothcases,butasshowningurem,thealphaparticlemust beheadingalmoststraightforthecoppernucleustogetthesame angleofdeectionthatwouldhaveoccurredwithanalphathat wasmuchfartherothemark;thegoldnucleus'chargeissomuch greaterthanthecopper'sthatitexertsastrongforceonthealpha particleevenfromfaro.Thesituationisverymuchlikethatofa blindfoldedpersonplayingdarts.Justasitisimpossibletoaiman alphaparticleatanindividualnucleusinthetarget,theblindfolded personcannotreallyaimthedarts.Achievingaverycloseencounter withthecopperatomwouldbeakintohittinganinnercircleonthe dartboard.It'smuchmorelikelythatonewouldhavetheluckto hittheoutercircle,whichcoversagreaternumberofsquareinches. Byanalogy,ifyoumeasurethefrequencywithwhichalphasare scatteredbycopperatsomeparticularangle,saybetween19and 20degrees,andthenperformthesamemeasurementatthesame anglewithgold,yougetamuchhigherpercentageforgoldthanfor copper. m / Analphaparticlemustbe headedfortheringonthefront oftheimaginarycylindricalpipe inordertoproducescatteringat ananglebetween19and20degrees.Theareaofthisring iscalledthecross-sectionfor scatteringat19-20 becauseitis thecross-sectionalareaofacut throughthepipe. 50 Chapter2TheNucleus

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Infact,thenumericalratioofthetwonuclei'schargescanbe derivedfromthissameexperimentallydeterminedratio.Usingthe standardnotation Z fortheatomicnumberchargeofthenucleus dividedby e ,thefollowingequationcanbeprovedexample1: Z 2 gold Z 2 copper = numberofalphasscatteredbygoldat19-20 numberofalphasscatteredbycopperat19-20 Bymakingsuchmeasurementsfortargetsconstructedfromallthe elements,onecaninfertheratiosofalltheatomicnumbers,and sincetheatomicnumbersofthelightelementswerealreadyknown, atomicnumberscouldbeassignedtotheentireperiodictable.AccordingtoMosely,theatomicnumbersofcopper,silverandplatinumwere29,47,and78,whichcorrespondedwellwiththeirpositionsontheperiodictable.Chadwick'sguresforthesameelements were29.3,46.3,and77.4,witherrorbarsofabout1.5timesthefundamentalcharge,sothetwoexperimentswereingoodagreement. Thepointhereisabsolutelynotthatyoushouldbereadytoplug numbersintotheaboveequationforahomeworkorexamquestion! Myoverallgoalinthischapteristoexplainhowweknowwhatwe knowaboutatoms.AnaddedbonusofdescribingChadwick'sexperimentisthattheapproachisverysimilartothatusedinmodern particlephysicsexperiments,andtheideasusedintheanalysisare closelyrelatedtothenow-ubiquitousconceptofacross-section." Inthedartboardanalogy,thecross-sectionwouldbetheareaofthe circularringyouhavetohit.Thereasoningbehindtheinventionof thetermcross-section"canbevisualizedasshowningurem.In thislanguage,Rutherford'sinventionoftheplanetarymodelcame fromhisunexpecteddiscoverythattherewasanonzerocross-section foralphascatteringfromgoldatlargeangles,andChadwickconrmedMosely'sdeterminationsoftheatomicnumbersbymeasuring cross-sectionsforalphascattering. ProofoftherelationshipbetweenZandscatteringexample1 Theequationabovecanbederivedbythefollowingnotveryrigorousproof.Todeectthealphaparticlebyacertainanglerequires thatitacquireacertainmomentumcomponentinthedirection perpendiculartoitsoriginalmomentum.Althoughthenucleus's forceonthealphaparticleisnotconstant,wecanpretendthat itisapproximatelyconstantduringthetimewhenthealphais withinadistanceequalto,say,150%ofitsdistanceofclosest approach,andthattheforceiszerobeforeandafterthatpartof themotion.Ifwechose120%or200%,itshouldn'tmakeany differenceinthenalresult,becausethenalresultisaratio, andtheeffectsonthenumeratoranddenominatorshouldcancel eachother.Intheapproximationofconstantforce,thechange inthealpha'sperpendicularmomentumcomponentisthenequal Section2.3Atomicnumber 51

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to F t .TheCoulombforcelawsaystheforceisproportionalto Z = r 2 .Although r doeschangesomewhatduringthetimeinterval ofinterest,it'sgoodenoughtotreatitasaconstantnumber,since we'reonlycomputingtheratiobetweenthetwoexperiments'results.Sinceweareapproximatingtheforceasactingoverthe timeduringwhichthedistanceisnottoomuchgreaterthanthe distanceofclosestapproach,thetimeinterval t mustbeproportionalto r ,andthesidewaysmomentumimpartedtothealpha, F t ,isproportionalto Z = r 2 r ,or Z = r .Ifwe'recomparingalphas scatteredatthesameanglefromgoldandfromcopper,then p isthesameinbothcases,andtheproportionality p / Z = r tells usthattheonesscatteredfromcopperatthatanglehadtobe headedinalongalineclosertothecentralaxisbyafactorequaling Z gold = Z copper .Ifyouimagineadartboardringthatthealphas havetohit,thentheringforthegoldexperimenthasthesame proportionsastheoneforcopper,butitisenlargedbyafactor equalto Z gold = Z copper .Thatis,notonlyistheradiusofthering greaterbythatfactor,butunliketheringsonanormaldartboard, thethicknessoftheouterringisalsogreaterinproportiontoits radius.Whenyoutakeageometricshapeandscaleitupinsize likeaphotographicenlargement,itsareaisincreasedinproportiontothesquareoftheenlargementfactor,sotheareaofthe dartboardringinthegoldexperimentisgreaterbyafactorequal to Z gold = Z copper 2 .Sincethealphasareaimedentirelyrandomly, thechancesofanalphahittingtheringareinproportiontothe areaofthering,whichprovestheequationgivenabove. Asanexampleofthemodernuseofscatteringexperimentsand cross-sectionmeasurements,youmayhaveheardoftherecentexperimentalevidencefortheexistenceofaparticlecalledthetop quark.Ofthetwelvesubatomicparticlescurrentlybelievedtobethe smallestconstituentsofmatter,sixformafamilycalledthequarks, distinguishedfromtheothersixbytheintenseattractiveforcesthat makethequarkssticktoeachother.Theothersixconsistofthe electronplusveother,moreexoticparticles.Theonlytwotypesof quarksfoundinnaturallyoccurringmatteraretheupquark"and downquark,"whicharewhatprotonsandneutronsaremadeof, butfourothertypesweretheoreticallypredictedtoexist,foratotal ofsix.Thewhimsicaltermquark"comesfromalinebyJames JoycereadingThreequarksformasterMark."Untilrecently,only vetypesofquarkshadbeenproventoexistviaexperiments,and thesixth,thetopquark,wasonlytheorized.Therewasnohope ofeverdetectingatopquarkdirectly,sinceitisradioactive,and onlyexistsforazillionthofasecondbeforeevaporating.Instead, theresearcherssearchingforitattheFermiNationalAccelerator LaboratorynearChicagomeasuredcross-sectionsforscatteringof nucleioofothernuclei.Theexperimentwasmuchlikethoseof RutherfordandChadwick,exceptthattheincomingnucleihadto 52 Chapter2TheNucleus

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beboostedtomuchhigherspeedsinaparticleaccelerator.The resultingencounterwithatargetnucleuswassoviolentthatboth nucleiwerecompletelydemolished,but,asEinsteinproved,energy canbeconvertedintomatter,andtheenergyofthecollisioncreates asprayofexotic,radioactiveparticles,likethedeadlyshowerof woodfragmentsproducedbyacannonballinanoldnavalbattle. Amongthoseparticlesweresometopquarks.Thecross-sections beingmeasuredwerethecross-sectionsfortheproductionofcertain combinationsofthesesecondaryparticles.Howeverdierentthe details,theprinciplewasthesameasthatemployedattheturnof thecentury:yousmashthingstogetherandlookatthefragments thatyotoseewhatwasinsidethem.Theapproachhasbeen comparedtoshootingaclockwitharieandthenstudyingthe piecesthatyotogureouthowtheclockworked. DiscussionQuestions A Thediagram,showingalphaparticlesbeingdeectedbyagold nucleus,wasdrawnwiththeassumptionthatalphaparticlescameinon linesatmanydifferentdistancesfromthenucleus.Whywouldn'ttheyall comeinalongthesameline,sincetheyallcameoutthroughthesame tube? B Whydoesitmakesensethat,asshowninthegure,thetrajectories thatresultin19 and20 scatteringcrosseachother? C Rutherfordknewthevelocityofthealphaparticlesemittedbyradium, andguessedthatthepositivelychargedpartofagoldatomhadacharge ofabout+100 e wenowknowitis+79 e .Consideringthefactthatsome alphaparticlesweredeectedby180 ,howcouldhethenuseconservationofenergytoderiveanupperlimitonthesizeofagoldnucleus?For simplicity,assumethesizeofthealphaparticleisnegligiblecomparedto thatofthegoldnucleus,andignorethefactthatthegoldnucleusrecoils alittlefromthecollision,pickingupalittlekineticenergy. 2.4Thestructureofnuclei Theproton Thefactthatthenuclearchargeswereallintegermultiplesof e suggestedtomanyphysiciststhatratherthanbeingapointlikeobject,thenucleusmightcontainsmallerparticleshavingindividual chargesof+ e .Evidenceinfavorofthisideawasnotlonginarriving.Rutherfordreasonedthatifhebombardedtheatomsofavery lightelementwithalphaparticles,thesmallchargeofthetarget nucleiwouldgiveaveryweakrepulsion.Perhapsthosefewalpha particlesthathappenedtoarriveonhead-oncollisioncourseswould getsoclosethattheywouldphysicallycrashintosomeofthetarget nuclei.Analphaparticleisitselfanucleus,sothiswouldbeacollisionbetweentwonuclei,andaviolentoneduetothehighspeeds involved.Rutherfordhitpaydirtinanexperimentwithalphaparticlesstrikingatargetcontainingnitrogenatoms.Chargedparticles Section2.4Thestructureofnuclei 53

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n / Examplesoftheconstruction ofatoms:hydrogentopand heliumbottom.Onthisscale, theelectrons'orbitswouldbethe sizeofacollegecampus. weredetectedyingoutofthetargetlikepartsyingoofcarsin ahigh-speedcrash.Measurementsofthedeectionoftheseparticlesinelectricandmagneticeldsshowedthattheyhadthesame charge-to-massratioassingly-ionizedhydrogenatoms.Rutherford concludedthattheseweretheconjecturedsingly-chargedparticles thatheldthechargeofthenucleus,andtheywerelaternamed protons.Thehydrogennucleusconsistsofasingleproton,andin general,anelement'satomicnumbergivesthenumberofprotons containedineachofitsnuclei.Themassoftheprotonisabout1800 timesgreaterthanthemassoftheelectron. Theneutron Itwouldhavebeenniceandsimpleifallthenucleicouldhave beenbuiltonlyfromprotons,butthatcouldn'tbethecase.Ifyou spendalittletimelookingataperiodictable,youwillsoonnotice thatalthoughsomeoftheatomicmassesareverynearlyinteger multiplesofhydrogen'smass,manyothersarenot.Evenwherethe massesareclosewholenumbers,themassesofanelementother thanhydrogenisalwaysgreaterthanitsatomicnumber,notequal toit.Helium,forinstance,hastwoprotons,butitsmassisfour timesgreaterthanthatofhydrogen. Chadwickcleareduptheconfusionbyprovingtheexistenceof anewsubatomicparticle.Unliketheelectronandproton,which areelectricallycharged,thisparticleiselectricallyneutral,andhe namedittheneutron.Chadwick'sexperimenthasbeendescribed indetailinchapter4ofbook2ofthisseries,butbrieythemethod wastoexposeasampleofthelightelementberylliumtoastreamof alphaparticlesfromalumpofradium.Berylliumhasonlyfourprotons,soanalphathathappenstobeaimeddirectlyataberyllium nucleuscanactuallyhititratherthanbeingstoppedshortofacollisionbyelectricalrepulsion.Neutronswereobservedasanewform ofradiationemergingfromthecollisions,andChadwickcorrectly inferredthattheywerepreviouslyunsuspectedcomponentsofthe nucleusthathadbeenknockedout.Asdescribedin Conservation Laws ,Chadwickalsodeterminedthemassoftheneutron;itisvery nearlythesameasthatoftheproton. Tosummarize,atomsaremadeofthreetypesofparticles: charge massinunitsof theproton'smass locationinatom proton + e 1 innucleus neutron 0 1.001 innucleus electron )]TJ/F20 10.9091 Tf 8.485 0 Td [(e 1/1836 orbitingnucleus Theexistenceofneutronsexplainedthemysteriousmassesof theelements.Helium,forinstance,hasamassveryclosetofour timesgreaterthanthatofhydrogen.Thisisbecauseitcontains twoneutronsinadditiontoitstwoprotons.Themassofanatomis 54 Chapter2TheNucleus

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o / AversionoftheThomson apparatusmodiedformeasuring themass-to-chargeratiosof ionsratherthanelectrons.A smallsampleoftheelementin question,copperinourexample, isboiledintheoventocreate athinvapor.Avacuumpump iscontinuouslysuckingonthe mainchambertokeepitfrom accumulatingenoughgastostop thebeamofions.Someofthe atomsofthevaporareionizedby asparkorbyultravioletlight.Ions thatwanderoutofthenozzle andintotheregionbetween thechargedplatesarethen acceleratedtowardthetopofthe gure.AsintheThomsonexperiment,mass-to-chargeratiosare inferredfromthedeectionofthe beam. essentiallydeterminedbythetotalnumberofneutronsandprotons. Thetotalnumberofneutronsplusprotonsisthereforereferredto astheatom's massnumber Isotopes Wenowhaveaclearinterpretationofthefactthatheliumis closetofourtimesmoremassivethanhydrogen,andsimilarlyfor alltheatomicmassesthatareclosetoanintegermultipleofthe massofhydrogen.Butwhataboutcopper,forinstance,whichhad anatomicmass63.5timesthatofhydrogen?Itdidn'tseemreasonabletothinkthatitpossessedanextrahalfofaneutron!The solutionwasfoundbymeasuringthemass-to-chargeratiosofsinglyionizedatomsatomswithoneelectronremoved.Thetechnique isessentiallythatsameastheoneusedbyThomsonforcathode rays,exceptthatwholeatomsdonotspontaneouslyleapoutofthe surfaceofanobjectaselectronssometimesdo.Figureoshowsan exampleofhowtheionscanbecreatedandinjectedbetweenthe chargedplatesforacceleration. Injectingastreamofcopperionsintothedevice,wendasurprise|thebeamsplitsintotwoparts!Chemistshadelevatedto dogmatheassumptionthatalltheatomsofagivenelementwere identical,butwendthat69%ofcopperatomshaveonemass,and 31%haveanother.Notonlythat,butbothmassesareverynearly integermultiplesofthemassofhydrogenand65,respectively. Coppergetsitschemicalidentityfromthenumberofprotonsinits nucleus,29,sincechemicalreactionsworkbyelectricforces.But apparentlysomecopperatomshave63 )]TJ/F15 10.9091 Tf 11.569 0 Td [(29=34neutronswhile othershave65 )]TJ/F15 10.9091 Tf 10.652 0 Td [(29=36.Theatomicmassofcopper,63.5,reects theproportionsofthemixtureofthemass-63andmass-65varieties. Thedierentmassvarietiesofagivenelementarecalled isotopesof thatelement. Isotopescanbenamedbygivingthemassnumberasasubscript totheleftofthechemicalsymbol,e.g., 65 Cu.Examples: protons neutrons massnumber 1 H 1 0 0+1=1 4 He 2 2 2+2=4 12 C 6 6 6+6=12 14 C 6 8 6+8=14 262 Ha 105 157 105+157=262 self-checkA Whyarethepositiveandnegativechargesoftheacceleratingplates reversedintheisotope-separatingapparatuscomparedtotheThomson apparatus? Answer,p.203 Chemicalreactionsareallabouttheexchangeandsharingof electrons:thenucleihavetositoutthisdancebecausetheforces ofelectricalrepulsionpreventthemfromevergettingcloseenough Section2.4Thestructureofnuclei 55

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q / Thestrongnuclearforce cutsoffverysharplyatarangeof about1fm. tomakecontactwitheachother.Althoughtheprotonsdohavea vitallyimportanteectonchemicalprocessesbecauseoftheirelectricalforces,theneutronscanhavenoeectontheatom'schemical reactions.Itisnotpossible,forinstance,toseparate 63 Cufrom 65 Cu bychemicalreactions.Thisiswhychemistshadneverrealizedthat dierentisotopesexisted.Tobeperfectlyaccurate,dierentisotopesdobehaveslightlydierentlybecausethemoremassiveatoms movemoresluggishlyandthereforereactwithatinybitlessintensity.Thistinydierenceisused,forinstance,toseparateoutthe isotopesofuraniumneededtobuildanuclearbomb.Thesmallness ofthiseectmakestheseparationprocessaslowanddicultone, whichiswhatwehavetothankforthefactthatnuclearweapons havenotbeenbuiltbyeveryterroristcabalontheplanet. Sizesandshapesofnuclei Matterisnearlyallnucleiifyoucountbyweight,butinterms ofvolumenucleidon'tamounttomuch.Theradiusofanindividual neutronorprotonisverycloseto1fmfm=10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(15 m,soevenabig leadnucleuswithamassnumberof208stillhasadiameterofonly about13fm,whichistenthousandtimessmallerthanthediameter ofatypicalatom.Contrarytotheusualimageryofthenucleusasa smallsphere,itturnsoutthatmanynucleiaresomewhatelongated, likeanAmericanfootball,andafewhaveexoticasymmetricshapes likepearsorkiwifruits. DiscussionQuestions A Supposetheentireuniversewasinaverylargecerealbox,and thenutritionallabelingwassupposedtotellagodlikeconsumerwhatpercentageofthecontentswasnuclei.Roughlywhatwouldthepercentage belikeifthelabelingwasaccordingtomass?Whatifitwasbyvolume? 2.5Thestrongnuclearforce,alphadecayand ssion Oncephysicistsrealizedthatnucleiconsistedofpositivelycharged protonsandunchargedneutrons,theyhadaproblemontheirhands. Theelectricalforcesamongtheprotonsareallrepulsive,sothe nucleusshouldsimplyyapart!Thereasonallthenucleiinyour bodyarenotspontaneouslyexplodingatthismomentisthatthere isanotherforceacting.Thisforce,calledthe strongnuclearforce ,is alwaysattractive,andactsbetweenneutronsandneutrons,neutrons andprotons,andprotonsandprotonswithroughlyequalstrength. Thestrongnuclearforcedoesnothaveanyeectonelectrons,which iswhyitdoesnotinuencechemicalreactions. Unlikeelectricforces,whosestrengthsaregivenbythesimple Coulombforcelaw,thereisnosimpleformulaforhowthestrong nuclearforcedependsondistance.Roughlyspeaking,itiseec56 Chapter2TheNucleus

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p / AnuclearpowerplantatCattenom,France.Unlikethecoal andoilplantsthatsupplymost oftheU.S.'selectricalpower,a nuclearpowerplantlikethisone releasesnopollutionorgreenhousegasesintotheEarth'satmosphere,andthereforedoesn't contributetoglobalwarming.The whitestuffpufngoutofthis plantisnon-radioactivewatervapor.Althoughnuclearpower plantsgeneratelong-livednuclear waste,thiswastearguablyposes muchlessofathreattothebiospherethangreenhousegases would. tiveoverrangesof 1fm,butfallsoextremelyquicklyatlarger distancesmuchfasterthan1 =r 2 .Sincetheradiusofaneutronor protonisabout1fm,thatmeansthatwhenabunchofneutronsand protonsarepackedtogethertoformanucleus,thestrongnuclear forceiseectiveonlybetweenneighbors. Figurerillustrateshowthestrongnuclearforceactstokeep ordinarynucleitogether,butisnotabletokeepveryheavynuclei frombreakingapart.Inr/1,aprotoninthemiddleofacarbon nucleusfeelsanattractivestrongnuclearforcearrowsfromeach ofitsnearestneighbors.Theforcesareallindierentdirections, andtendtocancelout.Thesameistruefortherepulsiveelectrical forcesnotshown.Ingurer/2,aprotonattheedgeofthenucleus hasneighborsonlyononeside,andthereforeallthestrongnuclear forcesactingonitaretendingtopullitbackin.Althoughallthe electricalforcesfromtheotherveprotonsdarkarrowsareall pushingitoutofthenucleus,theyarenotsucienttoovercome thestrongnuclearforces. Inaveryheavynucleus,r/3,aprotonthatndsitselfnearthe edgehasonlyafewneighborscloseenoughtoattractitsignicantly viathestrongnuclearforce,buteveryotherprotoninthenucleus exertsarepulsiveelectricalforceonit.Ifthenucleusislargeenough, thetotalelectricalrepulsionmaybesucienttoovercometheattractionofthestrongforce,andthenucleusmayspitoutaproton. Protonemissionisfairlyrare,however;amorecommontypeofradioactivedecay 1 inheavynucleiisalphadecay,showninr/4.The 1 Alphadecayismorecommonbecauseanalphaparticlehappenstobea verystablearrangementofprotonsandneutrons. Section2.5Thestrongnuclearforce,alphadecayandssion 57

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r / 1.Theforcescancel.2.Theforcesdon'tcancel.3.Inaheavy nucleus,thelargenumberofelectricalrepulsionscanadduptoaforce thatiscomparabletothestrongnuclearattraction.4.Alphaemission.5. Fission. imbalanceoftheforcesissimilar,butthechunkthatisejectedisan alphaparticletwoprotonsandtwoneutronsratherthanasingle proton. Itisalsopossibleforthenucleustosplitintotwopiecesof roughlyequalsize,r/5,aprocessknownasssion.Notethatin additiontothetwolargefragments,thereisasprayofindividual neutrons.Inanuclearssionbomboranuclearssionreactor, someoftheseneutronsyoandhitothernuclei,causingthemto undergossionaswell.Theresultisachainreaction. Whenanucleusisabletoundergooneoftheseprocesses,itis saidtoberadioactive,andtoundergoradioactivedecay.Someof thenaturallyoccurringnucleionearthareradioactive.Theterm radioactive"comesfromBecquerel'simageofraysradiatingout fromsomething,notfromradiowaves,whichareawholedierentphenomenon.Thetermdecay"canalsobealittlemisleading, sinceitimpliesthatthenucleusturnstodustorsimplydisappears {actuallyitissplittingintotwonewnucleiwithanthesametotal numberofneutronsandprotons,sothetermradioactivetransformation"wouldhavebeenmoreappropriate.Althoughtheoriginal atom'selectronsaremerespectatorsintheprocessofweakradioac58 Chapter2TheNucleus

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tivedecay,weoftenspeaklooselyofradioactiveatoms"ratherthan radioactivenuclei." Randomnessinphysics Howdoesanatomdecidewhentodecay?Wemightimagine thatitislikeatermite-infestedhousethatgetsweakerandweaker, untilnallyitreachesthedayonwhichitisdestinedtofallapart. Experiments,however,havenotsucceededindetectingsuchtickingclock"hiddenbelowthesurface;theevidenceisthatallatoms ofagivenisotopeareabsolutelyidentical.Why,then,wouldone uraniumatomdecaytodaywhileanotherlivesforanothermillion years?Theanswerappearstobethatitisentirelyrandom.We canmakegeneralstatementsabouttheaveragetimerequiredfora certainisotopetodecay,orhowlongitwilltakeforhalftheatoms inasampletodecayitshalf-life,butwecanneverpredictthe behaviorofaparticularatom. Thisistherstexamplewehaveencounteredofaninescapable randomnessinthelawsofphysics.Ifthiskindofrandomnessmakes youuneasy,you'reingoodcompany.Einstein'sfamousquoteis ...IamconvincedthatHe[God]doesnotplaydice.Einstein's distasteforrandomness,andhisassociationofdeterminismwith divinity,goesbacktotheEnlightenmentconceptionoftheuniverse asagiganticpieceofclockworkthatonlyhadtobesetinmotion initiallybytheBuilder.Physicshadtobeentirelyrebuiltinthe 20thcenturytoincorporatethefundamentalrandomnessofphysics, andthismodernrevolutionisthetopicofbook6inthisseries.In particular,wewilldelaythemathematicaldevelopmentofthehalflifeconceptuntilthen. 2.6Theweaknuclearforce;betadecay Allthenuclearprocesseswe'vediscussedsofarhaveinvolvedrearrangementsofneutronsandprotons,withnochangeinthetotal numberofneutronsorthetotalnumberofprotons.Nowconsider theproportionsofneutronsandprotonsinyourbodyandinthe planetearth:neutronsandprotonsareroughlyequallynumerous inyourbody'scarbonandoxygennuclei,andalsointhenickeland ironthatmakeupmostoftheearth.Theproportionsareabout 50-50.But,asdiscussedinmoredetailinoptionalsection2.10,the onlychemicalelementsproducedinanysignicantquantitiesbythe bigbang 2 werehydrogenabout90%andheliumabout10%.If theearlyuniversewasalmostnothingbuthydrogenatoms,whose nucleiareprotons,wheredidallthoseneutronscomefrom? Theansweristhatthereisanothernuclearforce,theweaknuclearforce,thatiscapableoftransformingneutronsintoprotons 2 Theevidenceforthebigbangtheoryoftheoriginoftheuniversewasdiscussedinbook3ofthisseries. Section2.6Theweaknuclearforce;betadecay 59

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andvice-versa.Twopossiblereactionsare n p+e )]TJ/F15 10.9091 Tf 9.509 -4.504 Td [(+ [electrondecay] and p n+e + + .[positrondecay] Thereisalsoathirdtypecalledelectroncapture,inwhichaproton grabsoneoftheatom'selectronsandtheyproduceaneutronand aneutrino. Whereasalphadecayandssionarejustaredivisionofthepreviouslyexistingparticles,thesereactionsinvolvethedestructionof oneparticleandthecreationofthreenewparticlesthatdidnot existbefore. Therearethreenewparticlesherethatyouhaveneverpreviouslyencountered.Thesymbole + standsforanantielectron,which isaparticlejustliketheelectronineveryway,exceptthatitselectricchargeispositiveratherthannegative.Antielectronsarealso knownaspositrons.Nobodyknowswhyelectronsaresocommonin theuniverseandantielectronsarescarce.Whenanantielectronencountersanelectron,theyannihilateeachother,producinggamma rays,andthisisthefateofalltheantielectronsthatareproduced bynaturalradioactivityonearth.Antielectronsareanexampleof antimatter.Acompleteatomofantimatterwouldconsistofantiprotons,antielectrons,andantineutrons.Althoughindividualparticles ofantimatteroccurcommonlyinnatureduetonaturalradioactivity andcosmicrays,onlyafewcompleteatomsofantihydrogenhave everbeenproducedarticially. Thenotation standsforaparticlecalledaneutrino,and meansanantineutrino.Neutrinosandantineutrinoshavenoelectric chargehencethename. Wecannowlistallfouroftheknownfundamentalforcesof physics: gravity electromagnetism strongnuclearforce weaknuclearforce Theotherforceswehavelearnedabout,suchasfrictionandthe normalforce,allarisefromelectromagneticinteractionsbetween atoms,andthereforearenotconsideredtobefundamentalforcesof physics. 60 Chapter2TheNucleus

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Decayof 212 Pb example2 Asanexample,considertheradioactiveisotopeoflead 212 Pb.It contains82protonsand130neutrons.Itdecaysbytheprocess n p + e )]TJ/F39 10.9091 Tf 10.409 -3.959 Td [(+ .Thenewlycreatedprotonisheldinsidethe nucleusbythestrongnuclearforce,sothenewnucleuscontains 83protonsand129neutrons.Having83protonsmakesitthe elementbismuth,soitwillbeanatomof 212 Bi. Inareactionlikethisone,theelectroniesoathighspeed typicallyclosetothespeedoflight,andtheescapingelectrons arethethingsthatmakelargeamountsofthistypeofradioactivity dangerous.Theoutgoingelectronwastherstthingthattipped oscientistsintheearly1900stotheexistenceofthistypeofradioactivity.Sincetheydidn'tknowthattheoutgoingparticleswere electrons,theycalledthembetaparticles,andthistypeofradioactivedecaywasthereforeknownasbetadecay.Aclearerbutless commonterminologyistocallthetwoprocesseselectrondecayand positrondecay. Theneutrinoorantineutrinoemittedinsuchareactionpretty muchignoresallmatter,becauseitslackofchargemakesitimmune toelectricalforces,anditalsoremainsalooffromstrongnuclear interactions.Evenifithappenstoyogoingstraightdown,it isalmostcertaintomakeitthroughtheentireearthwithoutinteractingwithanyatomsinanyway.Itendsupyingthrough outerspaceforever.Theneutrino'sbehaviormakesitexceedingly diculttodetect,andwhenbetadecaywasrstdiscoverednobody realizedthatneutrinosevenexisted.Wenowknowthattheneutrinocarriesosomeoftheenergyproducedinthereaction,butat thetimeitseemedthatthetotalenergyafterwardsnotcounting theunsuspectedneutrino'senergywasgreaterthanthetotalenergybeforethereaction,violatingconservationofenergy.Physicists weregettingreadytothrowconservationofenergyoutthewindow asabasiclawofphysicswhenindirectevidenceledthemtothe conclusionthatneutrinosexisted. Thesolarneutrinoproblem Whatabouttheseneutrinos?Whyhaven'tyouheardofthem before?It'snotbecausethey'rerare|abillionneutrinospass throughyourbodyeverymicrosecond,butuntilrecentlyalmost nothingwasknownaboutthem.Producedasaside-eectofthe nuclearreactionsthatpoweroursunandotherstars,theseghostlike bitsofmatterarebelievedtobethemostnumerousparticlesinthe universe.Buttheyinteractsoweaklywithordinarymatterthat nearlyalltheneutrinosthatentertheearthononesidewillemerge fromtheothersideofourplanetwithoutevenslowingdown. Ourrstrealpeekatthepropertiesoftheelusiveneutrinohas comefromahugedetectorinaplayed-outJapanesezincmine,s.An internationalteamofphysicistsoutttedthemineshaftwithwallSection2.6Theweaknuclearforce;betadecay 61

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s / Thisneutrinodetectoris intheprocessofbeinglledwith ultrapurewater. to-walllightsensors,andthenlledthewholethingwithwaterso purethatyoucanseethroughitforahundredmeters,comparedto onlyafewmetersfortypicaltapwater.Neutrinosstreamthrough the50millionlitersofwatercontinually,justastheyoodeverythingelsearoundus,andthevastmajorityneverinteractwitha watermolecule.Averysmallpercentage,however,doannihilate themselvesinthewater,andthetinyashesoflighttheyproduce canbedetectedbythebeachball-sizedvacuumtubesthatlinethe darkenedmineshaft.Mostoftheneutrinosarounduscomefrom thesun,butfortechnicalreasonsthistypeofwater-baseddetector ismoresensitivetothelesscommonbutmoreenergeticneutrinos producedwhencosmicrayparticlesstriketheearth'satmosphere. Neutrinoswerealreadyknowntocomeinthreeavors,"which canbedistinguishedfromeachotherbytheparticlescreatedwhen theycollidewithmatter.Anelectron-avoredneutrino"createsan ordinaryelectronwhenitisannihilated,whilethetwoothertypes createmoreexoticparticlescalledmuandtauparticles.Thinkof thethreetypesofneutrinosaschocolate,vanilla,andstrawberry. Whenyoubuyachocolateicecreamcone,youexpectthatitwill keepbeingchocolateasyoueatit.Theunexpectedndingfrom theJapaneseexperimentisthatsomeoftheneutrinosarechanging avorbetweenthetimewhentheyareproducedbyacosmicrayand themomentwhentheywinkoutofexistenceinthewater.It'sas thoughyourchocolateicecreamconetransformeditselfmagically intostrawberrywhileyourbackwasturned. Howdidthephysicistsgureoutthechangeinavor?The experimentdetectssomeneutrinosoriginatingintheatmosphere aboveJapan,andalsomanyneutrinoscomingfromdistantpartsof theearth.AneutrinocreatedabovetheAtlanticOceanarrivesin Japanfromunderneath,andtheexperimentcandistinguishthese upward-travelingneutrinosfromthedownward-movinglocalvariety.Theyfoundthatthemixtureofneutrinoscomingfrombelow wasdierentfromthemixturearrivingfromabove,withsomeof theelectron-avoredandtau-avoredneutrinoshavingapparently changedintomu-avoredneutrinosduringtheirvoyagethroughthe earth.Theonescomingfromabovedidn'thavetimetochange avorsontheirmuchshorterjourney. Thisisinterpretedasevidencethattheneutrinosareconstantly changingbackandforthamongthethreeavors.Ontheoretical grounds,itisbelievedthatsuchavibrationcanonlyoccurifneutrinoshavemass.Onlyaroughestimateofthemassispossibleat thispoint:itappearsthatneutrinoshaveamasssomewhereinthe neighborhoodofonebillionthofthemassofanelectron,orabout 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(39 kg. Iftheneutrino'smassissotiny,doesitevenmatter?Itmatters toastronomers.Neutrinosaretheonlyparticlesthatcanbeused 62 Chapter2TheNucleus

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t / Adetectorbeinglowered downashaftattheIceCube neutrinotelescopeinAntarctica. toprobecertainphenomena.Forexample,theyaretheonlydirect probeswehavefortestingourmodelsofthecoreofourownsun, whichisthesourceofenergyforalllifeonearth.Onceastronomers haveagoodhandleonthebasicpropertiesoftheneutrino,they canstartthinkingseriouslyaboutusingthemforastronomy.Asof 2006,themassoftheneutrinohasbeenconrmedbyanacceleratorbasedexperiment,andneutrinoobservatorieshavebeenoperating forafewyearsinAntarctica,usinghugevolumesofnaturalicein thesamewaythatthewaterwasusedintheJapaneseexperiment. A Inthereactionsn p+e )]TJ/F39 9.9626 Tf 9.637 -3.616 Td [(+ andp n+e + + ,verifythat chargeisconserved.Inbetadecay,whenoneofthesereactionshappens toaneutronorprotonwithinanucleus,oneormoregammaraysmay alsobeemitted.Doesthisaffectconservationofcharge?Woulditbe possibleforsomeextraelectronstobereleasedwithoutviolatingcharge conservation? B Whenanantielectronandanelectronannihilateeachother,they producetwogammarays.Ischargeconservedinthisreaction? 2.7Fusion Aswehaveseen,heavynucleitendtoyapartbecauseeachproton isbeingrepelledbyeveryotherprotoninthenucleus,butisonly attractedbyitsnearestneighbors.Thenucleussplitsupintotwo parts,andassoonasthosetwopartsaremorethanabout1fm apart,thestrongnuclearforcenolongercausesthetwofragments toattracteachother.Theelectricalrepulsionthenacceleratesthem, causingthemtogainalargeamountofkineticenergy.Thisrelease ofkineticenergyiswhatpowersnuclearreactorsandssionbombs. Itmightseem,then,thatthelightestnucleiwouldbethemost stable,butthatisnotthecase.Let'scompareanextremelylight nucleuslike 4 Hewithasomewhatheavierone, 16 O.Aneutronor protonin 4 Hecanbeattractedbythethreeothers,butin 16 O,it mighthaveveorsixneighborsattractingit.The 16 Onucleusis thereforemorestable. Itturnsoutthatthemoststablenucleiofallarethosearound nickelandiron,havingabout30protonsand30neutrons.Justasa nucleusthatistooheavytobestablecanreleaseenergybysplitting apartintopiecesthatareclosertothemoststablesize,lightnuclei canreleaseenergyifyoustickthemtogethertomakebiggernuclei thatareclosertothemoststablesize.Fusingonenucleuswith anotheriscallednuclearfusion.Nuclearfusioniswhatpowersour sunandotherstars. Section2.7Fusion 63

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u / 1.Oursun'ssourceofenergyisnuclearfusion,sonuclearfusionisalsothesourceofpowerforall lifeonearth,including,2,thisrainforestinFatu-Hiva.3.Therstreleaseofenergybynuclearfusionthrough humantechnologywasthe1952IvyMiketestattheEnewetakAtoll.4.Thisarrayofgamma-raydetectorsis calledGAMMASPHERE.Duringoperation,thearrayisclosedup,andabeamofionsproducedbyaparticle acceleratorstrikesatargetatitscenter,producingnuclearfusionreactions.Thegammarayscanbestudiedfor informationaboutthestructureofthefusednuclei,whicharetypicallyvarietiesnotfoundinnature.5.Nuclear fusionpromisestobeaclean,inexhaustiblesourceofenergy.However,thegoalofcommerciallyviablenuclear fusionpowerhasremainedelusive,duetotheengineeringdifcultiesinvolvedinmagneticallycontaininga plasmaionizedgasatasufcientlyhightemperatureanddensity.ThisphotoshowstheexperimentalJET reactor,withthedeviceopenedupontheleft,andinactionontheright. 2.8Nuclearenergyandbindingenergies Inthesamewaythatchemicalreactionscanbeclassiedasexothermicreleasingenergyorendothermicrequiringenergytoreact,so nuclearreactionsmayeitherreleaseoruseupenergy.Theenergies involvedinnuclearreactionsaregreaterbyahugefactor.Thousandsoftonsofcoalwouldhavetobeburnedtoproduceasmuch 64 Chapter2TheNucleus

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energyaswouldbeproducedinanuclearpowerplantbyonekgof fuel. Althoughnuclearreactionsthatuseupenergyendothermic reactionscanbeinitiatedinaccelerators,whereonenucleusis rammedintoanotherathighspeed,theydonotoccurinnature,not eveninthesun.Theamountofkineticenergyrequiredissimply notavailable. Tondtheamountofenergyconsumedorreleasedinanuclear reaction,youneedtoknowhowmuchnuclearinteractionenergy, U nuc ,wasstoredorreleased.Experimentalistshavedeterminedthe amountofnuclearenergystoredinthenucleusofeverystableelement,aswellasmanyunstableelements.Thisistheamountof mechanicalworkthatwouldberequiredtopullthenucleusapart intoitsindividualneutronsandprotons,andisknownasthenuclear bindingenergy. Areactionoccurringinthesunexample3 Thesunproducesitsenergythroughaseriesofnuclearfusion reactions.Oneofthereactionsis 1 H+ 2 H 3 He+ Theexcessenergyisalmostallcarriedoffbythegammaraynot bythekineticenergyofthehelium-3atom.Thebindingenergies inunitsofpJpicojoulesare: 1 H0J 2 H0.35593pJ 3 He1.23489pJ Thetotalinitial nuclearenergyis0pJ+0.35593pJ,andthenalnuclearenergy is1.23489pJ,sobyconservationofenergy,thegammaraymust carryoff0.87896pJofenergy.Thegammarayisthenabsorbed bythesunandconvertedtoheat. self-checkB Whyisthebindingenergyof 1 Hexactlyequaltozero? Answer,p. 203 Conversionofmasstoenergyandenergytomass Ifyouaddupthemassesofthethreeparticlesproducedinthe reactionn p+e )]TJ/F39 9.9626 Tf 8.781 -3.615 Td [(+ ,youwillndthattheydonotequalthemassof theneutron,somassisnotconserved.Anevenmoreblatantexampleis theannihilationofanelectronwithapositron,e )]TJ/F39 9.9626 Tf 7.244 -3.615 Td [(+e + 2 ,inwhichthe originalmassiscompletelydestroyed,sincegammarayshavenomass. Nonconservationofmassisnotjustapropertyofnuclearreactions.It alsooccursinchemicalreactions,butthechangeinmassistoosmall todetectwithordinarylaboratorybalances. Thereasonwhymassisnotbeingconservedisthatmassisbeingconvertedtoenergy,accordingtoEinstein'scelebratedequation E = mc 2 ,inwhich c standsforthespeedoflight.Inthereaction e )]TJ/F39 9.9626 Tf 9.139 -3.615 Td [(+e + 2 ,forinstance,imagineforsimplicitythattheelectronand Section2.8Nuclearenergyandbindingenergies 65

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positronaremovingveryslowlywhentheycollide,sothereisnosignificantamountofenergytostartwith.Wearestartingwithmassandno energy,andendingupwithtwogammaraysthatpossessenergybut nomass.Einstein's E = mc 2 tellsusthattheconversionfactorbetween massandenergyisequaltothesquareofthespeedoflight.Since c isabignumber,theamountofenergyconsumedorreleasedbya chemicalreactiononlyshowsupasatinychangeinmass.Butinnuclearreactions,whichinvolvelargeamountsofenergy,thechangein massmayamounttoasmuchasonepartperthousand.Notethatin thiscontext, c isnotnecessarilythespeedofanyoftheparticles.We arejustusingitsnumericalvalueasaconversionfactor.Notealsothat E = mc 2 doesnotmeanthatanobjectofmass m hasakineticenergy equalto mc 2 ;theenergybeingdescribedby E = mc 2 istheenergy youcouldreleaseifyoudestroyedtheparticleandconverteditsmass entirelyintoenergy,andthatenergywouldbeinadditiontoanykinetic orpotentialenergytheparticlehad. Havewenowbeencheatedoutoftwoperfectlygoodconservation laws,thelawsofconservationofmassandofenergy?No,it'sjust thataccordingtoEinstein,theconservedquantityis E + mc 2 ,not E or m individually.Thequantity E + mc 2 isreferredtoasthemass-energy, andnoviolationofthelawofconservationofmass-energyhasyetbeen observed.Inmostpracticalsituations,itisaperfectlyreasonableto treatmassandenergyasseparatelyconservedquantities. Itisnoweasytoexplainwhyisolatedprotonshydrogennucleiare foundinnature,butneutronsareonlyencounteredintheinteriorof anucleus,notbythemselves.Intheprocessn p+e )]TJ/F39 9.9626 Tf 10.007 -3.615 Td [(+ ,the totalnalmassislessthanthemassoftheneutron,somassisbeing convertedintoenergy.Inthebetadecayofaproton,p n+e + + ,,the nalmassisgreaterthantheinitialmass,sosomeenergyneedstobe suppliedforconversionintomass.Aprotonsittingbyitselfinahydrogen atomcannotdecay,sinceithasnosourceofenergy.Onlyprotons sittinginsidenucleicandecay,andonlythenifthedifferenceinpotential energybetweentheoriginalnucleusandthenewnucleuswouldresult inareleaseofenergy.Butanyisolatedneutronthatiscreatedinnatural orarticialreactionswilldecaywithinamatterofseconds,releasing someenergy. Theequation E = mc 2 occursnaturallyaspartofEinstein'stheory ofspecialrelativity,whichisnotwhatwearestudyingrightnow.This brieftreatmentisonlymeanttoclearuptheissueofwherethemass wasgoinginsomeofthenuclearreactionswewerediscussing. Figurevisacompactwayofshowingthevastvarietyofthe nuclei.Eachboxrepresentsaparticularnumberofneutronsand protons.Theblackboxesarenucleithatarestable,i.e.,thatwould requireaninputofenergyinordertochangeintoanother.The grayboxesshowalltheunstablenucleithathavebeenstudiedexperimentally.Someoftheselastforbillionsofyearsontheaveragebeforedecayingandarefoundinnature,butmosthavemuch shorteraveragelifetimes,andcanonlybecreatedandstudiedin thelaboratory. 66 Chapter2TheNucleus

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v / Theknownnuclei,representedonachartofprotonnumberversusneutronnumber.Notethetwo nucleiinthebottomrowwithzeroprotons.Oneissimplyasingleneutron.Theotherisaclusteroffour neutrons.Thistetraneutronwasreported,unexpectedly,tobeaboundsysteminresultsfroma2002 experiment.Theresultiscontroversial.Ifcorrect,itimpliestheexistenceofaheretoforeunsuspectedtypeof matter,theneutrondroplet,whichwecanthinkofasanatomwithnoprotonsorelectrons. Thecurvealongwhichthestablenucleilieiscalledthelineof stability.Nucleialongthislinehavethemoststableproportion ofneutronstoprotons.Forlightnucleithemoststablemixture isabout50-50,butwecanseethatstableheavynucleihavetwo orthreetimesmoreneutronsthanprotons.Thisisbecausethe electricalrepulsionsofalltheprotonsinaheavynucleusaddup toapowerfulforcethatwouldtendtotearitapart.Thepresence ofalargenumberofneutronsincreasesthedistancesamongthe protons,andalsoincreasesthenumberofattractionsduetothe strongnuclearforce. 2.9Biologicaleffectsofionizingradiation Asascienceeducator,InditfrustratingthatnowhereinthemassiveamountofjournalismdevotedtotheChernobyldisasterdoes oneeverndanynumericalstatementsabouttheamountofradiationtowhichpeoplehavebeenexposed.Anyonementallycapableof Section2.9Biologicaleffectsofionizingradiation 67

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w / Amapshowinglevelsof radiationnearthesiteofthe Chernobyldisaster.Atthe boundaryofthemosthighly contaminatedbrightredareas, peoplewouldbeexposedto about1300milliremperyear, oraboutfourtimesthenatural backgroundlevel.Inthepink areas,whicharestilldensely populated,theexposureiscomparabletothenaturallevelfound inahigh-altitudecitysuchas Denver. understandingsportsstatisticsorweatherreportsoughttobeable tounderstandsuchmeasurements,aslongassomethinglikethe followingexplanatorytextwasinsertedsomewhereinthearticle: Radiationexposureismeasuredinunitsofmillirems.Theaveragepersonisexposedtoabout200milliremseachyearfromnatural backgroundsources. Withthiscontext,peoplewouldbeabletocometoinformedconclusionsbasedonstatementssuchas,"ChildreninFinlandreceived anaveragedoseof milliremsabovenaturalbackground levelsbecauseoftheChernobyldisaster. Amillirem,ormrem,is,ofcourse,athousandthofarem,but whatisarem?Itmeasurestheamountofenergyperkilogramdepositedinthebodybyionizingradiation,multipliedbyaquality factor"toaccountforthedierenthealthhazardsposedbyalphas, betas,gammas,neutrons,andothertypesofradiation.Onlyionizingradiationiscounted,sincenonionizingradiationsimplyheats one'sbodyratherthankillingcellsoralteringDNA.Forinstance, alphaparticlesaretypicallymovingsofastthattheirkineticenergy issucienttoionizethousandsofatoms,butitispossibleforan alphaparticletobemovingsoslowlythatitwouldnothaveenough kineticenergytoionizeevenoneatom. NotwithstandingthepopcultureimagesoftheIncredibleHulk andGodzilla,itisnotpossibleforamulticellularanimaltobecome mutated"asawhole.Inmostcases,aparticleofionizingradiation willnotevenhittheDNA,andevenifitdoes,itwillonlyaect theDNAofasinglecell,noteverycellintheanimal'sbody.Typically,thatcellissimplykilled,becausetheDNAbecomesunable tofunctionproperly.Onceinawhile,however,theDNAmaybe alteredsoastomakethatcellcancerous.Forinstance,skincancer canbecausedbyUVlighthittingasingleskincellinthebodyof asunbather.Ifthatcellbecomescancerousandbeginsreproducing uncontrollably,shewillendupwithatumortwentyyearslater. Otherthancancer,theonlyotherdramaticeectthatcanresult fromalteringasinglecell'sDNAisifthatcellhappenstobea spermorovum,whichcanresultinnonviableormutatedospring. Menarerelativelyimmunetoreproductiveharmfromradiation, becausetheirspermcellsarereplacedfrequently.Womenaremore vulnerablebecausetheykeepthesamesetofovaaslongasthey live. Awhole-bodyexposureof500,000mremwillkillapersonwithin aweekorso.Luckily,onlyasmallnumberofhumanshaveeverbeen exposedtosuchlevels:onescientistworkingontheManhattan Project,somevictimsoftheNagasakiandHiroshimaexplosions, and31workersatChernobyl.Deathoccursbymassivekillingof cells,especiallyintheblood-producingcellsofthebonemarrow. 68 Chapter2TheNucleus

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x / WildPrzewalski'shorses prosperintheChernobylarea. y / Fossilfuelshavedoneincomparablymoredamagetothe environmentthannuclearpower everhas.Polarbears'habitatis rapidlybeingdestroyedbyglobal warming. Lowerlevels,ontheorderof100,000mrem,wereinictedon somepeopleatNagasakiandHiroshima.Noacutesymptomsresult fromthislevelofexposure,butcertaintypesofcanceraresignicantlymorecommonamongthesepeople.Itwasoriginallyexpected thattheradiationwouldcausemanymutationsresultinginbirth defects,butveryfewsuchinheritedeectshavebeenobserved. Agreatdealoftimehasbeenspentdebatingtheeectsofvery lowlevelsofionizingradiation.Amedicalx-ray,forinstance,may resultinadoseontheorderofa100mremabovebackground,i.e., lessthanadoublingofthenormalbackgroundlevel.Similardoses inexcessoftheaveragebackgroundlevelmaybereceivedbypeople livingathighaltitudesorpeoplewithhighconcentrationsofradon gasintheirhouses.Unfortunatelyorfortunately,dependingonhow youlookatit,theaddedrisksofcancerorbirthdefectsresulting fromtheselevelsofexposureareextremelysmall,andtherefore nearlyimpossibletomeasure.Aswithmanysuspectedcarcinogenic chemicals,theonlypracticalmethodofestimatingrisksistogive laboratoryanimalsdosesmanyordersofmagnitudegreater,and thenassumethatthehealthriskisdirectlyproportionaltothedose. Undertheseassumptions,theaddedriskposedbyadentalx-rayor radoninone'sbasementisnegligibleonapersonallevel,andisonly signicantintermsofaslightincreaseincancerthroughoutthe population.Asamatterofsocialpolicy,excessradiationexposure isnotasignicantpublichealthproblemcomparedtocaraccidents ortobaccosmoking. Inthelatetwentiethcentury,antinuclearactivistslargelysucceededinbringingconstructionofnewnuclearpowerplantstoa haltintheU.S.Ironically,wenowknowthattheburningoffossil fuels,whichleadstoglobalwarming,isafarmoregravethreatto theenvironmentthaneventheChernobyldisaster.Ateamofbiologistswrites:"DuringrecentvisitstoChernobyl,weexperienced numeroussightingsofmooseAlcesalces,roedeerCapreolcapreolus,RussianwildboarSusscrofa,foxesVulpesvulpes,river otterLutracanadensis,andrabbitsLepuseuropaeus...Diversityofowersandotherplantsinthehighlyradioactiveregionsis impressiveandequalsthatobservedinprotectedhabitatsoutside thezone...Theobservationthattypicalhumanactivityindustrialization,farming,cattleraising,collectionofrewood,hunting,etc. ismoredevastatingtobiodiversityandabundanceoflocaloraand faunathanistheworstnuclearpowerplantdisastervalidatesthe negativeimpacttheexponentialgrowthofhumanpopulationshas onwildlife." 3 Nuclearpoweristheonlysourceofenergythatis 3 BakerandChesser,Env.ToxicologyandChem.192000.Similar eectshavebeenseenattheBikiniAtoll,thesiteofa1954hydrogenbombtest. Althoughsomespecieshavedisappearedfromthearea,thecoralreefisinmany wayshealthierthansimilarreefselsewhere,becausehumanshavetendedtostay awayforfearofradiationRichardsetal.,MarinePollutionBulletin56 503. Section2.9Biologicaleffectsofionizingradiation 69

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z / TheCrabNebulaisaremnant ofasupernovaexplosion.Almostalltheelementsourplanet ismadeoforiginatedinsuch explosions. sucienttoreplaceanysignicantpercentageofenergyfromfossil fuelsontherapidscheduledemandedbythespeedatwhichglobal warmingisprogressing.Peopleworriedaboutthedownsideofnuclearenergymightbebetteroputtingtheirenergyintoissues relatedtonuclearweapons:thepoorstewardshipoftheformerSovietUnion'swarheads;nuclearproliferationinunstablestatessuch asPakistan;andthepoorsafetyandenvironmentalhistoryofthe superpowers'nuclearweaponsprograms,includingthelossofseveralwarheadsinplanecrashes,andtheenvironmentaldisasterat theHanford,Washingtonweaponsplant. DiscussionQuestions A Shouldthequalityfactorforneutrinosbeverysmall,becausethey mostlydon'tinteractwithyourbody? B Wouldanalphasourcebelikelytocausedifferenttypesofcancer dependingonwhetherthesourcewasexternaltothebodyorswallowed incontaminatedfood?Whataboutagammasource? 2.10 ? Thecreationoftheelements CreationofhydrogenandheliumintheBigbang Didallthechemicalelementswe'remadeofcomeintobeingin thebigbang? 4 Temperaturesintherstmicrosecondsafterthebig bangweresohighthatatomsandnucleicouldnotholdtogether atall.Afterthingshadcooleddownenoughfornucleiandatoms toexist,therewasaperiodofaboutthreeminutesduringwhich thetemperatureanddensitywerehighenoughforfusiontooccur, butnotsohighthatatomscouldholdtogether.Wehaveagood, detailedunderstandingofthelawsofphysicsthatapplyunderthese conditions,sotheoristsareabletosaywithcondencethatthe onlyelementheavierthanhydrogenthatwascreatedinsignicant quantitieswashelium. Wearestardust Inthatcase,wheredidalltheotherelementscomefrom?Astronomerscameupwiththeanswer.Bystudyingthecombinations ofwavelengthsoflight,calledspectra,emittedbyvariousstars, theyhadbeenabletodeterminewhatkindsofatomstheycontained.Wewillhavemoretosayaboutspectraattheendofthis book.Theyfoundthatthestarsfellintotwogroups.Onetype wasnearly100%hydrogenandhelium,whiletheothercontained 99%hydrogenandheliumand1%otherelements.Theyinterpreted theseastwogenerationsofstars.Therstgenerationhadformed outofcloudsofgasthatcamefreshfromthebigbang,andtheir compositionreectedthatoftheearlyuniverse.Thenuclearfusion reactionsbywhichtheyshinehavemainlyjustincreasedthepro4 Theevidenceforthebigbangtheoryoftheoriginoftheuniversewasdiscussedinbook3ofthisseries. 70 Chapter2TheNucleus

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aa / ConstructionoftheUNILACacceleratorinGermany,one ofwhoseusesisforexperiments tocreateveryheavyarticial elements.Insuchanexperiment, fusionproductsrecoilthrougha devicecalledSHIPnotshown thatseparatesthembasedon theircharge-to-massratios itisessentiallyjustascaled-up versionofThomson'sapparatus.Atypicalexperimentruns forseveralmonths,andoutof thebillionsoffusionreactions inducedduringthistime,only oneortwomayresultinthe productionofsuperheavyatoms. Inalltherest,thefusednucleus breaksupimmediately.SHIP isusedtoidentifythesmall numberofgoodreactionsand separatethemfromthisintense background. portionofheliumrelativetohydrogen,withoutmakinganyheavier elements. Themembersoftherstgenerationthatweseetoday,however, areonlythosethatlivedalongtime.Smallstarsaremoremiserly withtheirfuelthanlargestars,whichhaveshortlives.Thelarge starsoftherstgenerationhavealreadynishedtheirlives.Near theendofitslifetime,astarrunsoutofhydrogenfuelandundergoes aseriesofviolentandspectacularreorganizationsasitfusesheavier andheavierelements.Verylargestarsnishthissequenceofevents byundergoingsupernovaexplosions,inwhichsomeoftheirmaterial isungointothevoidwhiletherestcollapsesintoanexoticobject suchasablackholeorneutronstar. Thesecondgenerationofstars,ofwhichourownsunisanexample,condensedoutofcloudsofgasthathadbeenenrichedinheavy elementsduetosupernovaexplosions.Itisthoseheavyelements thatmakeupourplanetandourbodies. Articialsynthesisofheavyelements Elementsuptouranium,atomicnumber92,werecreatedby theseastronomicalprocesses.Beyondthat,theincreasingelectrical repulsionoftheprotonsleadstoshorterandshorterhalf-lives.Even ifasupernovaabillionyearsagodidcreatesomequantityofan elementsuchasBerkelium,number97,therewouldbenoneleftin theEarth'scrusttoday.Theheaviestelementshaveallbeencreated byarticialfusionreactionsinaccelerators.Asof2006,theheaviest elementthathasbeencreatedis116. 5 Althoughthecreationofanewelement,i.e.,anatomwitha novelnumberofprotons,hashistoricallybeenconsideredaglamorousaccomplishment,tothenuclearphysicistthecreationofan atomwithahithertounobservednumberofneutronsisequallyimportant.Thegreatestneutronnumberreachedsofaris179.One tantalizinggoalofthistypeofresearchisthetheoreticalprediction thattheremightbeanislandofstabilitybeyondthepreviouslyexploredtipofthechartofthenucleishowninsection2.8.Justas certainnumbersofelectronsleadtothechemicalstabilityofthenoblegaseshelium,neon,argon,...,certainnumbersofneutronsand protonsleadtoaparticularlystablepackingoforbits.Calculations datingbacktothe1960'shavehintedthattheremightberelatively stablenucleihavingapproximately114protonsand184neutrons. Theisotopesofelements114and116thathavebeenproducedso farhavehadhalf-livesinthesecondormillosecondrange.This maynotseemlikeverylong,butlifetimesinthemicrosecondrange aremoretypicalforthesuperheavyelementsthathavepreviously beendiscovered.Thereisevenspeculationthatcertainsuperheavy 5 Anearlierclaimofthecreationofelement116byagroupatBerkeleyturned outtobeacaseofscienticfraud,buttheelementwaslaterproducedbya dierentgroup,atDubna,Russia. Section2.10 ? Thecreationoftheelements 71

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isotopeswouldbestableenoughtobeproducedinquantitiesthat couldforinstancebeweighedandusedinchemicalreactions. 72 Chapter2TheNucleus

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Summary SelectedVocabulary alphaparticle..aformofradioactivityconsistingofheliumnuclei betaparticle...aformofradioactivityconsistingofelectrons gammaray....aformofradioactivityconsistingofavery high-frequencyformoflight proton......apositivelychargedparticle,oneofthetypes thatnucleiaremadeof neutron......anunchargedparticle,theothertypesthatnucleiaremadeof isotope......oneofthepossiblevarietiesofatomsofagiven element,havingacertainnumberofneutrons atomicnumber.thenumberofprotonsinanatom'snucleus; determineswhatelementitis atomicmass...themassofanatom massnumber..thenumberofprotonsplusthenumberofneutronsinanucleus;approximatelyproportional toitsatomicmass strongnuclear force........ theforcethatholdsnucleitogetheragainst electricalrepulsion weaknuclear force........ theforceresponsibleforbetadecay betadecay....theradioactivedecayofanucleusviathereactionn p+e )]TJ/F15 10.9091 Tf 10.34 -3.959 Td [(+ orp n+e + + ; socalledbecauseanelectronorantielectronis alsoknownasabetaparticle alphadecay...theradioactivedecayofanucleusviaemission ofanalphaparticle ssion.......theradioactivedecayofanucleusbysplitting intotwoparts fusion.......anuclearreactioninwhichtwonucleistick togethertoformonebiggernucleus millirem.....aunitformeasuringaperson'sexposureto radioactivity Notation e )]TJ/F15 10.9091 Tf 12.727 -3.959 Td [(.........anelectron e + .........anantielectron;justlikeanelectron,butwith positivecharge n..........aneutron p..........aproton ..........aneutrino ..........anantineutrino Summary 73

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OtherTerminologyandNotation Z .........atomicnumbernumberofprotonsinanucleus N .........numberofneutronsinanucleus A .........massnumber N + Z Summary RutherfordandMarsdenobservedthatsomealphaparticles fromabeamstrikingathingoldfoilcamebackatanglesupto 180degrees.Thiscouldnotbeexplainedinthethen-favoredraisin -cookiemodeloftheatom,andledtotheadoptionoftheplanetary modeloftheatom,inwhichtheelectronsorbitatiny,positivelychargednucleus.Furtherexperimentsshowedthatthenucleusitself wasaclusterofpositively-chargedprotonsandunchargedneutrons. Radioactivenucleiarethosethatcanreleaseenergy.Themost commontypesofradioactivityarealphadecaytheemissionofa heliumnucleus,betadecaythetransformationofaneutroninto aprotonorvice-versa,andgammadecaytheemissionofatype ofvery-high-frequencylight.Starsarepoweredbynuclearfusion reactions,inwhichtwolightnucleicollideandformabiggernucleus, withthereleaseofenergy. Humanexposuretoionizingradiationismeasuredinunitsof millirem.Thetypicalpersonisexposedtoabout200mremworth ofnaturalbackgroundradiationperyear. ExploringFurther TheFirstThreeMinutes ,StevenWeinberg.Thisbookdescribestherstthreeminutesoftheuniverse'sexistence. 74 Chapter2TheNucleus

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Problem1. Problem2. Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Aheliumatomndsitselfmomentarilyinthisarrangement. Findthedirectionandmagnitudeoftheforceactingontherighthandelectron.Thetwoprotonsinthenucleusaresoclosetogether 1fmthatyoucanconsiderthemasbeingrightontopofeach other. p 2 Theheliumatomofproblem1hassomenewexperiences,goes throughsomelifechanges,andlateronndsitselfinthecongurationshownhere.Whatarethedirectionandmagnitudeoftheforce actingonthebottomelectron?Drawasketchtomakeclearthe denitionyouareusingfortheanglethatgivesdirection. p 3 Supposeyouareholdingyourhandsinfrontofyou,10cm apart. aEstimatethetotalnumberofelectronsineachhand. p bEstimatethetotalrepulsiveforceofalltheelectronsinonehand onalltheelectronsintheother. p cWhydon'tyoufeelyourhandsrepellingeachother? dEstimatehowmuchthechargeofaprotoncoulddierinmagnitudefromthechargeofanelectronwithoutcreatinganoticeable forcebetweenyourhands. 4 Supposethataprotoninaleadnucleuswandersouttothe surfaceofthenucleus,andexperiencesastrongnuclearforceof about8kNfromthenearbyneutronsandprotonspullingitback in.Comparethisnumericallytotherepulsiveelectricalforcefrom theotherprotons,andverifythatthenetforceisattractive.Alead nucleusisverynearlyspherical,andisabout6.5fminradius. p 5 Thesubatomicparticlescalledmuonsbehaveexactlylikeelectrons,exceptthatamuon'smassisgreaterbyafactorof206.77. MuonsarecontinuallybombardingtheEarthaspartofthestream ofparticlesfromspaceknownascosmicrays.Whenamuonstrikes anatom,itcandisplaceoneofitselectrons.Iftheatomhappens tobeahydrogenatom,thenthemuontakesupanorbitthatison theaverage206.77timesclosertotheprotonthantheorbitofthe ejectedelectron.Howmanytimesgreateristheelectricforceexperiencedbythemuonthanthatpreviouslyfeltbytheelectron? p 6 Thenuclearprocessofbetadecaybyelectroncaptureisdescribedparentheticallyinsection2.6.Thereactionisp+e )]TJ/F23 10.9091 Tf 10.115 -3.959 Td [(! n+ aShowthatchargeisconservedinthisreaction. bConversionbetweenenergyandmassisdiscussedintheoptional topiconpage65.Basedontheseideas,explainwhyelectroncapturedoesn'toccurinhydrogenatoms.Ifitdid,matterwouldn't Problems 75

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exist! Solution,p.204 7 234 Pudecayseitherbyelectrondecayorbyalphadecay.A given 234 Punucleusmaydoeitherone;it'srandom.Whatarethe isotopescreatedasproductsofthesetwomodesofdecay? 76 Chapter2TheNucleus

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Chapter3 Circuits,Part1 Madam,whatgoodisababy? MichaelFaraday,whenaskedby QueenVictoriawhattheelectricaldevicesinhislabweregoodfor Afewyearsago,mywifeandIboughtahousewithCharacter, Characterbeingasurvivalmechanismthathouseshaveevolvedin ordertoconvincehumanstoagreetomuchlargermortgagepaymentsthanthey'doriginallyenvisioned.Anyway,oneofthefeaturesthatgivesourhouseCharacteristhatitpossesses,builtinto thewallofthefamilyroom,asetofthreepachinkomachines.These areJapanesegamblingdevicessortoflikeverticalpinballmachines. Thelegalpaperswegotfromthesellershastenedtotellusthat theywereforamusementpurposesonly."Unfortunately,onlyone ofthethreemachineswasworkingwhenwemovedin,anditsoon diedonus.Havingbecomeapachinkoaddict,Idecidedtoxit, butthatwaseasiersaidthandone.TheinsideisaveritableRube Goldbergmechanismoflevers,hooks,springs,andchutes.Myhormonalpride,combinedwithmyPh.D.inphysics,mademecertain ofsuccess,andrenderedmyeventualutterfailureallthemoredemoralizing. Contemplatingmydefeat,IrealizedhowfewcomplexmechanicaldevicesIusedfromdaytoday.Apartfromourcarsandmy 77

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a / Gymnotuscarapo ,aknifesh, useselectricalsignalstosense itsenvironmentandtocommunicatewithothersofitsspecies. saxophone,everytechnologicaltoolinourmodernlife-supportsystemwaselectronicratherthanmechanical. 3.1Current Unityofalltypesofelectricity Wearesurroundedbythingswehavebeen told areelectrical," butit'sfarfromobviouswhattheyhaveincommontojustifybeing groupedtogether.Whatrelationshipistherebetweenthewaysocks clingtogetherandthewayabatterylightsalightbulb?Wehave beentoldthatbothanelectriceelandourownbrainsaresomehow electricalinnature,butwhatdotheyhaveincommon? BritishphysicistMichaelFaraday-1867setouttoaddress thisproblem.Heinvestigatedelectricityfromavarietyofsources |includingelectriceels!|toseewhethertheycouldallproduce thesameeects,suchasshocksandsparks,attractionandrepulsion.Heating"refers,forexample,tothewayalightbulblament getshotenoughtoglowandemitlight.Magneticinductionisan eectdiscoveredbyFaradayhimselfthatconnectselectricityand magnetism.Wewillnotstudythiseect,whichisthebasisforthe electricgenerator,indetailuntillaterinthebook. s ource e ect attractionand shockssparksrepulsionheating rubbing pppp battery pppp animal pp p p magnetically induced pppp ThetableshowsasummaryofsomeofFaraday'sresults.Check marksindicatethatFaradayorhisclosecontemporarieswereableto verifythataparticularsourceofelectricitywascapableofproducing acertaineect.Theyevidentlyfailedtodemonstrateattraction andrepulsionbetweenobjectschargedbyelectriceels,although modernworkershavestudiedthesespeciesindetailandbeenable tounderstandalltheirelectricalcharacteristicsonthesamefooting asotherformsofelectricity. Faraday'sresultsindicatethatthereisnothingfundamentally dierentaboutthetypesofelectricitysuppliedbythevarioussources. Theyareallabletoproduceawidevarietyofidenticaleects.Wrote Faraday,Thegeneralconclusionwhichmustbedrawnfromthis collectionoffactsisthatelectricity,whatevermaybeitssource,is identicalinitsnature." 78 Chapter3Circuits,Part1

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b / Andr eMarieAmp ere1836. Ifthetypesofelectricityarethesamething,whatthingisthat? Theanswerisprovidedbythefactthatallthesourcesofelectricity cancauseobjectstorepelorattracteachother.Weusetheword charge"todescribethepropertyofanobjectthatallowsitto participateinsuchelectricalforces,andwehavelearnedthatcharge ispresentinmatterintheformofnucleiandelectrons.Evidently alltheseelectricalphenomenaboildowntothemotionofcharged particlesinmatter. Electriccurrent Ifthefundamentalphenomenonisthemotionofchargedparticles,thenhowcanwedeneausefulnumericalmeasurementofit? Wemightdescribetheowofariversimplybythevelocityofthe water,butvelocitywillnotbeappropriateforelectricalpurposes becauseweneedtotakeintoaccounthowmuchchargethemoving particleshave,andinanycasetherearenopracticaldevicessold atRadioShackthatcantellusthevelocityofchargedparticles. Experimentsshowthattheintensityofvariouselectricaleectsis relatedtoadierentquantity:thenumberofcoulombsofcharge thatpassbyacertainpointpersecond.Byanalogywiththeow ofwater,thisquantityiscalledtheelectric current I .Itsunits ofcoulombs/secondaremoreconvenientlyabbreviatedasamperes, 1A=1C/s.Ininformalspeech,oneusuallysaysamps." Themainsubtletyinvolvedinthisdenitionishowtoaccount forthetwotypesofcharge.Thestreamofwatercomingfroma hoseismadeofatomscontainingchargedparticles,butitproduces noneoftheeectsweassociatewithelectriccurrents.Forexample, youdonotgetanelectricalshockwhenyouaresprayedbyahose. Thistypeofexperimentshowsthattheeectcreatedbythemotion ofonetypeofchargedparticlecanbecanceledoutbythemotionof theoppositetypeofchargeinthesamedirection.Inwater,every oxygenatomwithachargeof+8 e issurroundedbyeightelectrons withchargesof )]TJ/F20 10.9091 Tf 8.485 0 Td [(e ,andlikewiseforthehydrogenatoms. Wethereforereneourdenitionofcurrentasfollows: denitionofelectriccurrent Whenchargedparticlesareexchangedbetweenregionsofspace AandB,theelectriccurrentowingfromAtoBis I = q t where q isthechangeinregionB'stotalchargeoccurring overaperiodoftime t Inthegardenhoseexample,yourbodypicksupequalamountsof positiveandnegativecharge,resultinginnochangeinyourtotal charge,sotheelectricalcurrentowingintoyouiszero. Section3.1Current 79

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Interpretationof q= t example1 Howshouldtheexpression q = t beinterpretedwhenthecurrentisn'tconstant? You'veseenlotsofequationsofthisformbefore: v = x = t F = p = t ,etc.Thesearealldescriptionsofratesofchange, andtheyallrequirethattherateofchangebeconstant.Ifthe rateofchangeisn'tconstant,youinsteadhavetousetheslope ofthetangentlineonagraph.Theslopeofatangentlineis equivalenttoaderivativeincalculus;applicationsofcalculusare discussedinsection3.6. Ionsmovingacrossacellmembraneexample2 Figurecshowsions,labeledwiththeircharges,movinginor outthroughthemembranesofthreecells.Iftheionsallcross themembranesduringthesameintervaloftime,howwouldthe currentsintothecellscomparewitheachother? CellAhaspositivecurrentgoingintoitbecauseitschargeis increased,i.e.,hasapositivevalueof q CellBhasthesamecurrentascellA,becausebylosingoneunit ofnegativechargeitalsoendsupincreasingitsowntotalcharge byoneunit. CellC'stotalchargeisreducedbythreeunits,soithasalarge negativecurrentgoingintoit. CellDlosesoneunitofcharge,soithasasmallnegativecurrent intoit. c / Example2 Itmayseemstrangetosaythatanegativelychargedparticle goingonewaycreatesacurrentgoingtheotherway,butthisis quiteordinary.Aswewillsee,currentsowthroughmetalwires viathemotionofelectrons,whicharenegativelycharged,sothe directionofmotionoftheelectronsinacircuitisalwaysoppositeto thedirectionofthecurrent.Ofcourseitwouldhavebeenconvenient ofBenjaminFranklinhaddenedthepositiveandnegativesignsof chargetheoppositeway,sincesomanyelectricaldevicesarebased onmetalwires. 80 Chapter3Circuits,Part1

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d / 1.Staticelectricityruns outquickly.2.Apracticalcircuit. 3.Anopencircuit.4.Howan ammeterworks.5.Measuring thecurrentwithanammeter. Numberofelectronsowingthroughalightbulbexample3 Ifalightbulbhas1.0Aowingthroughit,howmanyelectrons willpassthroughthelamentin1.0s? Weareonlycalculatingthenumberofelectronsthatow,sowe canignorethepositiveandnegativesigns.Solvingfor q = I t givesachargeof1.0Cowinginthistimeinterval.Thenumber ofelectronsis numberofelectrons=coulombs electrons coulomb =coulombs = coulombs electron =1.0C = e =6.2 10 18 3.2Circuits Howcanweputelectriccurrentstowork?Theonlymethodof controllingelectricchargewehavestudiedsofaristochargedierentsubstances,e.g.,rubberandfur,byrubbingthemagainsteach other.Figured/1showsanattempttousethistechniquetolight alightbulb.Thismethodisunsatisfactory.True,currentwillow throughthebulb,sinceelectronscanmovethroughmetalwires,and theexcesselectronsontherubberrodwillthereforecomethrough thewiresandbulbduetotheattractionofthepositivelycharged furandtherepulsionoftheotherelectrons.Theproblemisthat afterazillionthofasecondofcurrent,therodandfurwillbothhave runoutofcharge.Nomorecurrentwillow,andthelightbulbwill goout. Figured/2showsasetupthatworks.Thebatterypushescharge throughthecircuit,andrecyclesitoverandoveragain.Wewill havemoretosaylaterinthischapterabouthowbatterieswork. Thisiscalleda completecircuit .Today,theelectricaluseofthe wordcircuit"istheonlyonethatspringstomindformostpeople, buttheoriginalmeaningwastotravelaroundandmakearound trip,aswhenacircuitcourtjudgewouldridearoundtheboondocks, dispensingjusticeineachtownonacertaindate. Notethatanexampleliked/3doesnotwork.Thewirewill quicklybeginacquiringanetcharge,becauseithasnowaytoget ridofthechargeowingintoit.Therepulsionofthischargewill makeitmoreandmorediculttosendanymorechargein,and soontheelectricalforcesexertedbythebatterywillbecanceled outcompletely.Thewholeprocesswouldbeoversoquicklythat thelamentwouldnotevenhaveenoughtimetogethotandglow. Thisisknownasan opencircuit .Exactlythesamethingwould happenifthecompletecircuitofgured/2wascutsomewherewith apairofscissors,andinfactthatisessentiallyhowanordinary Section3.2Circuits 81

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lightswitchworks:byopeningupagapinthecircuit. Thedenitionofelectriccurrentwehavedevelopedhasthegreat virtuethatitiseasytomeasure.Inpracticalelectricalwork,one almostalwaysmeasurescurrent,notcharge.Theinstrumentusedto measurecurrentiscalledan ammeter .Asimpliedammeter,d/4, simplyconsistsofacoiled-wiremagnetwhoseforcetwistsaniron needleagainsttheresistanceofaspring.Thegreaterthecurrent, thegreatertheforce.Althoughtheconstructionofammetersmay dier,theiruseisalwaysthesame.Webreakintothepathofthe electriccurrentandinterposethemeterlikeatollboothonaroad, d/5.Thereisstillacompletecircuit,andasfarasthebatteryand bulbareconcerned,theammeterisjustanothersegmentofwire. Doesitmatterwhereinthecircuitweplacetheammeter?Could we,forinstance,haveputitintheleftsideofthecircuitinstead oftheright?Conservationofchargetellsusthatthiscanmakeno dierence.Chargeisnotdestroyedorusedup"bythelightbulb, sowewillgetthesamecurrentreadingoneithersideofit.Whatis usedup"isenergystoredinthebattery,whichisbeingconverted intoheatandlightenergy. 3.3Voltage Thevoltunit Electricalcircuitscanbeusedforsendingsignals,storinginformation,ordoingcalculations,buttheirmostcommonpurposeby faristomanipulateenergy,asinthebattery-and-bulbexampleof theprevioussection.Weknowthatlightbulbsareratedinunitsof watts,i.e.,howmanyjoulespersecondofenergytheycanconvert intoheatandlight,buthowwouldthisrelatetotheowofchargeas measuredinamperes?Bywayofanalogy,supposeyourfriend,who didn'ttakephysics,can'tndanyjobbetterthanpitchingbalesof hay.Thenumberofcaloriesheburnsperhourwillcertainlydepend onhowmanybaleshepitchesperminute,butitwillalsobeproportionaltohowmuchmechanicalworkhehastodooneachbale. Ifhisjobistotossthemupintoahayloft,hewillgottiredalot morequicklythansomeonewhomerelytipsbalesoaloadingdock intotrucks.Inmetricunits, joules second = haybales second joules haybale Similarly,therateofenergytransformationbyabatterywillnot justdependonhowmanycoulombsperseconditpushesthrougha circuitbutalsoonhowmuchmechanicalworkithastodooneach coulombofcharge: joules second = coulombs second joules coulomb 82 Chapter3Circuits,Part1

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e / AlessandroVolta-1827. or power=current workperunitcharge. Unitsofjoulespercoulombareabbreviatedas volts ,1V=1J/C, namedaftertheItalianphysicistAlessandroVolta.Everyoneknows thatbatteriesareratedinunitsofvolts,butthevoltageconceptis moregeneralthanthat;itturnsoutthatvoltageisapropertyof everypointinspace.Togainmoreinsight,let'sthinkmorecarefully aboutwhatgoesoninthebatteryandbulbcircuit. Thevoltageconceptingeneral Todoworkonachargedparticle,thebatteryapparentlymustbe exertingforcesonit.Howdoesitdothis?Well,theonlythingthat canexertanelectricalforceonachargedparticleisanothercharged particle.It'sasthoughthehaybaleswerepushingandpullingeach otherintothehayloft!Thisispotentiallyahorriblycomplicated situation.Evenifweknewhowmuchexcesspositiveornegative chargetherewasateverypointinthecircuitwhichrealisticallywe don'twewouldhavetocalculatezillionsofforcesusingCoulomb's law,performallthevectoradditions,andnallycalculatehowmuch workwasbeingdoneonthechargesastheymovedalong.Tomake thingsevenmorescary,thereismorethanonetypeofcharged particlethatmoves:electronsarewhatmoveinthewiresandthe bulb'slament,butionsarethemovingchargecarriersinsidethe battery.Luckily,therearetwowaysinwhichwecansimplifythings: Thesituationisunchanging. Unliketheimaginarysetup inwhichweattemptedtolightabulbusingarubberrodanda pieceoffur,thiscircuitmaintainsitselfinasteadystateafter perhapsamicrosecond-longperiodofsettlingdownafterthe circuitisrstassembled.Thecurrentissteady,andascharge owsoutofanyareaofthecircuititisreplacedbythesame amountofchargeowingin.Theamountofexcesspositive ornegativechargeinanypartofthecircuitthereforestays constant.Similarly,whenwewatchariverowing,thewater goesbybuttheriverdoesn'tdisappear. Forcedependsonlyonposition. Sincethechargedistributionisnotchanging,thetotalelectricalforceonacharged particledependsonlyonitsownchargeandonitslocation. Ifanotherchargedparticleofthesametypevisitsthesame locationlateron,itwillfeelexactlythesameforce. Thesecondobservationtellsusthatthereisnothingallthat dierentabouttheexperienceofonechargedparticleascompared toanother's.Ifwesingleoutoneparticletopayattentionto,and gureouttheamountofworkdoneonitbyelectricalforcesasit goesfrompointAtopoint B alongacertainpath,thenthisis thesameamountofworkthatwillbedoneonanyothercharged particlesofthesametypeasitfollowsthesamepath.Forthesakeof Section3.3Voltage 83

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f / Example4. visualization,let'sthinkaboutthepaththatstartsatoneterminal ofthebattery,goesthroughthelightbulb'slament,andendsat theotherterminal.Whenanobjectexperiencesaforcethatdepends onlyonitspositionandwhencertainother,technicalconditions aresatised,wecandeneanelectricalenergyassociatedwiththe positionofthatobject.Theamountofworkdoneontheparticle byelectricalforcesasitmovesfromAtoBequalsthedropin electricalenergybetweenAandB.Thiselectricalenergyiswhatis beingconvertedintootherformsofenergysuchasheatandlight. Wethereforedenevoltageingeneralaselectricalenergyperunit charge: denitionofvoltagedierence Thedierenceinvoltagebetweentwopointsinspaceisdenedas V = U elec =q where U elec isthechangeintheelectricalenergyofaparticle withcharge q asitmovesfromtheinitialpointtothenal point. Theamountofpowerdissipatedi.e.,rateatwhichenergy istransformedbytheowofelectricityisthengivenbythe equation P = I V Energystoredinabatteryexample4 The1.2Vrechargeablebatteryingurefislabeled1800milliamphours.Whatisthemaximumamountofenergythebatterycan store? Anampere-hourisaunitofcurrentmultipliedbyaunitoftime. Currentischargeperunittime,soanampere-hourisinfacta funnyunitof charge : Ahour=C/ss =3600C 1800milliamp-hoursistherefore1800 10 )]TJ/F39 7.9701 Tf 6.586 0 Td [(3 3600C=6.5 10 3 C.That'sahugenumberofchargedparticles,butthetotal lossofelectricalenergywilljustbetheirtotalchargemultipliedby thevoltagedifferenceacrosswhichtheymove: U elec = q V =.5 10 3 C.2V =7.8kJ Unitsofvolt-ampsexample5 Doorbellsareoftenratedinvolt-amps.Whatdoesthiscombinationofunitsmean? 84 Chapter3Circuits,Part1

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. Currenttimesvoltagegivesunitsofpower, P = I V ,sovoltampsarereallyjustanonstandardwayofwritingwatts.Theyare tellingyouhowmuchpowerthedoorbellrequires. Powerdissipatedbyabatteryandbulbexample6 Ifa9.0-voltbatterycauses1.0Atoowthroughalightbulb,how muchpowerisdissipated? Thevoltageratingofabatterytellsuswhatvoltagedifference V itisdesignedtomaintainbetweenitsterminals. P = I V =9.0A V =9.0 C s J C =9.0J/s =9.0W Theonlynontrivialthinginthisproblemwasdealingwiththeunits. Onequicklygetsusedtotranslatingcommoncombinationslike A Vintosimplerterms. Hereareafewquestionsandanswersaboutthevoltageconcept. Question: OK,sowhat is voltage,really? Answer: Adevicelikeabatteryhaspositiveandnegativecharges insideitthatpushotherchargesaroundtheoutsidecircuit.A higher-voltagebatteryhasdenserchargesinit,whichwilldomore workoneachchargedparticlethatmovesthroughtheoutsidecircuit. Touseagravitationalanalogy,wecanputapaddlewheelatthe bottomofeitheratallwaterfallorashortone,butakgofwater thatfallsthroughthegreatergravitationalenergydierencewill havemoreenergytogiveuptothepaddlewheelatthebottom. Question: Whydowedenevoltageaselectricalenergydividedby charge,insteadofjustdeningitaselectricalenergy? Answer: Oneansweristhatit'stheonlydenitionthatmakesthe equation P = I V work.Amoregeneralansweristhatwewant tobeabletodeneavoltagedierencebetweenanytwopoints inspacewithouthavingtoknowinadvancehowmuchchargethe particlesmovingbetweenthemwillhave.Ifyouputanine-volt batteryonyourtongue,thenthechargedparticlesthatmoveacross yourtongueandgiveyouthattinglysensationarenotelectronsbut ions,whichmayhavechargesof+ e )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 e ,orpracticallyanything. Themanufacturerprobablyexpectedthebatterytobeusedmostly incircuitswithmetalwires,wherethechargedparticlesthatowed wouldbeelectronswithchargesof )]TJ/F20 10.9091 Tf 8.485 0 Td [(e .Iftheonesowingacross yourtonguehappentohavechargesof )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 e ,theelectricalenergy dierenceforthemwillbetwiceasmuch,butdividingbytheir Section3.3Voltage 85

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chargeof )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 e inthedenitionofvoltagewillstillgivearesultof9 V Question: Aretheretwoseparaterolesforthechargedparticlesin thecircuit,atypethatsitsstillandexertstheforces,andanother thatmovesundertheinuenceofthoseforces? Answer: No.Everychargedparticlesimultaneouslyplaysboth roles.Newton'sthirdlawsaysthatanyparticlethathasanelectricalforcesactingonitmustalsobeexertinganelectricalforce backontheotherparticle.Therearenodesignatedmovers"or designatedforce-makers." Question: Whydoesthedenitionofvoltageonlyrefertovoltage dierences ? Answer: It'sperfectlyOKtodenevoltageas V = U elec =q .But recallthatitisonly dierences ininteractionenergy, U ,thathave directphysicalmeaninginphysics.Similarly,voltagedierencesare reallymoreusefulthanabsolutevoltages.Avoltmetermeasures voltagedierences,notabsolutevoltages. DiscussionQuestions A Arollercoasterissortoflikeanelectriccircuit,butitusesgravitationalforcesonthecarsinsteadofelectricones.Whatwouldahighvoltagerollercoasterbelike?Whatwouldahigh-currentrollercoasterbe like? B Criticizethefollowingstatements: Hetouchedthewire,and10000voltswentthroughhim. Thatbatteryhasachargeof9volts. Youusedupthechargeofthebattery. C Whenyoutoucha9-voltbatterytoyourtongue,bothpositiveand negativeionsmovethroughyoursaliva.Whichionsgowhichway? D Ioncetouchedapieceofphysicsapparatusthathadbeenwired incorrectly,andgotaseveral-thousand-voltvoltagedifferenceacrossmy hand.Iwasnotinjured.Forwhatpossiblereasonwouldtheshockhave hadinsufcientpowertohurtme? 86 Chapter3Circuits,Part1

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g / GeorgSimonOhm1854. 3.4Resistance Resistance Sofarwehavesimplypresenteditasanobservedfactthata battery-and-bulbcircuitquicklysettlesdowntoasteadyow,but whyshouldit?Newton'ssecondlaw, a = F=m ,wouldseemto predictthatthesteadyforcesonthechargedparticlesshouldmake themwhiparoundthecircuitfasterandfaster.Theansweristhatas chargedparticlesmovethroughmatter,therearealwaysforces,analogoustofrictionalforces,thatresistthemotion.Theseforcesneed tobeincludedinNewton'ssecondlaw,whichisreally a = F total =m not a = F=m .If,byanalogy,youpushacrateacrosstheoorat constantspeed,i.e.,withzeroacceleration,thetotalforceonitmust bezero.Afteryougetthecrategoing,theoor'sfrictionalforceis exactlycancelingoutyourforce.Thechemicalenergystoredin yourbodyisbeingtransformedintoheatinthecrateandtheoor, andnolongerintoanincreaseinthecrate'skineticenergy.Similarly,thebattery'sinternalchemicalenergyisconvertedintoheat, notintoperpetuallyincreasingthechargedparticles'kineticenergy. Changingenergyintoheatmaybeanuisanceinsomecircuits,such asacomputerchip,butitisvitalinalightbulb,whichmustgethot enoughtoglow.Whetherwelikeitornot,thiskindofheatingeect isgoingtooccuranytimechargedparticlesmovethroughmatter. Whatdeterminestheamountofheating?Oneashlightbulb designedtoworkwitha9-voltbatterymightbelabeled1.0watts, another5.0.Howdoesthiswork?Evenwithoutknowingthedetails ofthistypeoffrictionattheatomiclevel,wecanrelatetheheat dissipationtotheamountofcurrentthatowsviatheequation P = I V.Ifthetwoashlightbulbscanhavetwodierentvalues of P whenusedwithabatterythatmaintainsthesame V ,it mustbethatthe5.0-wattbulballowsvetimesmorecurrentto owthroughit. Formanysubstances,includingthetungstenfromwhichlightbulblamentsaremade,experimentsshowthattheamountofcurrentthatwillowthroughitisdirectlyproportionaltothevoltage dierenceplacedacrossit.Foranobjectmadeofsuchasubstance, wedeneitselectrical resistance asfollows: denitionofresistance Ifanobjectinsertedinacircuitdisplaysacurrentowproportionaltothevoltagedierenceacrossit,thenwedeneits resistanceastheconstantratio R = V=I Theunitsofresistancearevolts/ampere,usuallyabbreviatedas ohms,symbolizedwiththecapitalGreekletteromega,. Section3.4Resistance 87

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h / Fourobjectsmadeofthe samesubstancehavedifferent resistances. Resistanceofalightbulbexample7 Aashlightbulbpoweredbya9-voltbatteryhasaresistanceof 10 .Howmuchcurrentwillitdraw? Solvingthedenitionofresistancefor I ,wend I = V = R =0.9V = =0.9V = V = A =0.9A Ohm'slawstatesthatmanysubstances,includingmanysolids andsomeliquids,displaythiskindofbehavior,atleastforvoltages thatarenottoolarge.ThefactthatOhm'slawiscalledalaw" shouldnotbetakentomeanthatallmaterialsobeyit,orthatithas thesamefundamentalimportanceasNewton'slaws,forexample. Materialsarecalled ohmic or nonohmic ,dependingonwhetherthey obeyOhm'slaw. Ifobjectsofthesamesizeandshapemadefromtwodierent ohmicmaterialshavedierentresistances,wecansaythatonematerialismoreresistivethantheother,orequivalentlythatitisless conductive.Materials,suchasmetals,thatareveryconductiveare saidtobegood conductors .Thosethatareextremelypoorconductors,forexamplewoodorrubber,areclassiedas insulators .There isnosharpdistinctionbetweenthetwoclassesofmaterials.Some, suchassilicon,liemidwaybetweenthetwoextremes,andarecalled semiconductors. Onanintuitivelevel,wecanunderstandtheideaofresistance bymakingthesoundshhhhhh"and."Tomakeairowout ofyourmouth,youuseyourdiaphragmtocompresstheairinyour chest.Thepressuredierencebetweenyourchestandtheairoutside yourmouthisanalogoustoavoltagedierence.Whenyoumakethe h"sound,youformyourmouthandthroatinawaythatallowsair tooweasily.Thelargeowofairislikealargecurrent.Dividing byalargecurrentinthedenitionofresistancemeansthatweget asmallresistance.Wesaythatthesmallresistanceofyourmouth andthroatallowsalargecurrenttoow.Whenyoumakethef" sound,youincreasetheresistanceandcauseasmallercurrentto ow. Notethatalthoughtheresistanceofanobjectdependsonthe substanceitismadeof,wecannotspeaksimplyoftheresistance ofgold"ortheresistanceofwood."Figurehshowsfourexamples ofobjectsthathavehadwiresattachedattheendsaselectrical connections.Iftheyweremadeofthesamesubstance,theywould allneverthelesshavedierentresistancesbecauseoftheirdierent sizesandshapes.Amoredetaileddiscussionwillbemorenatural inthecontextofthefollowingchapter,butitshouldnotbetoo surprisingthattheresistanceofh/2willbegreaterthanthatofh/1 88 Chapter3Circuits,Part1

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|theimageofwaterowingthroughapipe,howeverincorrect, givesustherightintuition.Objecth/3willhaveasmallerresistance thanh/1becausethechargedparticleshavelessofittogetthrough. Superconductors Allmaterialsdisplaysomevariationinresistanceaccordingto temperatureafactthatisusedinthermostatstomakeathermometerthatcanbeeasilyinterfacedtoanelectriccircuit.More spectacularly,mostmetalshavebeenfoundtoexhibitasudden changeto zero resistancewhencooledtoacertaincriticaltemperature.Theyarethensaidtobesuperconductors.Theoretically, superconductorsshouldmakeagreatmanyexcitingdevicespossible,forexamplecoiled-wiremagnetsthatcouldbeusedtolevitate trains.Inpractice,thecriticaltemperaturesofallmetalsarevery low,andtheresultingneedforextremerefrigerationhasmadetheir useuneconomicalexceptforsuchspecializedapplicationsasparticle acceleratorsforphysicsresearch. Butscientistshaverecentlymadethesurprisingdiscoverythat certainceramicsaresuperconductorsatlessextremetemperatures. Thetechnologicalbarrierisnowinndingpracticalmethodsfor makingwireoutofthesebrittlematerials.WallStreetiscurrently investingbillionsofdollarsindevelopingsuperconductingdevices forcellularphonerelaystationsbasedonthesematerials.In2001, thecityofCopenhagenreplacedashortsectionofitselectricalpower trunkswithsuperconducingcables,andtheyarenowinoperation andsupplyingpowertocustomers. Thereiscurrentlynosatisfactorytheoryofsuperconductivityin general,althoughsuperconductivityinmetalsisunderstoodfairly well.UnfortunatelyIhaveyettondafundamentalexplanationof superconductivityinmetalsthatworksattheintroductorylevel. i / AsuperconductingsegmentoftheATLASacceleratoratArgonneNationalLaboratorynearChicago.Itisusedtoacceleratebeamsofions toafewpercentofthespeedoflightfornuclearphysicsresearch.The shinysilver-coloredsurfacesaremadeoftheelementniobium,whichisa superconductoratrelativelyhightemperaturescomparedtoothermetals relativelyhighmeaningthetemperatureofliquidhelium!Thebeam ofionspassesthroughtheholesinthetwosmallcylindersontheends ofthecurvedrods.Chargeisshufedbackandforthbetweenthemat afrequencyof12millioncyclespersecond,sothattheytaketurnsbeingpositiveandnegative.Thepositivelychargedbeamconsistsofshort spurts,eachtimedsothatwhenitisinoneofthesegmentsitwillbe pulledforwardbynegativechargeonthecylinderinfrontofitandpushed forwardbythepositivelychargedonebehind.Thehugecurrentsinvolved seeexample9onpage98wouldquicklymeltanymetalthatwasnot superconducting,butinasuperconductortheyproducenoheatatall. Section3.4Resistance 89

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Constantvoltagethroughoutaconductor Theideaofasuperconductorleadsustothequestionofhow weshouldexpectanobjecttobehaveifitismadeofaverygood conductor.Superconductorsareanextremecase,butoftenametal wirecanbethoughtofasaperfectconductor,forexampleifthe partsofthecircuitotherthanthewirearemadeofmuchlessconductivematerials.Whathappensif R equalszerointheequation R = V=I ?Theresultofdividingtwonumberscanonlybezeroif thenumberontopequalszero.Thistellsusthatifwepickanytwo pointsinaperfectconductor,thevoltagedierencebetweenthem mustbezero.Inotherwords,theentireconductormustbeatthe samevoltage. Constantvoltagemeansthatnoworkwouldbedoneonacharge asitmovedfromonepointintheconductortoanother.Ifzerowork wasdoneonlyalongacertainpathbetweentwospecicpoints,it mightmeanthatpositiveworkwasdonealongpartofthepathand negativeworkalongtherest,resultinginacancellation.Butthereis nowaythattheworkcouldcomeouttobezeroforallpossiblepaths unlesstheelectricalforceonachargewasinfactzeroateverypoint. Suppose,forexample,thatyoubuildupastaticchargebyscung yourfeetonacarpet,andthenyoudepositsomeofthatchargeonto adoorknob,whichisagoodconductor.Howcanallthatchargebe inthedoorknobwithoutcreatinganyelectricalforceatanypoint insideit?Theonlypossibleansweristhatthechargemovesaround untilithasspreaditselfintojusttherightcongurationsothatthe forcesexertedbyallthelittlebitsofexcesssurfacechargeonany chargedparticlewithinthedoorknobexactlycanceledout. Wecanexplainthisbehaviorifweassumethatthechargeplaced onthedoorknobeventuallysettlesdownintoastableequilibrium. Sincethedoorknobisaconductor,thechargeisfreetomovethrough it.Ifitwasfreetomoveandanypartofitdidexperienceanonzero totalforcefromtherestofthecharge,thenitwouldmove,andwe wouldnothaveanequilibrium. Italsoturnsoutthatchargeplacedonaconductor,onceit reachesitsequilibriumconguration,isentirelyonthesurface,not ontheinterior.Wewillnotprovethisfactformally,butitisintuitivelyreasonable.Suppose,forinstance,thatthenetchargeon theconductorisnegative,i.e.,ithasanexcessofelectrons.These electronsallrepeleachother,andthisrepulsionwilltendtopush themontothesurface,sincebeingonthesurfaceallowsthemtobe asfarapartaspossible. 90 Chapter3Circuits,Part1

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k / Thesymbolusedinschematicstorepresentaresistor. j / Short-circuitingabattery. Warning:youcanburnyourself thiswayorstartare!Ifyou wanttotrythis,trymakingthe connectiononlyverybriey,use alow-voltagebattery,andavoid touchingthebatteryorthewire, bothofwhichwillgethot. Shortcircuits Sofarwehavebeenassumingaperfectconductor.Whatifit isagoodconductor,butnotaperfectone?Thenwecansolvefor V = IR .Anordinary-sizedcurrentwillmakeaverysmallresult whenwemultiplyitbytheresistanceofagoodconductorsuchas ametalwire.Thevoltagethroughoutthewirewillthenbenearly constant.If,ontheotherhand,thecurrentisextremelylarge,we canhaveasignicantvoltagedierence.Thisiswhathappensina short-circuit: acircuitinwhichalow-resistancepathwayconnects thetwosidesofavoltagesource.Notethatthisismuchmore specicthanthepopularuseofthetermtoindicateanyelectrical malfunctionatall.If,forexample,youshort-circuita9-voltbattery asshowningurej,youwillproduceperhapsathousandamperes ofcurrent,leadingtoaverylargevalueof P = I V .Thewiregets hot! self-checkA Whatwouldhappentothebatteryinthiskindofshortcircuit? Answer,p.203 Resistors Insideanyelectronicgadgetyouwillseequiteafewlittlecircuit elementsliketheoneshownbelow.These resistors aresimplya cylinderofohmicmaterialwithwiresattachedtotheend. Atthisstage,moststudentshaveahardtimeunderstandingwhy resistorswouldbeusedinsidearadiooracomputer.Weobviously wantalightbulboranelectricstovetohaveacircuitelementthat resiststheowofelectricityandheatsup,butheatingisundesirable inradiosandcomputers.Withoutgoingtoofaraeld,let'susea mechanicalanalogytogetageneralideaofwhyaresistorwouldbe usedinaradio. Themainpartsofaradioreceiverareanantenna,atunerfor selectingthefrequency,andanampliertostrengthenthesignal sucientlytodriveaspeaker.Thetunerresonatesattheselected frequency,justasintheexamplesofmechanicalresonancediscussed inbook3ofthisseries.Thebehaviorofamechanicalresonatordependsonthreethings:itsinertia,itsstiness,andtheamountof frictionordamping.Thersttwoparameterslocatethepeakof theresonancecurve,whilethedampingdeterminesthewidthofthe resonance.Intheradiotunerwehaveanelectricallyvibratingsystemthatresonatesataparticularfrequency.Insteadofaphysical objectmovingbackandforth,thesevibrationsconsistofelectrical currentsthatowrstinonedirectionandthenintheother.In amechanicalsystem,dampingmeanstakingenergyoutofthevibrationintheformofheat,andexactlythesameideaappliestoan electricalsystem:theresistorsuppliesthedamping,andtherefore Section3.4Resistance 91

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l / Anexampleofaresistor withacolorcode. colormeaning black0 brown1 red2 orange3 yellow4 green5 blue6 violet7 gray8 white9 silver 10% gold 5% m / Colorcodesusedonresistors. controlsthewidthoftheresonance.Ifwesetouttoeliminateall resistanceinthetunercircuit,bynotbuildinginaresistorandby somehowgettingridofalltheinherentelectricalresistanceofthe wires,wewouldhaveauselessradio.Thetuner'sresonancewould besonarrowthatwecouldnevergetcloseenoughtotherightfrequencytobringinthestation.Therolesofinertiaandstinessare playedbyothercircuitelementswehavenotdiscussesacapacitor andacoil. Manyelectricaldevicesarebasedonelectricalresistanceand Ohm'slaw,eveniftheydonothavelittlecomponentsinthemthat lookliketheusualresistor.Thefollowingaresomeexamples. Lightbulb Thereisnothingspecialaboutalightbulblament|youcan easilymakealightbulbbycuttinganarrowwaistintoametallic gumwrapperandconnectingthewrapperacrosstheterminalsof a9-voltbattery.Thetroubleisthatitwillinstantlyburnout. Edisonsolvedthistechnicalchallengebyencasingthelamentin anevacuatedbulb,whichpreventedburning,sinceburningrequires oxygen. Polygraph Thepolygraph,orliedetector,"isreallyjustasetofmeters forrecordingphysicalmeasuresofthesubject'spsychologicalstress, suchassweatingandquickenedheartbeat.Thereal-timesweatmeasurementworksontheprinciplethatdryskinisagoodinsulator, butsweatyskinisaconductor.Ofcourseatruthfulsubjectmay becomenervoussimplybecauseofthesituation,andapracticed liarmaynotevenbreakasweat.Themethod'spractitionersclaim thattheycantellthedierence,butyoushouldthinktwicebefore allowingyourselftobepolygraphtested.MostU.S.courtsexclude allpolygraphevidence,butsomeemployersattempttoscreenout dishonestemployeesbypolygraphtestingjobapplicants,anabuse thatrankswithsuchpseudoscienceashandwritinganalysis. Fuse Afuseisadeviceinsertedinacircuittollbooth-styleinthesame mannerasanammeter.Itissimplyapieceofwiremadeofmetals havingarelativelylowmeltingpoint.Iftoomuchcurrentpasses throughthefuse,itmelts,openingthecircuit.Thepurposeisto makesurethatthebuilding'swiresdonotcarrysomuchcurrent thattheythemselveswillgethotenoughtostartare.Mostmodern housesusecircuitbreakersinsteadoffuses,althoughfusesarestill commonincarsandsmalldevices.Acircuitbreakerisaswitch operatedbyacoiled-wiremagnet,whichopensthecircuitwhen enoughcurrentows.Theadvantageisthatonceyouturnosome 92 Chapter3Circuits,Part1

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n / 1.Asimplieddiagramof howavoltmeterworks.2.Measuringthevoltagedifference acrossalightbulb.3.Thesame setupdrawninschematicform.4. Thesetupformeasuringcurrent isdifferent. oftheappliancesthatweresuckinguptoomuchcurrent,youcan immediatelyiptheswitchclosed.Inthedaysoffuses,onemight getcaughtwithoutareplacementfuse,orevenbetemptedtostu aluminumfoilinasareplacement,defeatingthesafetyfeature. Voltmeter Avoltmeterisnothingmorethananammeterwithanadditional high-valueresistorthroughwhichthecurrentisalsoforcedtoow. Ohm'slawrelatesthecurrentthroughtheresistorisrelateddirectly tothevoltagedierenceacrossit,sothemetercanbecalibrated inunitsofvoltsbasedontheknownvalueoftheresistor.The voltmeter'stwoprobesaretouchedtothetwolocationsinacircuit betweenwhichwewishtomeasurethevoltagedierence,n/2.Note howcumbersomethistypeofdrawingis,andhowdicultitcan betotellwhatisconnectedtowhat.Thisiswhyelectricaldrawing areusuallyshowninschematicform.Figuren/3isaschematic representationofguren/2. Thesetupsformeasuringcurrentandvoltagearedierent.When wearemeasuringcurrent,wearendinghowmuchstugoes through,"soweplacetheammeterwhereallthecurrentisforced togothroughit.Voltage,however,isnotstuthatgoesthrough," itisameasureofelectricalenergy.Ifanammeterislikethemeter thatmeasuresyourwateruse,avoltmeterislikeameasuringstick thattellsyouhowhighawaterfallis,sothatyoucandeterminehow muchenergywillbereleasedbyeachkilogramoffallingwater.We donotwanttoforcethewatertogothroughthemeasuringstick! Thearrangementinguren/3isa parallel circuit:oneinthereare forksintheroad"wheresomeofthecurrentwillowonewayand somewillowtheother.Figuren/4issaidtobewiredin series: allthecurrentwillvisitallthecircuitelementsoneaftertheother. Wewilldealwithseriesandparallelcircuitsinmoredetailinthe followingchapter. Ifyouinsertedavoltmeterincorrectly,inserieswiththebulband battery,itslargeinternalresistancewouldcutthecurrentdownso lowthatthebulbwouldgoout.Youwouldhaveseverelydisturbed thebehaviorofthecircuitbytryingtomeasuresomethingaboutit. Incorrectlyplacinganammeterinparallelislikelytobeeven moredisconcerting.Theammeterhasnothingbutwireinsideitto provideresistance,sogiventhechoice,mostofthecurrentwillow throughitratherthanthroughthebulb.Somuchcurrentwillow throughtheammeter,infact,thatthereisadangerofburningout thebatteryorthemeterorboth!Forthisreason,mostammeters havefusesorcircuitbreakersinside.Somemodelswilltriptheir circuitbreakersandmakeanaudiblealarminthissituation,while otherswillsimplyblowafuseandstopworkinguntilyoureplaceit. Section3.4Resistance 93

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DiscussionQuestions A Inguren/1,woulditmakeanydifferenceinthevoltagemeasurementifwetouchedthevoltmeter'sprobestodifferentpointsalongthe samesegmentsofwire? B Explainwhyitwouldbeincorrecttodeneresistanceastheamount ofchargetheresistorallowstoow. 3.5Current-conductingpropertiesofmaterials Ohm'slawhasaremarkableproperty,whichisthatcurrentwillow eveninresponsetoavoltagedierencethatisassmallaswecare tomakeit.Intheanalogyofpushingacrateacrossaoor,itis asthoughevenaeacouldslidethecrateacrosstheoor,albeit atsomeverylowspeed.Theeacannotdothisbecauseofstatic friction,whichwecanthinkofasaneectarisingfromthetendency ofthemicroscopicbumpsandvalleysinthecrateandoortolock together.ThefactthatOhm'slawholdsfornearlyallsolidshas aninterestinginterpretation:atleastsomeoftheelectronsarenot lockeddown"atalltoanyspecicatom. Moregenerallywecanaskhowchargeactuallyowsinvarious solids,liquids,andgases.Thiswillleadustotheexplanationsof manyinterestingphenomena,includinglightning,thebluishcrust thatbuildsupontheterminalsofcarbatteries,andtheneedfor electrolytesinsportsdrinks. Solids Inatomicterms,thedeningcharacteristicofasolidisthatits atomsarepackedtogether,andthenucleicannotmoveveryfarfrom theirequilibriumpositions.Itmakessense,then,thatelectrons,not ions,wouldbethechargecarrierswhencurrentsowinsolids.This factwasestablishedexperimentallybyTolmanandStewart,inan experimentinwhichtheyspunalargecoilofwireandthenabruptly stoppedit.Theyobservedacurrentinthewireimmediatelyafter thecoilwasstopped,whichindicatedthatchargedparticlesthat werenotpermanentlylockedtoaspecicatomhadcontinuedto movebecauseoftheirowninertia,evenafterthematerialofthe wireingeneralstopped.Thedirectionofthecurrentshowedthat itwasnegativelychargedparticlesthatkeptmoving.Thecurrent onlylastedforaninstant,however;asthenegativelychargedparticlescollectedatthedownstreamendofthewire,fartherparticles werepreventedjoiningthemduetotheirelectricalrepulsion,aswell astheattractionfromtheupstreamend,whichwasleftwithanet positivecharge.TolmanandStewartwereevenabletodetermine themass-to-chargeratiooftheparticles.Weneednotgointothe detailsoftheanalysishere,butparticleswithhighmasswouldbe diculttodecelerate,leadingtoastrongerandlongerpulseofcurrent,whileparticleswithhighchargewouldfeelstrongerelectrical 94 Chapter3Circuits,Part1

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forcesdeceleratingthem,whichwouldcauseaweakerandshorter pulse.Themass-to-chargeratiothusdeterminedwasconsistentwith the m=q oftheelectrontowithintheaccuracyoftheexperiment, whichessentiallyestablishedthattheparticleswereelectrons. Thefactthatonlyelectronscarrycurrentinsolids,notions,has manyimportantimplications.Foronething,itexplainswhywires don'tfrayorturntodustaftercarryingcurrentforalongtime. Electronsareverysmallperhapsevenpointlike,anditiseasyto imaginethempassingbetweenthecracksamongtheatomswithout creatingholesorfracturesintheatomicframework.Forthosewho knowalittlechemistry,italsoexplainswhyallthebestconductors areontheleftsideoftheperiodictable.Theelementsinthatarea aretheonesthathaveonlyaverylooseholdontheiroutermost electrons. Gases Themoleculesinagasspendmostoftheirtimeseparatedfrom eachotherbysignicantdistances,soitisnotpossibleforthemto conductelectricitythewaysolidsdo,byhandingoelectronsfrom atomtoatom.Itisthereforenotsurprisingthatgasesaregood insulators. Gasesarealsousuallynonohmic.Asoppositechargesbuildup onastormcloudandthegroundbelow,thevoltagedierencebecomesgreaterandgreater.Zerocurrentows,however,untilnally thevoltagereachesacertainthresholdandwehaveanimpressive exampleofwhatisknownasasparkorelectricaldischarge.Ifair wasohmic,thecurrentbetweenthecloudandthegroundwould simplyincreasesteadilyasthevoltagedierenceincreased,rather thanbeingzerountilathresholdwasreached.Thisbehaviorcanbe explainedasfollows.Atsomepoint,theelectricalforcesontheair electronsandnucleioftheairmoleculesbecomesostrongthatelectronsarerippedrightoofsomeofthemolecules.Theelectrons thenacceleratetowardeitherthecloudortheground,whichever ispositivelycharged,andthepositiveionsacceleratetheopposite way.Asthesechargecarriersaccelerate,theystrikeandionizeother molecules,whichproducesarapidlygrowingcascade. Liquids Moleculesinaliquidareabletoslidepasteachother,soions aswellaselectronscancarrycurrents.Purewaterisapoorconductorbecausethewatermoleculestendtoholdontotheirelectrons strongly,andtherearethereforenotmanyelectronsorionsavailable tomove.Watercanbecomequiteagoodconductor,however,with theadditionofevenasmallamountofcertainsubstancescalled electrolytes,whicharetypicallysalts.Forexample,ifweaddtable salt,NaCl,towater,theNaClmoleculesdissolveintoNa + andCl )]TJ/F15 10.9091 Tf -313.231 -17.508 Td [(ions,whichcanthenmoveandcreatecurrents.ThisiswhyelecSection3.5Current-conductingpropertiesofmaterials 95

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triccurrentscanowamongthecellsinourbodies:cellularuidis quitesalty.Whenwesweat,welosenotjustwaterbutelectrolytes, sodehydrationplayshavocwithourcells'electricalsystems.Itis forthisreasonthatelectrolytesareincludedinsportsdrinksand formulasforrehydratinginfantswhohavediarrhea. Sincecurrentowinliquidsinvolvesentireions,itisnotsurprisingthatwecanseephysicalevidencewhenithasoccurred.For example,afteracarbatteryhasbeeninuseforawhile,theH 2 SO 4 batteryacidbecomesdepletedofhydrogenions,whicharethemain chargecarriersthatcompletethecircuitontheinsideofthebattery.TheleftoverSO 4 thenformsavisiblebluecrustonthebattery posts. Speedofcurrentsandelectricalsignals WhenItalkonthephonetomymotherinlawtwothousand milesaway,Idonotnoticeanydelaywhilethesignalmakesitsway backandforth.Electricalsignalsthereforemusttravelveryquickly, buthowfastexactly?Theanswerisrathersubtle.Forthesake ofconcreteness,let'srestrictourselvestocurrentsinmetals,which consistofelectrons. Theelectronsthemselvesareonlymovingatspeedsofperhaps afewthousandmilesperhour,andtheirmotionismostlyrandom thermalmotion.Thisshowsthattheelectronsinmyphonecannot possiblybezippingbackandforthbetweenCaliforniaandNewYork fastenoughtocarrythesignals.Eveniftheirthousand-mile-an-hour motionwasorganizedratherthanrandom,itwouldstilltakethem manyminutestogetthere.Realistically,itwilltaketheaverage electronevenlongerthanthattomakethetrip.Thecurrentinthe wireconsistsonlyofaslowoveralldrift,ataspeedontheorder ofafewcentimeterspersecond,superimposedonthemorerapid randommotion.Wecancomparethiswiththeslowwestwarddrift inthepopulationoftheU.S.Ifwecouldmakeamovieofthemotion ofallthepeopleintheU.S.fromouterspace,andcouldwatchitat highspeedsothatthepeopleappearedtobescurryingaroundlike ants,wewouldthinkthatthemotionwasfairlyrandom,andwe wouldnotimmediatelynoticethewestwarddrift.Onlyaftermany yearswouldwerealizethatthenumberofpeopleheadingwestover theSierrashadexceededthenumbergoingeast,sothatCalifornia increaseditsshareofthecountry'spopulation. Sowhyareelectricalsignalssofastiftheaveragedriftspeedof electronsissoslow?Theansweristhatadisturbanceinanelectrical systemcanmovemuchmorequicklythanthechargesthemselves. Itisasthoughwelledapipewithgolfballsandtheninsertedan extraballatoneend,causingaballtofalloutattheotherend. Theforcepropagatedtotheotherendinafractionofasecond,but theballsthemselvesonlytraveledafewcentimetersinthattime. 96 Chapter3Circuits,Part1

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Becausetherealityofcurrentconductionissocomplex,weoften describethingsusingmentalshortcutsthataretechnicallyincorrect. ThisisOKaslongasweknowthattheyarejustshortcuts.For example,supposethepresidentsofFranceandRussiashakehands, andtheFrenchpoliticianhasinadvertentlypickedupapositiveelectricalcharge,whichshockstheRussian.Wemaysaythattheexcess positivelychargedparticlesintheFrenchleader'sbody,whichall repeleachother,takethehandshakeasanopportunitytogetfartherapartbyspreadingoutintotwobodiesratherthanone.In reality,itwouldbeamatterofminutesbeforetheionsinoneperson'sbodycouldactuallydriftdeepintotheother's.Whatreally happensisthatthroughoutthebodyoftherecipientoftheshock therearealreadyvariouspositiveandnegativeionswhicharefree tomove.Evenbeforetheperpetrator'schargedhandtouchesthe victim'ssweatypalm,thechargesintheshocker'sbodybeginto repelthepositiveionsandattractthenegativeionsintheother person.Thesplit-secondsensationofshockiscausedbythesudden jumpingofthevictim'sionsbydistancesofperhapsamicrometer, thiseectoccurringsimultaneouslythroughoutthewholebody,althoughmoreviolentlyinthehandandarm,whichareclosertothe otherperson. 3.6 R ApplicationsofCalculus Asdiscussedinexample1onpage80,thedenitionofcurrentasthe rateofchangeofchargewithrespecttotimemustbereexpressed asaderivativeinthecasewheretherateofchangeisnotconstant, I = d q d t Findingcurrentgivenchargeexample8 Achargedballoonfallstotheground,anditschargebegins leakingofftotheEarth.Supposethatthechargeontheballoon isgivenby q = ae )]TJ/F107 7.9701 Tf 6.587 0 Td [(bt .Findthecurrentasafunctionoftime,and interprettheanswer. Takingthederivative,wehave I = d q d t = )]TJ/F107 10.9091 Tf 8.485 0 Td [(abe )]TJ/F107 7.9701 Tf 6.587 0 Td [(bt Anexponentialfunctionapproacheszeroastheexponentgets moreandmorenegative.Thismeansthatboththechargeand thecurrentaredecreasinginmagnitudewithtime.Itmakessense thatthechargeapproacheszero,sincetheballoonislosingits charge.Italsomakessensethatthecurrentisdecreasingin magnitude,sincechargecannotowatthesamerateforever withoutovershootingzero. Section3.6 R ApplicationsofCalculus 97

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Findingchargegivencurrentexample9 InthesegmentoftheATLASacceleratorshowningureion page89,thecurrentowingbackandforthbetweenthetwocylindersisgivenby I = a cos bt .Whatisthechargeononeofthe cylindersasafunctionoftime? Wearegiventhecurrentand wanttondthecharge,i.e.wearegiventhederivativeandwe wanttondtheoriginalfunctionthatwouldgivethatderivative. Thismeansweneedtointegrate: q = Z I d t = Z a cos bt d t = a b sin bt + q o where q o isaconstantofintegration. Wecaninterpretthisinordertoexplainwhyasuperconductor needstobeused.Theconstant b mustbeverylarge,sincethe currentissupposedtooscillatebackandforthmillionsoftimesa second.Lookingatthenalresult,weseethatif b isaverylarge number,and q istobeasignicantamountofcharge,then a must beaverylargenumberaswell.If a isnumericallylarge,thenthe currentmustbeverylarge,soitwouldheattheacceleratortoo muchifitwasowingthroughanordinaryconductor. 98 Chapter3Circuits,Part1

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Summary SelectedVocabulary current......therateatwhichchargecrossesacertain boundary ampere......themetricunitofcurrent,onecoulombpesecond;alsoamp" ammeter.....adeviceformeasuringelectricalcurrent circuit.......anelectricaldeviceinwhichchargecancome backtoitsstartingpointandberecycled ratherthangettingstuckinadeadend opencircuit...acircuitthatdoesnotfunctionbecauseithas agapinit shortcircuit...acircuitthatdoesnotfunctionbecausecharge isgivenalow-resistanceshortcut"paththat itcanfollow,insteadofthepaththatmakes itdosomethinguseful voltage......electricalpotentialenergyperunitchargethat willbepossessedbyachargedparticleata certainpointinspace volt........themetricunitofvoltage,onejouleper coulomb voltmeter.....adeviceformeasuringvoltagedierences ohmic.......describesasubstanceinwhichtheowofcurrentbetweentwopointsisproportionaltothe voltagedierencebetweenthem resistance....theratioofthevoltagedierencetothecurrentinanobjectmadeofanohmicsubstance ohm........themetricunitofelectricalresistance,onevolt perampere Notation I ..........current A .........unitsofamperes V .........voltage V .........unitsofvolts R .........resistance .........unitsofohms OtherTerminologyandNotation electricpotentialratherthanthemoreinformalvoltage"used here;despitethemisleadingname,itisnotthe sameaselectricpotentialenergy eV.........aunitofenergy,equalto e multipliedby1 volt;1.6 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(19 joules Summary Allelectricalphenomenaarealikeinthatthatarisefromthe presenceormotionofcharge.Mostpracticalelectricaldevicesare basedonthemotionofchargearoundacompletecircuit,sothat Summary 99

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thechargecanberecycledanddoesnothitanydeadends.The mostusefulmeasureoftheowofchargeiscurrent, I = q= t Anelectricaldevicewhosejobistotransformenergyfromone formintoanother,e.g.,alightbulb,usespowerataratewhichdependsbothonhowrapidlychargeisowingthroughitandonhow muchworkisdoneoneachunitofcharge.Thelatterquantityis knownasthevoltagedierencebetweenthepointwherethecurrent entersthedeviceandthepointwherethecurrentleavesit.Since thereisatypeofpotentialenergyassociatedwithelectricalforces, theamountofworktheydoisequaltothedierenceinpotential energybetweenthetwopoints,andwethereforedenevoltagedifferencesdirectlyintermsofpotentialenergy, V = PE elec =q Therateofpowerdissipationis P = I V Manyimportantelectricalphenomenacanonlybeexplainedif weunderstandthemechanismsofcurrentowattheatomiclevel.In metals,currentsarecarriedbyelectrons,inliquidsbyions.Gases arenormallypoorconductorsunlesstheiratomsaresubjectedto suchintenseelectricalforcesthattheatomsbecomeionized. Manysubstances,includingallsolids,respondtoelectricalforces insuchawaythattheowofcurrentbetweentwopointsisproportionaltothevoltagedierencebetweenthosepoints.Sucha substanceiscalledohmic,andanobjectmadeoutofanohmic substancecanberatedintermsofitsresistance, R =V/I.An importantcorollaryisthataperfectconductor,with R =0,must haveconstantvoltageeverywherewithinit. 100 Chapter3Circuits,Part1

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Problem3. Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Aresistorhasavoltagedierence V acrossit,causinga current I toow. aFindanequationforthepoweritdissipatesasheatintermsof thevariables I and R only,eliminating V p bIfanelectricallinecomingtoyourhouseistocarryagiven amountofcurrent,interpretyourequationfrompartatoexplain whetherthewire'sresistanceshouldbesmall,orlarge. p 2 aExpressthepowerdissipatedbyaresistorintermsof R and V only,eliminating I p bElectricalreceptaclesinyourhomearemostly110V,butcircuitsforelectricstoves,airconditioners,andwashersanddriersare usually220V.Thetwotypesofcircuitshavedierentlyshapedreceptacles.Supposeyourewiretheplugofadriersothatitcanbe pluggedintoa110Vreceptacle.Theresistorthatformstheheatingelementofthedrierwouldnormallydraw200W.Howmuch powerdoesitactuallydrawnow? p 3 Asdiscussedinthetext,whenaconductorreachesanequilibriumwhereitschargeisatrest,thereisalwayszeroelectricforce onachargeinitsinterior,andanyexcesschargeconcentratesitself onthesurface.Thesurfacelayerofchargearrangesitselfsoasto producezerototalforceatanypointintheinterior.Otherwisethe freechargeintheinteriorcouldnotbeatrest.Supposeyouhave ateardrop-shapedconductorliketheoneshowninthegure.Since theteardropisaconductor,therearefreechargeseverywhereinside it,butconsiderafreechargedparticleatthelocationshownwith awhitecircle.Explainwhy,inordertoproducezeroforceonthis particle,thesurfacelayerofchargemustbedenserinthepointed partoftheteardrop.Similarreasoningshowswhylightningrods aremadewithpoints.Thechargedstormcloudsinducepositiveand negativechargestomovetooppositeendsoftherod.Atthepointed upperendoftherod,thechargetendstoconcentrateatthepoint, andthischargeattractsthelightning. 4 Usetheresultofproblem3onpage39tondanequationfor thevoltageatapointinspaceatadistance r fromapointcharge Q .Takeyour V =0distancetobeanywhereyoulike. p Problems 101

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Problem5. Problem6. 5 Referringbacktoproblem6onpage40aboutthesodiumchloridecrystal,supposethelithiumionisgoingtojumpfromthegapit isoccupyingtooneofthefourclosestneighboringgaps.Whichone willitjumpto,andifitstartsfromrest,howfastwillitbegoing bythetimeitgetsthere?Itwillkeeponmovingandaccelerating afterthat,butthatdoesnotconcernus.[Hint:Theapproachis similartotheoneusedfortheotherproblem,butyouwanttowork withvoltageandpotentialenergyratherthanforce.] p ? 6 Referringbacktoouroldfriendtheneuronfromproblem 1onpage39,let'snowconsiderwhathappenswhenthenerveis stimulatedtotransmitinformation.Whentheblobatthetopthe cellbodyisstimulated,itcausesNa + ionstorushintothetopof thetailaxon.Thiselectricalpulsewillthentraveldowntheaxon, likeaameburningdownfromtheendofafuse,withtheNa + ions ateachpointrstgoingoutandthencomingbackin.If10 10 Na + ionscrossthecellmembranein0.5ms,whatamountofcurrentis created? p 7 Ifatypicallightbulbdrawsabout900mAfroma110-V householdcircuit,whatisitsresistance?Don'tworryaboutthe factthatit'salternatingcurrent. p 8 Today,evenabigluxurycarlikeaCadillaccanhavean electricalsystemthatisrelativelylowinpower,sinceitdoesn't needtodomuchmorethanrunheadlights,powerwindows,etc. Inthenearfuture,however,manufacturersplantostartmaking carswithelectricalsystemsaboutvetimesmorepowerful.This willallowcertainenergy-wastingpartslikethewaterpumptobe runonelectricalmotorsandturnedowhenthey'renotneeded |currentlythey'rerundirectlyonshaftsfromthemotor,sothey can'tbeshuto.Itmayevenbepossibletomakeanenginethat canshutoatastoplightandthenturnbackonagainwithout cranking,sincethevalvescanbeelectricallypowered.Currentcars' electricalsystemshave12-voltbatterieswith14-voltchargers,but thenewsystemswillhave36-voltbatterieswith42-voltchargers. aSupposethebatteryinanewcarisusedtorunadevicethat requiresthesameamountofpowerasthecorrespondingdevicein theoldcar.Basedonthesampleguresabove,howwouldthe currentshandledbythewiresinoneofthenewcarscomparewith thecurrentsintheoldones? bTherealpurposeofthegreatervoltageistohandledevicesthat need more power.Canyouguesswhytheydecidedtochangeto36voltbatteriesratherthanincreasingthepowerwithoutincreasing thevoltage? p 9 aYoutakeanLPrecordoutofitssleeve,anditacquiresa staticchargeof1nC.Youplayitatthenormalspeedof33 1 3 r.p.m., andthechargemovinginacirclecreatesanelectriccurrent.What isthecurrent,inamperes? p 102 Chapter3Circuits,Part1

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Problem11. Problem13. Problem14. bAlthoughtheplanetarymodeloftheatomcanbemadetowork withanyvaluefortheradiusoftheelectrons'orbits,moreadvanced modelsthatwewillstudylaterinthiscoursepredictdeniteradii. Iftheelectronisimaginedascirclingaroundtheprotonataspeed of2.2 10 6 m/s,inanorbitwitharadiusof0.05nm,whatelectric currentiscreated? p ? 10 Wehavereferredtoresistors dissipating heat,i.e.wehave assumedthat P = I V isalwaysgreaterthanzero.Could I V comeouttobenegativeforaresistor?Ifso,couldonemakea refrigeratorbyhookinguparesistorinsuchawaythatitabsorbed heatinsteadofdissipatingit? 11 Youaregivenabattery,aashlightbulb,andasinglepiece ofwire.Drawatleasttwocongurationsoftheseitemsthatwould resultinlightingupthebulb,andatleasttwothatwouldnotlight it.Don'tdrawschematics.Ifyou'renotsurewhat'sgoingon, borrowthematerialsfromyourinstructorandtryit.Notethatthe bulbhastwoelectricalcontacts:oneisthethreadedmetaljacket, andtheotheristhetipatthebottominthegure.[Problemby ArnoldArons.] 12 Inawirecarryingacurrentof1.0pA,howlongdoyouhave towait,ontheaverage,forthenextelectrontopassagivenpoint? Expressyouranswerinunitsofmicroseconds. Solution,p.204 13 Thegureshowsasimplieddiagramofanelectrongunsuch astheoneusedintheThomsonexperiment,ortheonethatcreatestheelectronbeaminaTVtube.Electronsthatspontaneously emergefromthenegativeelectrodecathodearethenaccelerated tothepositiveelectrode,whichhasaholeinit.Oncetheyemerge throughthehole,theywillslowdown.However,ifthetwoelectrodes arefairlyclosetogether,thisslowingdownisasmalleect,because theattractiveandrepulsiveforcesexperiencedbytheelectrontend tocancel.aIfthevoltagedierencebetweentheelectrodesis V,whatisthevelocityofanelectronasitemergesatB?Assume itsinitialvelocity,atA,isnegiligible.bEvaluateyourexpressionnumericallyforthecasewhere V =10kV,andcomparetothe speedoflight. Solution,p.205 14 Thegureshowsasimplieddiagramofadevicecalleda tandemaccelerator,usedforacceleratingbeamsofionsuptospeeds ontheorderof1%ofthespeedoflight.Thenucleioftheseionscollidewiththenucleiofatomsinatarget,producingnuclearreactions forexperimentsstudyingthestructureofnuclei.Theoutershellof theacceleratorisaconductoratzerovoltagei.e.,thesamevoltage astheEarth.Theelectrodeatthecenter,knownastheterminal," isatahighpositivevoltage,perhapsmillionsofvolts.Negativeions withachargeof )]TJ/F15 10.9091 Tf 8.485 0 Td [(1uniti.e.,atomswithoneextraelectronare producedostageontheright,typicallybychemicalreactionswith Problems 103

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Aprintedcircuitboard,like thekindreferredtoinproblem 16. cesium,whichisachemicalelementthathasastrongtendencyto giveawayelectrons.Relativelyweakelectricandmagneticforces areusedtotransportthese )]TJ/F15 10.9091 Tf 8.485 0 Td [(1ionsintotheaccelerator,wherethey areattractedtotheterminal.Althoughthecenteroftheterminal hasaholeinittolettheionspassthrough,thereisaverythincarbonfoiltherethattheymustphysicallypenetrate.Passingthrough thefoilstripsosomenumberofelectrons,changingtheatominto apositiveion,withachargeof+ n timesthefundamentalcharge. Nowthattheatomispositive,itisrepelledbytheterminal,and acceleratessomemoreonitswayoutoftheaccelerator.aFind thevelocity, v ,oftheemergingbeamofpositiveions,intermsof n theirmass m ,theterminalvoltage V ,andfundamentalconstants. Neglectthesmallchangeinmasscausedbythelossofelectrons inthestripperfoil.bTofuseprotonswithprotons,aminimum beamvelocityofabout11%ofthespeedoflightisrequired.What terminalvoltagewouldbeneededinthiscase? 15 Threecharges,eachofstrength Q Q> 0formaxed equilateraltrianglewithsidesoflength b .Youthrowaparticleof mass m andpositivecharge q fromfaraway,withaninitialspeed v .Yourgoalistogettheparticletogotothecenterofthetriangle, youraimisperfect,andyouarefreetothrowfromanydirection youlike.Whatistheminimumpossiblevalueof v ? 16 Youhavetododierentthingswithacircuittomeasure currentthantomeasureavoltagedierence.Whichwouldbemore practicalforaprintedcircuitboard,inwhichthewiresareactually stripsofmetalembeddedinsidetheboard? Solution,p.205 104 Chapter3Circuits,Part1

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AnIntel486computerchipinitspackaging. Chapter4 Circuits,Part2 Inchapter3welimitedourselvestorelativelysimplecircuits,essentiallynothingmorethanabatteryandasinglelightbulb.The purposeofthischapteristointroduceyoutomorecomplexcircuits, containingmultipleresistorsorvoltagesourcesinseries,inparallel, orboth. Whydoyouneedtoknowthisstu?Afterall,ifyouwere planningonbeinganelectricalengineeryouprobablywouldn'tbe learningphysicsfromthisbook.Consider,however,thateverytime youpluginalamporaradioyouareaddingacircuitelementtoa householdcircuitandmakingitmorecomplex.Electricalsafety,as well,cannotreallybeunderstoodwithoutunderstandingmultipleelementcircuits,sincegettingshockedusuallyinvolvesatleasttwo parts:thedevicethatissupposedtousepowerplusthebodyofthe personindanger.Ifyouareastudentmajoringinthelifesciences, youshouldrealizeaswellthatallcellsareinherentlyelectrical,and inanymulticellularorganismtherewillthereforebevariousseries andparallelcircuits. Evenapartfromthesepracticalpurposes,thereisaveryfundamentalreasonforreadingthischapter:tounderstandchapter3 better.Atthispointintheirstudies,Ialwaysobservestudentsusing 105

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b / Thetwoshadedareas shapedliketheletterEareboth regionsofconstantvoltage. wordsandmodesofreasoningthatshowtheyhavenotyetbecome completelycomfortableanduentwiththeconceptsofvoltageand current.Theyask,aren'tvoltageandcurrentsortofthesame idea?"Theyspeakofvoltagegoingthrough"alightbulb.Once theybeginhoningtheirskillsonmorecomplicatedcircuitsIalways seetheircondenceandunderstandingincreaseimmeasurably. 4.1Schematics Iseeachessposition;KasparovseesaninterestingRuyLopezvariation.Totheuninitiatedaschematicmaylookasunintelligibleas Mayanhieroglyphs,butevenalittlebitofeyetrainingcangoalong waytowardmakingitsmeaningleapothepage.Aschematicisa stylizedandsimplieddrawingofacircuit.Thepurposeistoeliminateasmanyirrelevantfeaturesaspossible,sothattherelevant onesareeasiertopickout. a / 1.Wrong:Theshapesofthe wiresareirrelevant.2.Wrong: Rightanglesshouldbeused.3. Wrong:Asimplepatternismade tolookunfamiliarandcomplicated.4.Right. Anexampleofanirrelevantfeatureisthephysicalshape,length, anddiameterofawire.Innearlyallcircuits,itisagoodapproximationtoassumethatthewiresareperfectconductors,sothatany pieceofwireuninterruptedbyothercomponentshasconstantvoltagethroughoutit.Changingthelengthofthewire,forinstance, doesnotchangethisfact.Ofcourseifweusedmilesandmiles ofwire,asinatelephoneline,thewire'sresistancewouldstartto addup,anditslengthwouldstarttomatter.Theshapesofthe wiresarelikewiseirrelevant,sowedrawthemwithstandardized, stylizedshapesmadeonlyofverticalandhorizontallineswithrightanglebendsinthem.Thishastheeectofmakingsimilarcircuits lookmorealikeandhelpingustorecognizefamiliarpatterns,just aswordsinanewspaperareeasiertorecognizethanhandwritten ones.Figureashowssomeexamplesoftheseconcepts. Themostimportantrststepinlearningtoreadschematicsis tolearntorecognizecontiguouspiecesofwirewhichmusthaveconstantvoltagethroughout.Ingureb,forexample,thetwoshaded E-shapedpiecesofwiremusteachhaveconstantvoltage.Thisfocusesourattentionontwoofthemainunknownswe'dliketobe abletopredict:thevoltageoftheleft-handEandthevoltageof theoneontheright. 106 Chapter4Circuits,Part2

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4.2Parallelresistancesandthejunctionrule Oneofthesimplestexamplestoanalyzeistheparallelresistance circuit,ofwhichgurebwasanexample.Ingeneralwemayhave unequalresistances R 1 and R 2 ,asinc/1.Sincethereareonlytwo constant-voltageareasinthecircuit,c/2,allthreecomponentshave thesamevoltagedierenceacrossthem.Abatterynormallysucceedsinmaintainingthevoltagedierencesacrossitselfforwhichit wasdesigned,sothevoltagedrops V 1 and V 2 acrosstheresistors mustbothequalthevoltageofthebattery: V 1 = V 2 = V battery Eachresistancethusfeelsthesamevoltagedierenceasifitwas theonlyoneinthecircuit,andOhm'slawtellsusthattheamount ofcurrentowingthrougheachoneisalsothesameasitwould havebeeninaone-resistorcircuit.Thisiswhyhouseholdelectrical circuitsarewiredinparallel.Wewanteveryappliancetowork thesame,regardlessofwhetherotherappliancesarepluggedinor unplugged,turnedonorswitchedo.Theelectriccompanydoesn't usebatteriesofcourse,butouranalysiswouldbethesameforany devicethatmaintainsaconstantvoltage. c / 1.Tworesistorsinparallel. 2.Therearetwoconstant-voltage areas.3.Thecurrentthatcomes outofthebatterysplitsbetween thetworesistors,andlaterreunites.4.Thetworesistorsin parallelcanbetreatedasasingle resistorwithasmallerresistance value. Ofcoursetheelectriccompanycantellwhenweturnonevery lightinthehouse.Howdotheyknow?Theansweristhatwedraw morecurrent.Eachresistancedrawsacertainamountofcurrent, andtheamountthathastobesuppliedisthesumofthetwoindividualcurrents.Thecurrentislikeariverthatsplitsinhalf,c/3, andthenreunites.Thetotalcurrentis I total = I 1 + I 2 Thisisanexampleofageneralfactcalledthejunctionrule: thejunctionrule Inanycircuitthatisnotstoringorreleasingcharge,conservationofchargeimpliesthatthetotalcurrentowingoutof anyjunctionmustbethesameasthetotalowingin. Comingbacktotheanalysisofourcircuit,weapplyOhm'slaw Section4.2Parallelresistancesandthejunctionrule 107

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toeachresistance,resultingin I total = V=R 1 + V=R 2 = V 1 R 1 + 1 R 2 Asfarastheelectriccompanyisconcerned,yourwholehouse isjustoneresistorwithsomeresistance R ,calledthe equivalent resistance .TheywouldwriteOhm'slawas I total = V=R fromwhichwecandeterminetheequivalentresistancebycomparisonwiththepreviousexpression: 1 =R = 1 R 1 + 1 R 2 R = 1 R 1 + 1 R 2 )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 [equivalentresistanceoftworesistorsinparallel] Tworesistorsinparallel,c/4,areequivalenttoasingleresistorwith avaluegivenbytheaboveequation. Twolampsonthesamehouseholdcircuitexample1 Youturnontwolampsthatareonthesamehouseholdcircuit. Eachonehasaresistanceof1ohm.Whatistheequivalentresistance,andhowdoesthepowerdissipationcomparewiththe caseofasinglelamp? Theequivalentresistanceofthetwolampsinparallelis R = 1 R 1 + 1 R 2 )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 = 1 1 + 1 1 )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 = 1 )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 +1 )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 = 2 )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 =0.5 Thevoltagedifferenceacrossthewholecircuitisalwaysthe110 Vsetbytheelectriccompanyit'salternatingcurrent,butthat's irrelevant.Theresistanceofthewholecircuithasbeencutin halfbyturningonthesecondlamp,soaxedamountofvoltage willproducetwiceasmuchcurrent.Twicethecurrentowing acrossthesamevoltagedifferencemeanstwiceasmuchpower dissipation,whichmakessense. 108 Chapter4Circuits,Part2

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d / Threeresistorsinparallel. e / Unitingfourresistorsin parallelisequivalenttomaking asingleresistorwiththesame lengthbutfourtimesthecrosssectionalarea.Theresultisto makearesistorwithonequarter theresistance. Thecuttinginhalfoftheresistancesurprisesmanystudents, sinceweareaddingmoreresistance"tothecircuitbyputtingin thesecondlamp.Whydoestheequivalentresistancecomeouttobe lessthantheresistanceofasinglelamp?Thisisacasewherepurely verbalreasoningcanbemisleading.Aresistivecircuitelement,such asthelamentofalightbulb,isneitheraperfectinsulatornor aperfectconductor.Insteadofanalyzingthistypeofcircuitin termsofresistors,"i.e.,partialinsulators,wecouldhavespokenof conductors."Thisexamplewouldthenseemreasonable,sincewe addedmoreconductance,"butonewouldthenhavetheincorrect expectationaboutthecaseofresistorsinseries,discussedinthe followingsection. Perhapsamoreproductivewayofthinkingaboutitistouse mechanicalintuition.Byanalogy,yournostrilsresisttheowof airthroughthem,buthavingtwonostrilsmakesittwiceaseasyto breathe. Threeresistorsinparallelexample2 Whathappensifwehavethreeormoreresistorsinparallel? Thisisanimportantexample,becausethesolutioninvolves animportanttechniqueforunderstandingcircuits:breakingthem downintosmallerpartsandthemsimplifyingthoseparts.Inthe circuitd/1,withthreeresistorsinparallel,wecanthinkoftwo oftheresistorsasformingasingleresistor,d/2,withequivalent resistance R 12 = 1 R 1 + 1 R 2 )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 Wecanthensimplifythecircuitasshownind/3,sothatitcontainsonlytworesistances.Theequivalentresistanceofthewhole circuitisthengivenby R 123 = 1 R 12 + 1 R 3 )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 Substitutingfor R 12 andsimplifying,wendtheresult R 123 = 1 R 1 + 1 R 2 + 1 R 3 )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 whichyouprobablycouldhaveguessed.Theinterestingpoint hereisthedivide-and-conquerconcept,notthemathematicalresult. Anarbitrarynumberofidenticalresistorsinparallelexample3 Whatistheresistanceof N identicalresistorsinparallel? Generalizingtheresultsfortwoandthreeresistors,wehave R N = 1 R 1 + 1 R 2 + ::: )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 Section4.2Parallelresistancesandthejunctionrule 109

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g / Avoltmeterisreallyan ammeterwithaninternalresistor. Whenwemeasurethevoltage differenceacrossaresistor,1,we arereallyconstructingaparallel resistancecircuit,2. where...meansthatthesumincludesalltheresistors.Ifallthe resistorsareidentical,thisbecomes R N = N R )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 = R N Dependenceofresistanceoncross-sectionalareaexample4 Wehavealludedbrieytothefactthatanobject'selectricalresistancedependsonitssizeandshape,butnowweareready tobeginmakingmoremathematicalstatementsaboutit.Assuggestedbyguree,increasingaresistors'scross-sectionalareais equivalenttoaddingmoreresistorsinparallel,whichwillleadto anoveralldecreaseinresistance.Anyrealresistorwithstraight, parallelsidescanbeslicedupintoalargenumberofpieces,each withcross-sectionalareaof,say,1 m 2 .Thenumber, N ,ofsuch slicesisproportionaltothetotalcross-sectionalareaoftheresistor,andbyapplicationoftheresultofthepreviousexamplewe thereforendthattheresistanceofanobjectisinverselyproportionaltoitscross-sectionalarea. f / Afatpipehaslessresistance thanaskinnypipe. Ananalogousrelationshipholdsforwaterpipes,whichiswhy high-owtrunklineshavetohavelargecross-sectionalareas.To makelotsofwatercurrentowthroughaskinnypipe,we'dneed animpracticallylargepressurevoltagedifference. Incorrectreadingsfromavoltmeterexample5 Avoltmeterisreallyjustanammeterwithaninternalresistor,and weuseavoltmeterinparallelwiththethingthatwe'retryingto measurethevoltagedifferenceacross.Thismeansthatanytime wemeasurethevoltagedropacrossaresistor,we'reessentially puttingtworesistorsinparallel.Theammeterinsidethevoltmeter canbeignoredforthepurposeofanalyzingwhathowcurrent owsinthecircuit,sinceitisessentiallyjustsomecoiled-upwire withaverylowresistance. 110 Chapter4Circuits,Part2

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Nowifwearecarryingoutthismeasurementonaresistorthatis partofalargercircuit,wehavechangedthebehaviorofthecircuitthroughouractofmeasuring.Itisasthoughwehadmodied thecircuitbyreplacingtheresistance R withthesmallerequivalentresistanceof R and R v inparallel.Itisforthisreasonthat voltmetersarebuiltwiththelargestpossibleinternalresistance. Asanumericalexample,ifweuseavoltmeterwithaninternal resistanceof1 M tomeasurethevoltagedropacrossaoneohmresistor,theequivalentresistanceis0.999999 ,whichis notdifferentenoughtomakeanydifference.Butifwetriedtouse thesamevoltmetertomeasurethevoltagedropacrossa2 )]TJ/F107 10.9091 Tf 9.808 0 Td [(M resistor,wewouldbereducingtheresistanceofthatpartofthe circuitbyafactorofthree,whichwouldproduceadrasticchange inthebehaviorofthewholecircuit. Thisisthereasonwhyyoucan'tuseavoltmetertomeasurethe voltagedierencebetweentwodierentpointsinmid-air,orbetween theendsofapieceofwood.Thisisbynomeansastupidthingto wanttodo,sincetheworldaroundusisnotaconstant-voltage environment,themostextremeexamplebeingwhenanelectrical stormisbrewing.Butitwillnotworkwithanordinaryvoltmeter becausetheresistanceoftheairorthewoodismanygigaohms.The eectofwavingapairofvoltmeterprobesaroundintheairisthat weprovideareunitingpathforthepositiveandnegativecharges thathavebeenseparated|throughthevoltmeteritself,whichis agoodconductorcomparedtotheair.Thisreducestozerothe voltagedierenceweweretryingtomeasure. Ingeneral,avoltmeterthathasbeensetupwithanopencircuit oraverylargeresistancebetweenitsprobesissaidtobeoating."Anold-fashionedanalogvoltmeterofthetypedescribedhere willreadzerowhenleftoating,thesameaswhenitwassitting ontheshelf.Aoatingdigitalvoltmeterusuallyshowsanerror message. Section4.2Parallelresistancesandthejunctionrule 111

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h / 1.Abatterydrivescurrent throughtworesistorsinseries.2. Therearethreeconstant-voltage regions.3.Thethreevoltage differencesarerelated.4.If themetercrab-walksaroundthe circuitwithoutippingoveror crossingitslegs,theresulting voltageshaveplusandminus signsthatmakethemaddupto zero. 4.3Seriesresistances Thetwobasiccircuitlayoutsareparallelandseries,soapairof resistorsinseries,h/1,isanotherofthemostbasiccircuitswecan make.Byconservationofcharge,allthecurrentthatowsthrough oneresistormustalsoowthroughtheotheraswellasthroughthe battery: I 1 = I 2 Theonlywaytheinformationaboutthetworesistancevaluesis goingtobeusefulisifwecanapplyOhm'slaw,whichwillrelatethe resistanceofeachresistortothecurrentowingthroughitandthe voltagedierenceacrossit.Figureh/2showsthethreeconstantvoltageareas.Voltagedierencesaremorephysicallysignicant thanvoltages,sowedenesymbolsforthevoltagedierencesacross thetworesistorsingureh/3. Wehavethreeconstant-voltageareas,withsymbolsforthedifferenceinvoltagebetweeneverypossiblepairofthem.Thesethree voltagedierencesmustberelatedtoeachother.ItisasthoughI tellyouthatFredisafoottallerthanGinger,Gingerisafoottaller thanSally,andFredistwofeettallerthanSally.Theinformation isredundant,andyoureallyonlyneededtwoofthethreepiecesof datatoinferthethird.Inthecaseofourvoltagedierences,we have j V 1 j + j V 2 j = j V battery j Theabsolutevaluesignsarebecauseoftheambiguityinhowwe deneourvoltagedierences.Ifwereversedthetwoprobesofthe voltmeter,wewouldgetaresultwiththeoppositesign.Digital voltmeterswillactuallyprovideaminussignonthescreenifthe wireconnectedtotheV"plugislowerinvoltagethantheone connectedtotheCOM"plug.Analogvoltmeterspintheneedle againstapegifyoutrytousethemtomeasurenegativevoltages, soyouhavetoddletogettheleadsconnectedtherightway,and thensupplyanynecessaryminussignyourself. Figureh/4showsastandardwayoftakingcareoftheambiguity insigns.Foreachofthethreevoltagemeasurementsaroundthe loop,wekeepthesameprobethedarkeroneontheclockwise side.Itisasthoughthevoltmeterwassidlingaroundthecircuit likeacrab,withoutevercrossingitslegs."Withthisconvention, therelationshipamongthevoltagedropsbecomes V 1 + V 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [( V battery or,inmoresymmetricalform, V 1 + V 2 + V battery =0. Moregenerally,thisisknownastheloopruleforanalyzingcircuits: 112 Chapter4Circuits,Part2

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i / Example6. thelooprule Assumingthestandardconventionforplusandminussigns, thesumofthevoltagedropsaroundanyclosedloopina circuitmustbezero. Lookingforanexceptiontothelooprulewouldbelikeasking forahikethatwouldbedownhillallthewayandthatwouldcome backtoitsstartingpoint! Forthecircuitwesetouttoanalyze,theequation V 1 + V 2 + V battery =0 cannowberewrittenbyapplyingOhm'slawtoeachresistor: I 1 R 1 + I 2 R 2 + V battery =0. Thecurrentsarethesame,sowecanfactorthemout: I R 1 + R 2 + V battery =0, andthisisthesameresultwewouldhavegottenifwehadbeen analyzingaone-resistorcircuitwithresistance R 1 + R 2 .Thusthe equivalentresistanceofresistorsinseriesequalsthesumoftheir resistances. Twolightbulbsinseriesexample6 Iftwoidenticallightbulbsareplacedinseries,howdotheir brightnessescomparewiththebrightnessofasinglebulb? Takenasawhole,thepairofbulbsactlikeadoubledresistance, sotheywilldrawhalfasmuchcurrentfromthewall.Eachbulb willbedimmerthanasinglebulbwouldhavebeen. Thetotalpowerdissipatedbythecircuitis I V .Thevoltagedrop acrossthewholecircuitisthesameasbefore,butthecurrentis halved,sothetwo-bulbcircuitdrawshalfasmuchtotalpoweras theone-bulbcircuit.Eachbulbdrawsone-quarterofthenormal power. Roughlyspeaking,wemightexpectthistoresultinonequarter thelightbeingproducedbyeachbulb,butinrealitylightbulbs wastequiteahighpercentageoftheirpowerintheformofheat andwavelengthsoflightthatarenotvisibleinfraredandultraviolet.Lesslightwillbeproduced,butit'shardtopredictexactly howmuchless,sincetheefciencyofthebulbswillbechanged byoperatingthemunderdifferentconditions. Morethantwoequalresistancesinseriesexample7 Bystraightforwardapplicationofthedivide-and-conquertechnique discussedintheprevioussection,wendthattheequivalentresistanceof N identicalresistances R inserieswillbe NR Section4.3Seriesresistances 113

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j / Doublingthelengthofa resistorislikeputtingtworesistorsinseries.Theresistanceis doubled. Dependenceofresistanceonlengthexample8 Intheprevioussection,weprovedthatresistanceisinversely proportionaltocross-sectionalarea.Byequivalentreasonabout resistancesinseries,wendthatresistanceisproportionalto length.Analogously,itishardertoblowthroughalongstrawthan throughashortone. Puttingthetwoargumentstogether,wendthattheresistance ofanobjectwithstraight,parallelsidesisgivenby R =constant L=A Theproportionalityconstantiscalledtheresistivity,anditdepends onlyonthesubstanceofwhichtheobjectismade.Aresistivity measurementcouldbeused,forinstance,tohelpidentifyasample ofanunknownsubstance. Choiceofhighvoltageforpowerlinesexample9 ThomasEdisongotinvolvedinafamoustechnologicalcontroversyoverthevoltagedifferencethatshouldbeusedforelectrical powerlines.Atthistime,thepublicwasunfamiliarwithelectricity, andeasilyscaredbyit.ThepresidentoftheUnitedStates,for instance,refusedtohaveelectricallightingintheWhiteHouse whenitrstbecamecommerciallyavailablebecauseheconsidereditunsafe,preferringtheknownrehazardofoillampsto themysteriousdangersofelectricity.Mainlyasawaytoovercomepublicfear,Edisonbelievedthatpowershouldbetransmittedusingsmallvoltages,andhepublicizedhisopinionbygiving demonstrationsatwhichadogwasluredintopositiontobekilled byalargevoltagedifferencebetweentwosheetsofmetalonthe ground.Edison'sopponentsalsoadvocatedalternatingcurrent ratherthandirectcurrent,andACismoredangerousthanDCas well.Aswewilldiscusslater,ACcanbeeasilysteppedupand downtothedesiredvoltagelevelusingadevicecalledatransformer. Nowifwewanttodeliveracertainamountofpower P L toa loadsuchasanelectriclightbulb,weareconstrainedonlybythe equation P L = I V L .Wecandeliveranyamountofpowerwe wish,evenwithalowvoltage,ifwearewillingtouselargecurrents.Modernelectricaldistributionnetworks,however,usedangerouslyhighvoltagedifferencesoftensofthousandsofvolts. WhydidEdisonlosethedebate? Itboilsdowntomoney.Theelectriccompanymustdeliverthe amountofpower P L desiredbythecustomerthroughatransmissionlinewhoseresistance R T isxedbyeconomicsandgeography.Thesamecurrentowsthroughboththeloadandthetransmissionline,dissipatingpowerusefullyintheformerandwaste114 Chapter4Circuits,Part2

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fullyinthelatter.Theefciencyofthesystemis efciency= powerpaidforbythecustomer powerpaidforbytheutility = P L P L + P T = 1 1+ P T = P L Puttingourselvesintheshoesoftheelectriccompany,wewish togetridofthevariable P T ,sinceitissomethingwecontrolonly indirectlybyourchoiceof V T and I .Substituting P T = I V T ,we nd efciency= 1 1+ I V T P L Weassumethetransmissionlinebutnotnecessarilytheloadis ohmic,sosubstituting V T = IR T gives efciency = 1 1+ I 2 R T P L Thisquantitycanclearlybemaximizedbymaking I assmallas possible,sincewewillthenbedividingbythesmallestpossible quantityonthebottomofthefraction.Alow-currentcircuitcan onlydeliversignicantamountsofpowerifituseshighvoltages, whichiswhyelectricaltransmissionsystemsusedangeroushigh voltages. k / Example10. Acomplicatedcircuitexample10 Allsevenresistorsintheleft-handpanelofgurekareidentical.Initially,theswitchSisopenasshowninthegure,andthe currentthroughresistorAis I o .Theswitchisthenclosed.Find thecurrentthroughresistorB,aftertheswitchisclosed,interms of I o Section4.3Seriesresistances 115

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. Thesecondpanelshowsthecircuitredrawnforsimplicity,inthe initialconditionwiththeswitchopen.Whentheswitchisopen,no currentcanowthroughthecentralresistor,sowemayaswell ignoreit.I'vealsoredrawnthejunctions,withoutchangingwhat's connectedtowhat.Thisisthekindofmentalrearrangingthat you'lleventuallylearntodoautomaticallyfromexperiencewith analyzingcircuits.Theredrawnversionmakesiteasiertosee what'shappeningwiththecurrent.Chargeisconserved,soany chargethatowspastpoint1inthecircuitmustalsoowpast points2and3.Thiswouldhavebeenhardertoreasonaboutby applyingthejunctionruletotheoriginalversion,whichappears tohavenineseparatejunctions. Inthenewversion,it'salsoclearthatthecircuithasagreatdeal ofsymmetry.Wecouldipovereachparallelpairofidenticalresistorswithoutchangingwhat'sconnectedtowhat,sothatmakes itclearthatthevoltagedropsandcurrentsmustbeequalforthe membersofeachpair.Wecanalsoprovethisbyusingtheloop rule.Thelooprulesaysthatthetwovoltagedropsinloop4must beequal,andsimilarlyforloops5and6.Sincetheresistorsobey Ohm'slaw,equalvoltagedropsacrossthemalsoimplyequalcurrents.Thatmeansthatwhenthecurrentatpoint1comestothe topjunction,exactlyhalfofitgoesthrougheachresistor.Then thecurrentreunitesat2,splitsbetweenthenextpair,andsoon. Weconcludethateachofthesixresistorsinthecircuitexperiencesthesamevoltagedropandthesamecurrent.Applyingthe loopruletoloop7,wendthatthesumofthethreevoltagedrops acrossthethreeleft-handresistorsequalsthebattery'svoltage, V ,soeachresistorinthecircuitexperiencesavoltagedrop V = 3. Letting R standfortheresistanceofoneoftheresistors,wend thatthecurrentthroughresistorB,whichisthesameasthecurrentsthroughalltheothers,isgivenby I o = V = 3 R Wenowpasstothecasewheretheswitchisclosed,asshown inthethirdpanel.Thebattery'svoltageisthesameasbefore, andeachresistor'sresistanceisthesame,sowecanstillusethe samesymbols V and R forthem.Itisnolongertrue,however, thateachresistorfeelsavoltagedrop V = 3.Theequivalentresistanceofthewholecircuitis R = 2+ R = 3+ R = 2=4 R = 3,sothetotal currentdrawnfromthebatteryis3 V = 4 R .Inthemiddlegroup ofresistors,thiscurrentissplitthreeways,sothenewcurrent throughBis = 3 V = 4 R = V = 4 R =3 I o = 4. Interpretingthisresult,weseethatitcomesfromtwoeffectsthat partiallycancel.Closingtheswitchreducestheequivalentresistanceofthecircuitbygivingchargeanotherwaytoow,and increasestheamountofcurrentdrawnfromthebattery.Resistor B,however,onlygetsa1/3shareofthisgreatercurrent,not1/2. Thesecondeffectturnsouttobebiggerthanthesecondeffect, andthereforethecurrentthroughresistorBislessenedoverall. 116 Chapter4Circuits,Part2

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Gettingkilledbyyourammeterexample11 Aswithavoltmeter,anammetercangiveerroneousreadingsifit isusedinsuchawaythatitchangesthebehaviorthecircuit.An ammeterisusedinseries,soifitisusedtomeasurethecurrent througharesistor,theresistor'svaluewilleffectivelybechanged to R + R a ,where R a istheresistanceoftheammeter.Ammeters aredesignedwithverylowresistancesinordertomakeitunlikely that R + R a willbesignicantlydifferentfrom R Infact,therealhazardisdeath,notawrongreading!Virtually theonlycircuitswhoseresistancesaresignicantlylessthanthat ofanammeterarethosedesignedtocarryhugecurrents.An ammeterinsertedinsuchacircuitcaneasilymelt.WhenIwas workingatalaboratoryfundedbytheDepartmentofEnergy,we gotperiodicbulletinsfromtheDOEsafetyofceaboutseriousaccidentsatothersites,andtheyheldacertainghoulishfascination. OneofthesewasaboutaDOEworkerwhowascompletelyincineratedbytheexplosioncreatedwhenheinsertedanordinary RadioShackammeterintoahigh-currentcircuit.Laterestimates showedthattheheatwasprobablysointensethattheexplosion wasaballofplasmaagassohotthatitsatomshavebeen ionized. DiscussionQuestions A Wehavestatedtheloopruleinasymmetricformwhereaseries ofvoltagedropsaddsuptozero.Todothis,wehadtodeneastandard wayofconnectingthevoltmetertothecircuitsothattheplusandminus signswouldcomeoutright.Supposewewishtorestatethejunctionrule inasimilarsymmetricway,sothatinsteadofequatingthecurrentcoming intothecurrentgoingout,itsimplystatesthatacertainsumofcurrentsat ajunctionaddsuptozero.Whatstandardwayofinsertingtheammeter wouldwehavetousetomakethiswork? Section4.3Seriesresistances 117

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Summary Aschematicisadrawingofacircuitthatstandardizesandstylizes itsfeaturestomakeiteasiertounderstand.Anycircuitcanbe brokendownintosmallerparts.Forinstance,onebigcircuitmaybe understoodastwosmallcircuitsinseries,anotherasthreecircuits inparallel.Whencircuitelementsarecombinedinparallelandin series,wehavetwobasicrulestoguideusinunderstandinghowthe partsfunctionasawhole: thejunctionrule: Inanycircuitthatisnotstoringorreleasingcharge,conservationofchargeimpliesthatthetotal currentowingoutofanyjunctionmustbethesameasthe totalowingin. thelooprule: Assumingthestandardconventionforplus andminussigns,thesumofthevoltagedropsaroundany closedloopinacircuitmustbezero. Thesimplestapplicationoftheserulesistopairsofresistors combinedinseriesorparallel.Insuchcases,thepairofresistors actsjustlikeasingleunitwithacertainresistancevalue,called theirequivalentresistance.Resistancesinseriesaddtoproducea largerequivalentresistance, R series = R 1 + R 2 becausethecurrenthastoghtitswaythroughbothresistances. Parallelresistorscombinetoproduceanequivalentresistancethat issmallerthaneitherindividualresistance, R parallel = 1 R 1 + 1 R 2 )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 becausethecurrenthastwodierentpathsopentoit. Animportantexampleofresistancesinparallelandseriesisthe useofvoltmetersandammetersinresistivecircuits.Avoltmeter actsasalargeresistanceinparallelwiththeresistoracrosswhich thevoltagedropisbeingmeasured.Thefactthatitsresistance isnotinnitemeansthatitaltersthecircuititisbeingusedto investigate,producingalowerequivalentresistance.Anammeter actsasasmallresistanceinserieswiththecircuitthroughwhichthe currentistobedetermined.Itsresistanceisnotquitezero,which leadstoanincreaseintheresistanceofthecircuitbeingtested. 118 Chapter4Circuits,Part2

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Problem5. Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 aManybattery-operateddevicestakemorethanonebattery. Ifyoulookcloselyinthebatterycompartment,youwillseethatthe batteriesarewiredinseries.Consideraashlightcircuit.Whatdoes theloopruletellyouabouttheeectofputtingseveralbatteriesin seriesinthisway? bThecellsofanelectriceel'snervoussystemarenotthatdierent fromours|eachcellcandevelopavoltagedierenceacrossitof somewhereontheorderofonevolt.How,then,doyouthinkan electriceelcancreatevoltagesofthousandsofvoltsbetweendierent partsofitsbody? 2 Theheatingelementofanelectricstoveisconnectedinseries withaswitchthatopensandclosesmanytimespersecond.When youturntheknobupformorepower,thefractionofthetimethat theswitchisclosedincreases.Supposesomeonesuggestsasimpler alternativeforcontrollingthepowerbyputtingtheheatingelement inserieswithavariableresistorcontrolledbytheknob.Withthe knobturnedallthewayclockwise,thevariableresistor'sresistanceis nearlyzero,andwhenit'sallthewaycounterclockwise,itsresistance isessentiallyinnite.aDrawschematics.bWhywouldthe simplerdesignbeundesirable? 3 A1.0toasteranda2.0lampareconnectedinparallelwith the110-Vsupplyofyourhouse.Ignorethefactthatthevoltageis ACratherthanDC. aDrawaschematicofthecircuit. bForeachofthethreecomponentsinthecircuit,ndthecurrent passingthroughitandthevoltagedropacrossit. cSupposetheywereinsteadhookedupinseries.Drawaschematic andcalculatethesamethings. 4 Wireissoldinaseriesofstandarddiameters,calledgauges." Thedierenceindiameterbetweenonegaugeandthenextinthe seriesisabout20%.Howwouldtheresistanceofagivenlengthof wirecomparewiththeresistanceofthesamelengthofwireinthe nextgaugeintheseries? p 5 Thegureshowstwopossiblewaysofwiringaashlightwith aswitch.Bothwillservetoturnthebulbonando,althoughthe switchfunctionsintheoppositesense.Whyismethod1preferable? Problems 119

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Problem8. Problem6. 6 Inthegure,thebatteryis9V. aWhatarethevoltagedierencesacrosseachlightbulb? bWhatcurrentowsthrougheachofthethreecomponentsofthe circuit? cIfanewwireisaddedtoconnectpointsAandB,howwillthe appearancesofthebulbschange?Whatwillbethenewvoltages andcurrents? dSupposenowireisconnectedfromAtoB,butthetwobulbs areswitched.Howwilltheresultscomparewiththeresultsfrom theoriginalsetupasdrawn? 7 Youhaveacircuitconsistingoftwounknownresistorsinseries, andasecondcircuitconsistingoftwounknownresistorsinparallel. aWhat,ifanything,wouldyoulearnabouttheresistorsinthe seriescircuitbyndingthatthecurrentsthroughthemwereequal? bWhatifyoufoundoutthevoltagedierencesacrosstheresistors intheseriescircuitwereequal? cWhatwouldyoulearnabouttheresistorsintheparallelcircuit fromknowingthatthecurrentswereequal? dWhatifthevoltagesintheparallelcircuitwereequal? 8 Astudentinabiologylabisgiventhefollowinginstructions:ConnectthecerebraleraserC.E.andtheneuraldepolarizerN.D.inparallelwiththepowersupplyP.S..Underno circumstancesshouldyoueverallowthecerebralerasertocome within20cmofyourhead.Connectavoltmetertomeasurethe voltageacrossthecerebraleraser,andalsoinsertanammeterin thecircuitsothatyoucanmakesureyoudon'tputmorethan100 mAthroughtheneuraldepolarizer."Thediagramsshowtwolab groups'attemptstofollowtheinstructions.aTranslatediagram aintoastandard-styleschematic.Whatiscorrectandincorrect aboutthisgroup'ssetup?bDothesamefordiagramb. 9 Howmanydierentresistancevaluescanbecreatedbycombiningthreeunequalresistors?Don'tcountpossibilitieswherenot alltheresistorsareused. 10 Apersoninaruralareawhohasnoelectricityrunsan extremelylongextensioncordtoafriend'shousedowntheroadso shecanrunanelectriclight.Thecordissolongthatitsresistance, x ,isnotnegligible.Showthatthelamp'sbrightnessisgreatestif itsresistance, y ,isequalto x .Explainphysicallywhythelampis dimforvaluesof y thataretoosmallortoolarge. R 11 Whatresistancevaluescanbecreatedbycombininga1k resistoranda10kresistor? Solution,p.205 120 Chapter4Circuits,Part2

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Problems13and14. Problem15. Problem16. Problem17. 12 Supposesixidenticalresistors,eachwithresistance R ,are connectedsothattheyformtheedgesofatetrahedronapyramid withthreesidesinadditiontothebase,i.e.,onelesssidethanan Egyptianpyramid.Whatresistancevalueorvaluescanbeobtained bymakingconnectionsontoanytwopointsonthisarrangement? Solution,p.205 ? 13 Thegureshowsacircuitcontainingvelightbulbsconnectedtoabattery.Supposeyou'regoingtoconnectoneprobeofa voltmetertothecircuitatthepointmarkedwithadot.Howmany unique,nonzerovoltagedierencescouldyoumeasurebyconnecting theotherprobetootherwiresinthecircuit? 14 Thelightbulbsinthegureareallidentical.Ifyouwere insertinganammeteratvariousplacesinthecircuit,howmany uniquecurrentscouldyoumeasure?Ifyouknowthatthecurrent measurementwillgivethesamenumberinmorethanoneplace, onlycountthatasoneuniquecurrent. 15 Thebulbsareallidentical.Whichonedoesn'tlightup? ? 16 Eachbulbhasaresistanceofoneohm.Howmuchpoweris drawnfromtheone-voltbattery? ? 17 Thebulbsallhaveunequalresistances.Giventhethree currentsshowninthegure,ndthecurrentsthroughbulbsA,B, C,andD. Problems 121

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122 Chapter4Circuits,Part2

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Chapter5 FieldsofForce Okay.Yourdutiesareasfollows:GetBreen.Idon'tcarehowyouget him,butgethimsoon.Thatfaker!Heposedfortwentyyearsasascientist withouteverbeingapprehended.Well,I'mgoingtodosomeapprehendingthat'llmakeallpreviousapprehendinglooklikenoapprehensionatall. Youwithme? Yes,saidBattle,verymuchconfused.What'sthatthingyouhave? Piggy-backheat-ray.Youtransposetheairinitspathintoanunstable isotopewhichtendstocarryallenergyasheat.Thenyoushootyourjuice light,orwhateveralongtheisotopicpathandyouburnwhatever'sonthe receivingend.Youwantafew? No,saidBattle.Ihavemygats.Whatelsehaveyougotforoffense anddefense?Underbottamopenedacabinetandproudlywavedan arm.Everything,hesaid. Disintegraters,heat-rays,bombsofeverytype.Andimpenetrable shieldsofenergy,massiveandportable.WhatmoredoIneed? FromTHEREVERSIBLEREVOLUTIONSbyCecilCorwin,Cosmic Stories,March1941.ArtbyMorey,Bok,Kyle,Hunt,Forte.Copyright expired. Cutting-edgesciencereadilyinltratespopularculture,though sometimesingarbledform.TheNewtonianimaginationpopulated theuniversemostlywiththatnicesolidstucalledmatter,which wasmadeoflittlehardballscalledatoms.Intheearlytwentieth century,consumersofpulpctionandpopularizedsciencebegan tohearofanewimageoftheuniverse,fullofx-rays,N-rays,and Hertzianwaves.Whattheywerebeginningtosoakupthrough theirskinswasadrasticrevisionofNewton'sconceptofauniverse madeofchunksofmatterwhichhappenedtointeractviaforces.In thenewlyemergingpicture,theuniversewas made offorce,or,to bemoretechnicallyaccurate,ofripplesinuniversaleldsofforce. UnliketheaveragereaderofCosmicStoriesin1941,younowpossess enoughtechnicalbackgroundtounderstandwhataforceeld" reallyis. 5.1Whyelds? Timedelaysinforcesexertedatadistance Whatconvincedphysiciststhattheyneededthisnewconceptof aeldofforce?Althoughwehavebeendealingmostlywithelec123

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a / Abarmagnet'satomsare partiallyaligned. b / Abarmagnetinteracts withourmagneticplanet. c / Magnetsalignednorth-south. tricalforces,let'sstartwithamagneticexample.Infactthemain reasonI'vedelayedadetaileddiscussionofmagnetismforsolong isthatmathematicalcalculationsofmagneticeectsarehandled muchmoreeasilywiththeconceptofaeldofforce.Firstalittle backgroundleadinguptoourexample.Abarmagnet,a,hasanaxis aboutwhichmanyoftheelectrons'orbitsareoriented.Theearth itselfisalsoamagnet,althoughnotabar-shapedone.Theinteractionbetweentheearth-magnetandthebarmagnet,b,makesthem wanttolineuptheiraxesinopposingdirectionsinotherwords suchthattheirelectronsrotateinparallelplanes,butwithoneset rotatingclockwiseandtheothercounterclockwiseasseenlooking alongtheaxes.Onasmallerscale,anytwobarmagnetsplaced neareachotherwilltrytoalignthemselveshead-to-tail,c. Nowwegettotherelevantexample.Itisclearthattwopeople separatedbyapaper-thinwallcoulduseapairofbarmagnetsto signaltoeachother.Eachpersonwouldfeelherownmagnettrying totwistaroundinresponsetoanyrotationperformedbytheother person'smagnet.Thepracticalrangeofcommunicationwouldbe veryshortforthissetup,butasensitiveelectricalapparatuscould pickupmagneticsignalsfrommuchfartheraway.Infact,thisis notsodierentfromwhataradiodoes:theelectronsracingup anddownthetransmittingantennacreateforcesontheelectrons inthedistantreceivingantenna.Bothmagneticandelectricforces areinvolvedinrealradiosignals,butwedon'tneedtoworryabout thatyet. Aquestionnownaturallyarisesastowhetherthereisanytime delayinthiskindofcommunicationviamagneticandelectric forces.Newtonwouldhavethoughtnot,sinceheconceivedof physicsintermsofinstantaneousactionatadistance.Wenow know,however,thatthereissuchatimedelay.Ifyoumakea long-distancephonecallthatisroutedthroughacommunications satellite,youshouldeasilybeabletodetectadelayofabouthalfa secondoverthesignal'sroundtripof50,000miles.Modernmeasurementshaveshownthatelectric,magnetic,andgravitationalforces alltravelatthespeedoflight,3 10 8 m/s. 1 Infact,wewillsoon discusshowlightitselfismadeofelectricityandmagnetism. Ifittakessometimeforforcestobetransmittedthroughspace, thenapparentlythereissome thing thattravels through space.The factthatthephenomenontravelsoutwardatthesamespeedinall directionsstronglyevokeswavemetaphorssuchasripplesonapond. Moreevidencethateldsofforcearereal:theycarryenergy. Thesmoking-gunargumentforthisstrangenotionoftraveling forceripplescomesfromthefactthattheycarryenergy. 1 Asdiscussedinbook6ofthisseries,oneconsequenceofEinstein'stheoryof relativityisthatmaterialobjectscannevermovefasterthanthespeedoflight. Itcanalsobeshownthatsignalsorinformationaresubjecttothesamelimit. 124 Chapter5FieldsofForce

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f / Thewindpatternsina certainareaoftheoceancould bechartedinaseaofarrows representationlikethis.Each arrowrepresentsboththewind's strengthanditsdirectionata certainlocation. d / Thesecondmagnetisreversed. e / Bothmagnetsarereversed. Firstsupposethatthepersonholdingthebarmagnetonthe rightdecidestoreversehers,resultingincongurationd.Shehad todomechanicalworktotwistit,andifshereleasesthemagnet, energywillbereleasedasitipsbacktoc.Shehasapparentlystored energybygoingfromctod.Sofareverythingiseasilyexplained withouttheconceptofaeldofforce. Butnowimaginethatthetwopeoplestartinpositioncand thensimultaneouslyiptheirmagnetsextremelyquicklytoposition e,keepingthemlinedupwitheachotherthewholetime.Imagine, forthesakeofargument,thattheycandothissoquicklythat eachmagnetisreversedwhiletheforcesignalfromtheotheris stillintransit.Foramorerealisticexample,we'dhavetohave tworadioantennas,nottwomagnets,butthemagnetsareeasier tovisualize.Duringtheipping,eachmagnetisstillfeelingthe forcesarisingfromthewaytheothermagnet used tobeoriented. Eventhoughthetwomagnetsstayalignedduringtheip,thetime delaycauseseachpersontofeelresistanceasshetwistshermagnet around.Howcanthisbe?Bothofthemareapparentlydoing mechanicalwork,sotheymustbestoringmagneticenergysomehow. ButinthetraditionalNewtonianconceptionofmatterinteracting viainstantaneousforcesatadistance,interactionenergyarisesfrom therelativepositionsofobjectsthatareinteractingviaforces.If themagnetsneverchangedtheirorientationsrelativetoeachother, howcananymagneticenergyhavebeenstored? Theonlypossibleansweristhattheenergymusthavegone intothemagneticforceripplescrisscrossingthespacebetweenthe magnets.Fieldsofforceapparentlycarryenergyacrossspace,which isstrongevidencethattheyarerealthings. Thisisperhapsnotasradicalanideatousasitwastoour ancestors.Weareusedtotheideathataradiotransmittingantenna consumesagreatdealofpower,andsomehowspewsitoutintothe universe.Apersonworkingaroundsuchanantennaneedstobe carefulnottogettooclosetoit,sinceallthatenergycaneasily cookeshapainfulphenomenonknownasanRFburn". 5.2Thegravitationaleld Giventhateldsofforcearereal,howdowedene,measure, andcalculatethem?Afruitfulmetaphorwillbethewindpatterns experiencedbyasailingship.Wherevertheshipgoes,itwillfeela certainamountofforcefromthewind,andthatforcewillbeina certaindirection.Theweatherisever-changing,ofcourse,butfor nowlet'sjustimaginesteadywindpatterns.Denitionsinphysics areoperational,i.e.,theydescribehowtomeasurethethingbeing dened.Theship'scaptaincanmeasurethewind'seldofforce" bygoingtothelocationofinterestanddeterminingboththedirectionofthewindandthestrengthwithwhichitisblowing.Charting Section5.2Thegravitationaleld 125

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allthesemeasurementsonamapleadstoadepictionoftheeldof windforceliketheoneshowninthegure.Thisisknownasthe seaofarrows"methodofvisualizingaeld. Nowlet'sseehowtheseconceptsareappliedtothefundamental forceeldsoftheuniverse.We'llstartwiththegravitationaleld, whichistheeasiesttounderstand.Aswiththewindpatterns, we'llstartbyimagininggravityasastaticeld,eventhoughthe existenceofthetidesprovesthattherearecontinualchangesinthe gravityeldinourregionofspace.Deningthedirectionofthe gravitationaleldiseasyenough:wesimplygotothelocationof interestandmeasurethedirectionofthegravitationalforceonan object,suchasaweighttiedtotheendofastring. Buthowshouldwedenethestrengthofthegravitationaleld? Gravitationalforcesareweakeronthemoonthanontheearth,but wecannotspecifythestrengthofgravitysimplybygivingacertain numberofnewtons.Thenumberofnewtonsofgravitationalforce dependsnotjustonthestrengthofthelocalgravitationaleldbut alsoonthemassoftheobjectonwhichwe'retestinggravity,our testmass."Aboulderonthemoonfeelsastrongergravitational forcethanapebbleontheearth.Wecangetaroundthisproblem bydeningthestrengthofthegravitationaleldastheforceacting onanobject, dividedbytheobject'smass denitionofthegravitationaleld Thegravitationaleldvector, g ,atanylocationinspaceis foundbyplacingatestmass m t atthatpoint.Theeldvector isthengivenby g = F =m t ,where F isthegravitationalforce onthetestmass. Themagnitudeofthegravitationaleldnearthesurfaceofthe earthisabout9.8N/kg,andit'snocoincidencethatthisnumber looksfamiliar,orthatthesymbol g isthesameastheonefor gravitationalacceleration.Theforceofgravityonatestmasswill equal m t g ,where g isthegravitationalacceleration.Dividingby m t simplygivesthegravitationalacceleration.Whydeneanew nameandnewunitsforthesameoldquantity?Themainreasonis thatitpreparesuswiththerightapproachfordeningotherelds. Themostsubtlepointaboutallthisisthatthegravitational eldtellsusaboutwhatforces would beexertedonatestmassby theearth,sun,moon,andtherestoftheuniverse, if weinserteda testmassatthepointinquestion.Theeldstillexistsatallthe placeswherewedidn'tmeasureit. Gravitationaleldoftheearthexample1 Whatisthemagnitudeoftheearth'sgravitationaleld,interms ofitsmass, M ,andthedistance r fromitscenter? Substituting j F j = GMm t = r 2 intothedenitionofthegravitational eld,wend j g j = GM = r 2 .Thisexpressioncouldbeusedfor 126 Chapter5FieldsofForce

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g / Thegravitationaleldsurroundingaclumpofmasssuch astheearth. h / Thegravitationaleldsof theearthandmoonsuperpose. Notehowtheeldscancelatone point,andhowthereisnoboundarybetweentheinterpenetrating eldssurroundingthetwobodies. theeldofanysphericallysymmetricmassdistribution,sincethe equationweassumedforthegravitationalforcewouldapplyin anysuchcase. Sourcesandsinks Ifwemakeasea-of-arrowspictureofthegravitationalelds surroundingtheearth,g,theresultisevocativeofwatergoingdown adrain.Forthisreason,anythingthatcreatesaninward-pointing eldarounditselfiscalledasink.Theearthisagravitationalsink. Thetermsource"canreferspecicallytothingsthatmakeoutward elds,oritcanbeusedasamoregeneraltermforbothouties" andinnies."Howeverconfusingtheterminology,weknowthat gravitationaleldsareonlyattractive,sowewillneverndaregion ofspacewithanoutward-pointingeldpattern. Knowledgeoftheeldisinterchangeablewithknowledgeofits sourcesatleastinthecaseofastatic,unchangingeld.Ifaliens sawtheearth'sgravitationaleldpatterntheycouldimmediately infertheexistenceoftheplanet,andconverselyiftheyknewthe massoftheearththeycouldpredictitsinuenceonthesurrounding gravitationaleld. Superpositionofelds Averyimportantfactaboutalleldsofforceisthatwhenthere ismorethanonesourceorsink,theeldsaddaccordingtothe rulesofvectoraddition.Thegravitationaleldcertainlywillhave thisproperty,sinceitisdenedintermsoftheforceonatest mass,andforcesaddlikevectors.Superpositionisanimportant characteristicsofwaves,sothesuperpositionpropertyofeldsis consistentwiththeideathatdisturbancescanpropagateoutward aswavesinaeld. ReductioningravityonIoduetoJupiter'sgravityexample2 TheaveragegravitationaleldonJupiter'smoonIois1.81N/kg. ByhowmuchisthisreducedwhenJupiterisdirectlyoverhead? Io'sorbithasaradiusof4.22 10 8 m,andJupiter'smassis 1.899 10 27 kg. Bytheshelltheorem,wecantreattheJupiterasifitsmasswas allconcentratedatitscenter,andlikewiseforIo.IfwevisitIoand landatthepointwhereJupiterisoverhead,weareonthesame lineasthesetwocenters,sothewholeproblemcanbetreated one-dimensionally,andvectoradditionisjustlikescalaraddition. Let'susepositivenumbersfordownwardeldstowardthecenter ofIoandnegativeforupwardones.Pluggingtheappropriate dataintotheexpressionderivedinexample1,wendthatthe Jupiter'scontributiontotheeldis )]TJ/F39 10.9091 Tf 8.485 0 Td [(0.71N/kg.Superposition saysthatwecanndtheactualgravitationaleldbyaddingup theeldscreatedbyIoandJupiter:1.81 )]TJ/F39 10.9091 Tf 10.961 0 Td [(0.71N/kg=1.1N/kg. Youmightthinkthatthisreductionwouldcreatesomespectacular Section5.2Thegravitationaleld 127

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effects,andmakeIoanexcitingtouristdestination.Actuallyyou wouldnotdetectanydifferenceifyouewfromonesideofIo totheother.ThisisbecauseyourbodyandIobothexperience Jupiter'sgravity,soyoufollowthesameorbitalcurvethroughthe spacearoundJupiter. Gravitationalwaves Asourcethatsitsstillwillcreateastaticeldpattern,likeasteel ballsittingpeacefullyonasheetofrubber.Amovingsourcewillcreateaspreadingwavepatternintheeld,likeabugthrashingonthe surfaceofapond.Althoughwehavestartedwiththegravitational eldasthesimplestexampleofastaticeld,starsandplanetsdo morestatelyglidingthanthrashing,sogravitationalwavesarenot easytodetect.Newton'stheoryofgravitydoesnotdescribegravitationalwaves,buttheyarepredictedbyEinstein'sgeneraltheory ofrelativity.J.H.TaylorandR.A.HulsewereawardedtheNobel Prizein1993forgivingindirectevidencethatEinstein'swavesactuallyexist.Theydiscoveredapairofexotic,ultra-densestarscalled neutronstarsorbitingoneanotherveryclosely,andshowedthat theywerelosingorbitalenergyattheratepredictedbyEinstein's theory. i / ThepartoftheLIGOgravitywavedetectoratHanfordNuclearReservation,nearRichland, Washington.Theotherhalfofthe detectorisinLouisiana. ACaltech-MITcollaborationhasbuiltapairofgravitational wavedetectorscalledLIGOtosearchformoredirectevidenceof gravitationalwaves.Sincetheyareessentiallythemostsensitive vibrationdetectorsevermade,theyarelocatedinquietruralareas, andsignalswillbecomparedbetweenthemtomakesurethatthey werenotduetopassingtrucks.Theprojectbeganoperatingatfull 128 Chapter5FieldsofForce

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j / Example3. sensitivityin2005,andisnowabletodetectavibrationthatcauses achangeof10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(18 minthedistancebetweenthemirrorsattheends ofthe4-kmvacuumtunnels.Thisisathousandtimeslessthanthe sizeofanatomicnucleus!Thereisonlyenoughfundingtokeepthe detectorsoperatingforafewmoreyears,sothephysicistscanonly hopethatduringthattime,somewhereintheuniverse,asuciently violentcataclysmwilloccurtomakeadetectablegravitationalwave. Moreaccurately,theywantthewavetoarriveinoursolarsystem duringthattime,althoughitwillhavebeenproducedmillionsof yearsbefore. 5.3Theelectriceld Denition Thedenitionoftheelectriceldisdirectlyanalogousto,and hasthesamemotivationas,thedenitionofthegravitationaleld: denitionoftheelectriceld Theelectriceldvector, E ,atanylocationinspaceisfound byplacingatestcharge q t atthatpoint.Theelectriceld vectoristhengivenby E = F =q t ,where F istheelectricforce onthetestcharge. Chargesarewhatcreateelectricelds.Unlikegravity,whichis alwaysattractive,electricitydisplaysbothattractionandrepulsion. Apositivechargeisasourceofelectricelds,andanegativeoneis asink. Themostdicultpointaboutthedenitionoftheelectriceld isthattheforceonanegativechargeisintheoppositedirection comparedtotheeld.Thisfollowsfromthedenition,sincedividingavectorbyanegativenumberreversesitsdirection.It'sas thoughwehadsomeobjectsthatfellupwardinsteadofdown. self-checkA Findanequationforthemagnitudeoftheeldofasinglepointcharge Q Answer,p.203 Superpositionofelectriceldsexample3 Charges q and )]TJ/F107 10.9091 Tf 8.485 0 Td [(q areatadistance b fromeachother,as showninthegure.WhatistheelectriceldatthepointP,which liesatathirdcornerofthesquare? TheeldatPisthevectorsumoftheeldsthatwouldhave beencreatedbythetwochargesindependently.Letpositive x be totherightandletpositive y beup. Negativechargeshaveeldsthatpointatthem,sothecharge )]TJ/F107 10.9091 Tf 8.485 0 Td [(q makesaeldthatpointstotheright,i.e.,hasapositive x Section5.3Theelectriceld 129

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k / Adipoleeld.Electricelds divergefromapositivecharge andconvergeonanegative charge. l / Awatermoleculeisadipole. component.Usingtheanswertotheself-check,wehave E )]TJ/F107 7.9701 Tf 6.586 0 Td [(q x = kq b 2 E )]TJ/F107 7.9701 Tf 6.587 0 Td [(q y =0. Notethatifwehadblindlyignoredtheabsolutevaluesignsand pluggedin )]TJ/F107 10.9091 Tf 8.485 0 Td [(q totheequation,wewouldhaveincorrectlyconcludedthattheeldwenttotheleft. BythePythagoreantheorem,thepositivechargeisatadistance p 2 b fromP,sothemagnitudeofitscontributiontotheeldis E = kq = 2 b 2 .Positivechargeshaveeldsthatpointawayfrom them,sotheeldvectorisatanangleof135 counterclockwise fromthe x axis. E q x = kq 2 b 2 cos135 = )]TJ/F107 10.9091 Tf 17.582 7.38 Td [(kq 2 3/2 b 2 E q y = kq 2 b 2 sin135 = kq 2 3/2 b 2 Thetotaleldis E x = 1 )]TJ/F39 10.9091 Tf 10.909 0 Td [(2 )]TJ/F39 7.9701 Tf 6.586 0 Td [(3/2 kq b 2 E y = kq 2 3/2 b 2 Dipoles Thesimplestsetofsourcesthatcanoccurwithelectricitybut notwithgravityisthe dipole ,consistingofapositivechargeanda negativechargewithequalmagnitudes.Moregenerally,anelectric dipolecanbeanyobjectwithanimbalanceofpositivechargeon onesideandnegativeontheother.Awatermolecule,l,isadipole becausetheelectronstendtoshiftawayfromthehydrogenatoms andontotheoxygenatom. Yourmicrowaveovenactsonwatermoleculeswithelectricelds. Letusimaginewhathappensifwestartwithauniformelectriceld, m/1,madebysomeexternalcharges,andtheninsertadipole,m/2, consistingoftwochargesconnectedbyarigidrod.Thedipoledisturbstheeldpattern,butmoreimportantforourpresentpurposes isthatitexperiencesatorque.Inthisexample,thepositivecharge 130 Chapter5FieldsofForce

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feelsanupwardforce,butthenegativechargeispulleddown.The resultisthatthedipolewantstoalignitselfwiththeeld,m/3.The microwaveovenheatsfoodwithelectricalandmagneticwaves. Thealternationofthetorquecausesthemoleculestowiggleandincreasetheamountofrandommotion.Theslightlyvaguedenition ofadipolegivenabovecanbeimprovedbysayingthatadipoleis anyobjectthatexperiencesatorqueinanelectriceld. Whatdeterminesthetorqueonadipoleplacedinanexternally createdeld?Torquedependsontheforce,thedistancefromthe axisatwhichtheforceisapplied,andtheanglebetweentheforce andthelinefromtheaxistothepointofapplication.Letadipole consistingofcharges+ q and )]TJ/F20 10.9091 Tf 8.485 0 Td [(q separatedbyadistance ` beplaced inanexternaleldofmagnitude j E j ,atanangle withrespectto theeld.Thetotaltorqueonthedipoleis = ` 2 q j E j sin + ` 2 q j E j sin = `q j E j sin Notethateventhoughthetwoforcesareinoppositedirections, thetorquesdonotcancel,becausetheyarebothtryingtotwistthe dipoleinthesamedirection.Thequantity `q iscalledthedipole moment,notated D .Morecomplexdipolescanalsobeassigned adipolemoment|theyaredenedashavingthesamedipole momentasthetwo-chargedipolethatwouldexperiencethesame torque. DipolemomentofamoleculeofNaClgasexample4 InamoleculeofNaClgas,thecenter-to-centerdistancebetweenthetwoatomsisabout0.6nm.Assumingthatthechlorinecompletelystealsoneofthesodium'selectrons,computethe magnitudeofthismolecule'sdipolemoment. Thetotalchargeiszero,soitdoesn'tmatterwherewechoose theoriginofourcoordinatesystem.Forconvenience,let'schoose ittobeatoneoftheatoms,sothatthechargeonthatatom doesn'tcontributetothedipolemoment.Themagnitudeofthe m / 1.Auniformelectriceldcreatedbysomechargesoff-stage. 2.Adipoleisplacedintheeld.3.Thedipolealignswiththeeld. Section5.3Theelectriceld 131

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dipolemomentisthen D = 10 )]TJ/F39 7.9701 Tf 6.587 0 Td [(10 m e = 10 )]TJ/F39 7.9701 Tf 6.587 0 Td [(10 m.6 10 )]TJ/F39 7.9701 Tf 6.587 0 Td [(19 C =1 10 )]TJ/F39 7.9701 Tf 6.586 0 Td [(28 C m Alternativedenitionoftheelectriceld Thebehaviorofadipoleinanexternallycreatedeldleadsus toanalternativedenitionoftheelectriceld: alternativedenitionoftheelectriceld Theelectriceldvector, E ,atanylocationinspaceisdened byobservingthetorqueexertedonatestdipole D t placed there.Thedirectionoftheeldisthedirectioninwhichthe eldtendstoalignadipolefrom )]TJ/F15 10.9091 Tf 13.382 0 Td [(to+,andtheeld's magnitudeis j E j = =D t sin Themainreasonforintroducingaseconddenitionforthesame conceptisthatthemagneticeldismosteasilydenedusinga similarapproach. Voltagerelatedtoelectriceld Voltageispotentialenergyperunitcharge,andelectriceldis forceperunitcharge.Wecanthereforerelatevoltageandeldif westartfromtherelationshipbetweenpotentialenergyandforce, PE = )]TJ/F20 10.9091 Tf 8.485 0 Td [(Fd ,[assumingconstantforceand motionparalleltotheforce] anddividebycharge, PE = )]TJ/F20 10.9091 Tf 8.485 0 Td [(Fd ,[assumingconstantforceand motionparalleltotheforce] giving V = )]TJ/F20 10.9091 Tf 8.485 0 Td [(Ed ,[assumingconstantforceand motionparalleltotheforce] Inotherwords,thedierenceinvoltagebetweentwopointsequals theelectriceldstrengthmultipliedbythedistancebetweenthem. Theinterpretationisthatastrongelectriceldisaregionofspace wherethevoltageisrapidlychanging.Byanalogy,asteephillside isaplaceonthemapwherethealtitudeisrapidlychanging. Fieldgeneratedbyanelectriceelexample5 Supposeanelectriceelis1mlong,andgeneratesavoltage differenceof1000voltsbetweenitsheadandtail.Whatisthe electriceldinthewateraroundit? 132 Chapter5FieldsofForce

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n / Example6. Weareonlycalculatingtheamountofeld,notitsdirection,so weignorepositiveandnegativesigns.Subjecttothepossiblyinaccurateassumptionofaconstanteldparalleltotheeel'sbody, wehave j E j = V x =1000V/m. Thehammerheadsharkexample6 Oneofthereasonshammerheadsharkshavetheirheadsshaped thewaytheydoisthat,likequiteafewothersh,theycansense electriceldsasawayofndingprey,whichmayforexample behiddeninthesand.Fromtheequation E = V = x ,wecan seethatifthesharkissensingthevoltagedifferencebetween twopoints,itwillbeabletodetectsmallerelectriceldsifthose twopointsarefartherapart.Thesharkhasanetworkofsensory organs,calledtheampullaeofLorenzini,ontheskinofitshead. Sincethenetworkisspreadoverawiderhead,the x islarger. Somesharkscandetectelectriceldsasweakas50 pico volts permeter! Relatingtheunitsofelectriceldandvoltageexample7 Fromouroriginaldenitionoftheelectriceld,weexpectitto haveunitsofnewtonspercoulomb,N/C.Theexampleabove, however,cameoutinvoltspermeter,V/m.Aretheseinconsistent?Let'sreassureourselvesthatthisallworks.Inthiskind ofsituation,thebeststrategyisusuallytosimplifythemorecomplexunitssothattheyinvolveonlymksunitsandcoulombs.Since voltageisdenedaselectricalenergyperunitcharge,ithasunits ofJ/C: V m = J/C m = J C m Toconnectjoulestonewtons,werecallthatworkequalsforce timesdistance,soJ=N m,so V m = N m C m = N C Aswithothersuchdifcultieswithelectricalunits,onequickly beginstorecognizefrequentlyoccurringcombinations. Section5.3Theelectriceld 133

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o / DiscussionquestionH. DiscussionQuestions A Inthedenitionoftheelectriceld,doesthetestchargeneedtobe 1coulomb?Doesitneedtobepositive? B Doesachargedparticlesuchasanelectronorprotonfeelaforce fromitsownelectriceld? C Isthereanelectriceldsurroundingawallsocketthathasnothing pluggedintoit,orabatterythatisjustsittingonatable? D Inaashlightpoweredbyabattery,whichwaydotheelectricelds point?Whatwouldtheeldsbelikeinsidethewires?Insidethelament ofthebulb? E Criticizethefollowingstatement:Anelectriceldcanberepresented byaseaofarrowsshowinghowcurrentisowing. F Theeldofapointcharge, j E j = kQ = r 2 ,wasderivedintheselfcheckabove.Howwouldtheeldpatternofauniformlychargedsphere comparewiththeeldofapointcharge? G Theinteriorofaperfectelectricalconductorinequilibriummust havezeroelectriceld,sinceotherwisethefreechargeswithinitwould bedriftinginresponsetotheeld,anditwouldnotbeinequilibrium.What abouttheeldrightatthesurfaceofaperfectconductor?Considerthe possibilityofaeldperpendiculartothesurfaceorparalleltoit. H Comparethedipolemomentsofthemoleculesandmolecularions showninthegure. I Smallpiecesofpaperthathavenotbeenelectricallypreparedin anywaycanbepickedupwithachargedobjectsuchasachargedpiece oftape.Inournewterminology,wecoulddescribethetape'schargeas inducingadipolemomentinthepaper.Canasimilartechniquebeused toinducenotjustadipolemomentbutacharge? J Theearthandmoonarefairlyuneveninsizeandfarapart,likea baseballandaping-pongballheldinyouroutstretchedarms.Imagine insteadaplanetarysystemwiththecharacterofadoubleplanet:two planetsofequalsize,closetogether.Sketchaseaofarrowsdiagramof theirgravitationaleld. 5.4 R VoltageforNonuniformFields Thecalculus-savvyreaderwillhavenodicultygeneralizingthe eld-voltagerelationshiptothecaseofavaryingeld.Thepotential energyassociatedwithavaryingforceis PE = )]TJ/F26 10.9091 Tf 10.303 14.849 Td [(Z F d x ,[onedimension] soforelectriceldswedivideby q tond V = )]TJ/F26 10.9091 Tf 10.303 14.849 Td [(Z E d x ,[onedimension] Applyingthefundamentaltheoremofcalculusyields E = )]TJ/F15 10.9091 Tf 9.68 7.38 Td [(d V d x .[onedimension] 134 Chapter5FieldsofForce

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Voltageassociatedwithapointchargeexample8 Whatisthevoltageassociatedwithapointcharge? Asderivedpreviouslyinself-checkAonpage129,theeldis j E j = kQ r 2 Thedifferenceinvoltagebetweentwopointsonthesameradius lineis V = )]TJ/F26 10.9091 Tf 10.303 14.849 Td [(Z d V = )]TJ/F26 10.9091 Tf 10.303 14.848 Td [(Z E x d x Inthegeneraldiscussionabove, x wasjustagenericnamefor distancetraveledalongthelinefromonepointtotheother,soin thiscase x reallymeans r V = )]TJ/F26 10.9091 Tf 10.303 14.848 Td [(Z r 2 r 1 E r d r = )]TJ/F26 10.9091 Tf 10.303 14.848 Td [(Z r 2 r 1 kQ r 2 d r = kQ r r 2 r 1 = kQ r 2 )]TJ/F107 10.9091 Tf 12.105 7.38 Td [(kQ r 1 Thestandardconventionistouse r 1 = 1 asareferencepoint,so thatthevoltageatanydistance r fromthechargeis V = kQ r Theinterpretationisthatifyoubringapositivetestchargecloser toapositivecharge,itselectricalenergyisincreased;ifitwas released,itwouldspringaway,releasingthisaskineticenergy. self-checkB Showthatyoucanrecovertheexpressionfortheeldofapointcharge byevaluatingthederivative E x = )]TJ/F39 9.9626 Tf 7.749 0 Td [(d V = d x Answer,p.203 Section5.4 R VoltageforNonuniformFields 135

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q / Theconstant-voltagecurves surroundingapointcharge.Near thecharge,thecurvesareso closelyspacedthattheyblend togetheronthisdrawingdueto thenitewidthwithwhichthey weredrawn.Someelectricelds areshownasarrows. p / Left:AtopographicalmapofStowe,Vermont.Fromoneconstant-heightlinetothenextisaheight differenceof200feet.Linesfarapart,asinthelowervillage,indicaterelativelyatterrain,whilelines closetogether,liketheonestothewestofthemaintown,representasteepslope.Streamsowdownhill, perpendiculartotheconstant-heightlines.Right:Thesamemaphasbeenredrawninperspective,with shadingtosuggestrelief. 5.5TwoorThreeDimensions Thetopographicalmapshowningurepsuggestsagoodway tovisualizetherelationshipbetweeneldandvoltageintwodimensions.Eachcontouronthemapisalineofconstantheight;someof thesearelabeledwiththeirelevationsinunitsoffeet.Heightisrelatedtogravitationalpotentialenergy,soinagravitationalanalogy, wecanthinkofheightasrepresentingvoltage.Wherethecontour linesarefarapart,asinthetown,theslopeisgentle.Linesclose togetherindicateasteepslope. Ifwewalkalongastraightline,saystraighteastfromthetown, thenheightvoltageisafunctionoftheeast-westcoordinate x Usingtheusualmathematicaldenitionoftheslope,andwriting V fortheheightinordertoremindusoftheelectricalanalogy,the slopealongsuchalineis V= x.Iftheslopeisn'tconstant,we eitherneedtousetheslopeofthe V )]TJ/F20 10.9091 Tf 10.562 0 Td [(x graph,orusecalculusand talkaboutthederivatived V= d x Whatifeverythingisn'tconnedtoastraightline?Waterows downhill.Noticehowthestreamsonthemapcutperpendicularly throughthelinesofconstantheight. Itispossibletomapvoltagesinthesameway,asshownin gureq.Theelectriceldisstrongestwheretheconstant-voltage curvesareclosesttogether,andtheelectriceldvectorsalwayspoint perpendiculartotheconstant-voltagecurves. 136 Chapter5FieldsofForce

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r / Self-checkC. Figuresshowssomeexamplesofwaystovisualizeeldand voltagepatterns. Mathematically,thecalculusofsection5.4generalizestothree dimensionsasfollows: E x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(d V= d x E y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(d V= d y E z = )]TJ/F15 10.9091 Tf 8.485 0 Td [(d V= d z self-checkC Imaginethatthetopographicalmapingurerrepresentsvoltagerather thanheight.aConsiderthestreamthestartsnearthecenterofthe map.Determinethepositiveandnegativesignsof dV = dx and dV = dy andrelatethesetothedirectionoftheforcethatispushingthecurrent forwardagainsttheresistanceoffriction.bIfyouwantedtondalot ofelectricchargeonthismap,wherewouldyoulook? Answer,p. 204 s / Two-dimensionaleldandvoltagepatterns.Top:Auniformlychargedrod.Bottom: Adipole.Ineachcase,the diagramontheleftshowsthe eldvectorsandconstant-voltage curves,whiletheoneonthe rightshowsthevoltageup-down coordinateasafunctionofx andy.Interpretingtheelddiagrams:Eacharrowrepresents theeldatthepointwhereitstail hasbeenpositioned.Forclarity,someofthearrowsinregions ofverystrongeldstrengthare notshowntheywouldbetoo longtoshow.Interpretingthe constant-voltagecurves:Inregionsofverystrongelds,the curvesarenotshownbecause theywouldmergetogetherto makesolidblackregions.Interpretingtheperspectiveplots: Keepinmindthateventhough we'revisualizingthingsinthree dimensions,thesearereallytwodimensionalvoltagepatternsbeingrepresented.Thethirdupdowndimensionrepresentsvoltage,notposition. Section5.5TwoorThreeDimensions 137

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t / Example9. 5.6 R ? ElectricFieldofaContinuousCharge Distribution Chargereallycomesindiscretechunks,butoftenitismathematicallyconvenienttotreatasetofchargesasiftheywerelikea continuousuidspreadthroughoutaregionofspace.Forexample, achargedmetalballwillhavechargespreadnearlyuniformlyall overitssurface,andinformostpurposesitwillmakesensetoignorethefactthatthisuniformityisbrokenattheatomiclevel.The electriceldmadebysuchacontinuouschargedistributionisthe sumoftheeldscreatedbyeverypartofit.Ifwelettheparts" becomeinnitesimallysmall,wehaveasumofaninnitenumberof innitesimalnumbers,whichisanintegral.Ifitwasadiscretesum, wewouldhaveatotalelectriceldinthe x directionthatwasthe sumofallthe x componentsoftheindividualelds,andsimilarly we'dhavesumsforthe y and z components.Inthecontinuouscase, wehavethreeintegrals. Fieldofauniformlychargedrodexample9 Arodoflength L hascharge Q spreaduniformlyalongit.Find theelectriceldatapointadistance d fromthecenteroftherod, alongtherod'saxis. Thisisaone-dimensionalsituation,sowereallyonlyneedto doasingleintegralrepresentingthetotaleldalongtheaxis.We imaginebreakingtheroddownintoshortpiecesoflengthd z eachwithcharged q .Sincechargeisuniformlyspreadalongthe rod,wehaved q = d z ,where = Q = L Greeklambdaisthe chargeperunitlength,inunitsofcoulombspermeter.Since thepiecesareinnitesimallyshort,wecantreatthemaspoint chargesandusetheexpression k d q = r 2 fortheircontributionsto theeld,where r = d )]TJ/F107 10.9091 Tf 11.111 0 Td [(z isthedistancefromthechargeat z to thepointinwhichweareinterested. E z = Z k d q r 2 = Z + L = 2 )]TJ/F107 7.9701 Tf 6.587 0 Td [(L = 2 k d z r 2 = k Z + L = 2 )]TJ/F107 7.9701 Tf 6.586 0 Td [(L = 2 d z d )]TJ/F107 10.9091 Tf 10.909 0 Td [(z 2 Theintegralcanbelookedupinatable,orreducedtoanelementaryformbysubstitutinganewvariablefor d )]TJ/F107 10.9091 Tf 10.984 0 Td [(z .Theresult is E z = k 1 d )]TJ/F107 10.9091 Tf 10.909 0 Td [(z + L = 2 )]TJ/F107 7.9701 Tf 6.586 0 Td [(L = 2 = kQ L 1 d )]TJ/F107 10.9091 Tf 10.909 0 Td [(L = 2 )]TJ/F39 10.9091 Tf 27.03 7.38 Td [(1 d + L = 2 138 Chapter5FieldsofForce

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Forlargevaluesof d ,thisexpressiongetssmallerfortworeasons:thedenominatorsofthefractionsbecomelarge,and thetwofractionsbecomenearlythesame,andtendtocancel out.Thismakessense,sincetheeldshouldgetweakeraswe getfartherawayfromthecharge.Infact,theeldatlargedistancesmustapproach kQ = d 2 ,sincefromagreatdistance,the rodlookslikeapoint. It'salsointerestingtonotethattheeldbecomesinniteatthe endsoftherod,butisnotinniteontheinterioroftherod.Can youexplainphysicallywhythishappens? Section5.6 R ? ElectricFieldofaContinuousChargeDistribution 139

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Summary SelectedVocabulary eld........apropertyofapointinspacedescribingthe forcesthatwouldbeexertedonaparticleifit wasthere sink........apointatwhicheldvectorsconverge source.......apointfromwhicheldvectorsdiverge;oftenusedmoreinclusivelytorefertopointsof eitherconvergenceordivergence electriceld...theforceperunitchargeexertedonatest chargeatagivenpointinspace gravitationaleldtheforceperunitmassexertedonatestmass atagivenpointinspace electricdipole..anobjectthathasanimbalancebetweenpositivechargeononesideandnegativecharge ontheother;anobjectthatwillexperiencea torqueinanelectriceld Notation g ..........thegravitationaleld E .........theelectriceld D .........anelectricdipolemoment OtherTerminologyandNotation d p m ......othernotationsfortheelectricdipolemoment Summary Newtonconceivedofauniversewhereforcesreachedacrossspace instantaneously,butwenowknowthatthereisadelayintimebefore achangeinthecongurationofmassandchargeinonecornerofthe universewillmakeitselffeltasachangeintheforcesexperienced faraway.Weimaginetheoutwardspreadofsuchachangeasa rippleinaninvisibleuniverse-lling eldofforce Wedenethe gravitationaleld atagivenpointastheforceper unitmassexertedonobjectsinsertedatthatpoint,andlikewisethe electriceld isdenedastheforceperunitcharge.Theseeldsare vectors,andtheeldsgeneratedbymultiplesourcesaddaccording totherulesofvectoraddition. Whentheelectriceldisconstant,thevoltagedierencebetween twopointslyingonalineparalleltotheeldisrelatedtotheeld bytheequation V =| Ed ,where d isthedistancebetweenthe twopoints. 140 Chapter5FieldsofForce

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Problem1. Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Inourby-now-familiarneuron,thevoltagedierencebetweentheinnerandoutersurfacesofthecellmembraneisabout V out )]TJ/F20 10.9091 Tf 11.053 0 Td [(V in = )]TJ/F15 10.9091 Tf 8.485 0 Td [(70mVintherestingstate,andthethicknessofthe membraneisabout6.0nmi.e.onlyaboutahundredatomsthick. Whatistheelectriceldinsidethemembrane? p 2 Thegapbetweentheelectrodesinanautomobileengine's sparkplugis0.060cm.Toproduceanelectricsparkinagasolineairmixture,anelectriceldof3.0 10 6 V/mmustbeachieved. Onstartingacar,whatminimumvoltagemustbesuppliedbythe ignitioncircuit?Assumetheeldisuniform. p bThesmallsizeofthegapbetweentheelectrodesisinconvenient becauseitcangetblockedeasily,andspecialtoolsareneededto measureit.Whydon'ttheydesignsparkplugswithawidergap? 3 aAttime t =0,apositivelychargedparticleisplaced, atrest,inavacuum,inwhichthereisauniformelectriceldof magnitude E .Writeanequationgivingtheparticle'sspeed, v ,in termsof t E ,anditsmassandcharge m and q p bIfthisisdonewithtwodierentobjectsandtheyareobserved tohavethesamemotion,whatcanyouconcludeabouttheirmasses andcharges?Forinstance,whenradioactivitywasdiscovered,it wasfoundthatoneformofithadthesamemotionasanelectron inthistypeofexperiment. 4 Showthatthemagnitudeoftheelectriceldproducedbya simpletwo-chargedipole,atadistantpointalongthedipole'saxis, istoagoodapproximationproportionalto D=r 3 ,where r isthe distancefromthedipole.[Hint:Usetheapproximation+ p 1+ p ,whichisvalidforsmall .] ? 5 Giventhattheeldofadipoleisproportionalto D=r 3 see previousproblem,showthatitsvoltagevariesas D=r 2 .Ignore positiveandnegativesignsandnumericalconstantsofproportionality. R 6 AcarbondioxidemoleculeisstructuredlikeO-C-O,with allthreeatomsalongaline.Theoxygenatomsgrabalittlebitof extranegativecharge,leavingthecarbonpositive.Themolecule's symmetry,however,meansthatithasnooveralldipolemoment, unlikeaV-shapedwatermolecule,forinstance.Whereasthevoltage ofadipoleofmagnitude D isproportionalto D=r 2 problem5,it turnsoutthatthevoltageofacarbondioxidemoleculealongits axisequals k=r 3 ,where r isthedistancefromthemoleculeand k Problems 141

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Problem11. isaconstant.Whatwouldbetheelectriceldofacarbondioxide moleculeatadistance r ? R 7 Aprotonisinaregioninwhichtheelectriceldisgivenby E = a + bx 3 .Iftheprotonstartsatrestat x 1 =0,nditsspeed, v whenitreachesposition x 2 .Giveyouranswerintermsof a b x 2 and e and m ,thechargeandmassoftheproton. p R 8 Considertheelectriceldcreatedbyauniformringoftotalcharge q andradius b .aShowthattheeldatapointonthering'saxis atadistance a fromtheplaneoftheringis kqa a 2 + b 2 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 = 2 .b Showthatthisexpressionhastherightbehaviorfor a =0andfor a muchgreaterthan b ? 9 Considertheelectriceldcreatedbyaninniteuniformly chargedplane.Startingfromtheresultofproblem8,showthatthe eldatanypointis2 k ,where isthedensityofchargeonthe plane,inunitsofcoulombspersquaremeter.Notethattheresult isindependentofthedistancefromtheplane.[Hint:Slicetheplane intoinnitesimalconcentricrings,centeredatthepointintheplane closesttothepointatwhichtheeldisbeingevaluated.Integrate therings'contributionstotheeldatthispointtondthetotal eld.] Solution,p.205 R 10 Considertheelectriceldcreatedbyauniformlycharged cylinderthatextendstoinnityinonedirection.aStartingfrom theresultofproblem8,showthattheeldatthecenterofthe cylinder'smouthis2 k ,where isthedensityofchargeonthe cylinder,inunitsofcoulombspersquaremeter.[Hint:Youcanuse amethodsimilartotheoneinproblem9.]bThisexpressionis independentoftheradiusofthecylinder.Explainwhythisshould beso.Forexample,whatwouldhappenifyoudoubledthecylinder's radius? R 11 Threechargesarearrangedonasquareasshown.Allthree chargesarepositive.Whatvalueof q 2 =q 1 willproducezeroelectric eldatthecenterofthesquare? Solution,p.205 142 Chapter5FieldsofForce

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a / Thersttwohumanstoknow whatstarlightwas:JamesClerk MaxwellandKatherineMaxwell, 1869. Chapter6 Electromagnetism InthischapterwediscusstheintimaterelationshipbetweenmagnetismandelectricitydiscoveredbyJamesClerkMaxwell.Maxwell 143

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b / Breakingabarmagnetinhalf doesn'tcreatetwomonopoles,it createstwosmallerdipoles. c / Anexplanationattheatomic level. realizedthatlightwasawavemadeupofelectricandmagnetic eldslinkedtoeachother.Heissaidtohavegoneforawalkwith hiswifeonenightandtoldherthatshewastheonlyotherperson intheworldwhoknewwhatstarlightreallywas. 6.1Themagneticeld Nomagneticmonopoles Ifyoucouldplaywithahandfulofelectricdipolesandahandful ofbarmagnets,theywouldappearverysimilar.Forinstance,apair ofbarmagnetswantstoalignthemselveshead-to-tail,andapairof electricdipolesdoesthesamething.Itisunfortunatelynotthat easytomakeapermanentelectricdipolethatcanbehandledlike this,sincethechargetendstoleak. Youwouldeventuallynoticeanimportantdierencebetweenthe twotypesofobjects,however.Theelectricdipolescanbebroken aparttoformisolatedpositivechargesandnegativecharges.The two-endeddevicecanbebrokenintopartsthatarenottwo-ended. Butifyoubreakabarmagnetinhalf,b,youwillndthatyouhave simplymadetwosmallertwo-endedobjects. Thereasonforthisbehaviorisnothardtodivinefromourmicroscopicpictureofpermanentironmagnets.Anelectricdipolehas extrapositivestu"concentratedinoneendandextranegativein theother.Thebarmagnet,ontheotherhand,getsitsmagnetic propertiesnotfromanimbalanceofmagneticstu"atthetwo endsbutfromtheorientationoftherotationofitselectrons.One endistheonefromwhichwecouldlookdowntheaxisandseethe electronsrotatingclockwise,andtheotheristheonefromwhich theywouldappeartogocounterclockwise.Thereisnodierence betweenthestu"inoneendofthemagnetandtheother,c. Nobodyhaseversucceededinisolatingasinglemagneticpole. Intechnicallanguage,wesaythatmagneticmonopolesdonotseem toexist.Electricmonopoles do exist|that'swhatchargesare. Electricandmagneticforcesseemsimilarinmanyways.Both actatadistance,bothcanbeeitherattractiveorrepulsive,and bothareintimatelyrelatedtothepropertyofmattercalledcharge. Recallthatmagnetismisaninteractionbetweenmovingcharges. Physicists'saestheticsenseshavebeenoendedforalongtimebecausethisseemingsymmetryisbrokenbytheexistenceofelectricmonopolesandtheabsenceofmagneticones.Perhapssome exoticformofmatterexists,composedofparticlesthataremagneticmonopoles.Ifsuchparticlescouldbefoundincosmicrays ormoonrocks,itwouldbeevidencethattheapparentasymmetry wasonlyanasymmetryinthecompositionoftheuniverse,notin thelawsofphysics.Fortheseadmittedlysubjectivereasons,there havebeenseveralsearchesformagneticmonopoles.Experiments 144 Chapter6Electromagnetism

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d / Astandarddipolemade fromasquareloopofwireshortingacrossabattery.Itactsvery muchlikeabarmagnet,butits strengthismoreeasilyquantied. e / Adipoletendstoalignitselftothesurroundingmagnetic eld. havebeenperformed,withnegativeresults,tolookformagnetic monopolesembeddedinordinarymatter.Sovietphysicistsinthe 1960 s madeexcitingclaimsthattheyhadcreatedanddetectedmagneticmonopolesinparticleaccelerators,buttherewasnosuccess inattemptstoreproducetheresultsthereoratotheraccelerators. Themostrecentsearchformagneticmonopoles,donebyreanalyzingdatafromthesearchforthetopquarkatFermilab,turnedup nocandidates,whichshowsthateithermonopolesdon'texistin natureortheyareextremelymassiveandthushardtocreatein accelerators. Denitionofthemagneticeld Sincemagneticmonopolesdon'tseemtoexist,itwouldnotmake muchsensetodeneamagneticeldintermsoftheforceona testmonopole.Instead,wefollowthephilosophyofthealternative denitionoftheelectriceld,anddenetheeldintermsofthe torqueonamagnetictestdipole.Thisisexactlywhatamagnetic compassdoes:theneedleisalittleironmagnetwhichactslikea magneticdipoleandshowsusthedirectionoftheearth'smagnetic eld. Todenethestrengthofamagneticeld,however,weneedsome wayofdeningthestrengthofatestdipole,i.e.,weneedadenition ofthemagneticdipolemoment.Wecoulduseanironpermanent magnetconstructedaccordingtocertainspecications,butsuchan objectisreallyanextremelycomplexsystemconsistingofmany ironatoms,onlysomeofwhicharealigned.Amorefundamental standarddipoleisasquarecurrentloop.Thiscouldbelittleresistive circuitconsistingofasquareofwireshortingacrossabattery. Wewillndthatsuchaloop,whenplacedinamagneticeld, experiencesatorquethattendstoalignplanesothatitsfacepoints inacertaindirection.Sincetheloopissymmetric,itdoesn'tcare ifwerotateitlikeawheelwithoutchangingtheplaneinwhichit lies.Itisthispreferredfacingdirectionthatwewillendupdening asthedirectionofthemagneticeld. Experimentsshowiftheloopisoutofalignmentwiththeeld, thetorqueonitisproportionaltotheamountofcurrent,andalso totheinteriorareaoftheloop.Theproportionalitytocurrent makessense,sincemagneticforcesareinteractionsbetweenmoving charges,andcurrentisameasureofthemotionofcharge.The proportionalitytotheloop'sareaisalsonothardtounderstand, becauseincreasingthelengthofthesidesofthesquareincreases boththeamountofchargecontainedinthiscircularriver"and theamountofleveragesuppliedformakingtorque.Twoseparate physicalreasonsforaproportionalitytolengthresultinanoverall proportionalitytolengthsquared,whichisthesameastheareaof theloop.Forthesereasons,wedenethemagneticdipolemoment Section6.1Themagneticeld 145

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f / Theunitofmagneticeld, thetesla,isnamedafterSerbianAmericaninventorNikolaTesla. g / Themagneticeldpatternofabarmagnet.Thispicture wasmadebyputtingironlings onapieceofpaper,andbringing abarmagnetupunderneathit. Notehowtheeldpatternpasses acrossthebodyofthemagnet, formingclosedloops,asingure h/2.Therearenosourcesor sinks. ofasquarecurrentloopas D m = IA ,[denitionofthemagnetic dipolemomentofasquarecurrentloop] Wenowdenethemagneticeldinamannerentirelyanalogousto theseconddenitionoftheelectriceld: denitionofthemagneticeld Themagneticeldvector, B ,atanylocationinspaceisdenedbyobservingthetorqueexertedonamagnetictestdipole D mt consistingofasquarecurrentloop.Theeld'smagnitude is j B j = =D mt sin ,where istheanglebywhichtheloopis misaligned.Thedirectionoftheeldisperpendiculartothe loop;ofthetwoperpendiculars,wechoosetheonesuchthat ifwelookalongit,theloop'scurrentiscounterclockwise. Wendfromthisdenitionthatthemagneticeldhasunits ofN m = A m 2 =N = A m.Thisunwieldycombinationofunitsis abbreviatedasthetesla,1T=1N = A m.Refrainfrommemorizing thepartaboutthecounterclockwisedirectionattheend;insection 6.4we'llseehowtounderstandthisintermsofmorebasicprinciples. Thenonexistenceofmagneticmonopolesmeansthatunlikean electriceld,h/1,amagneticone,h/2,canneverhavesourcesor sinks.Themagneticeldvectorsleadinpathsthatloopbackon themselves,withouteverconvergingordivergingatapoint. 6.2Calculatingmagneticeldsandforces Magnetostatics Ourstudyoftheelectriceldbuiltonourpreviousunderstandingofelectricforces,whichwasultimatelybasedonCoulomb'slaw fortheelectricforcebetweentwopointcharges.Sincemagnetism isultimatelyaninteractionbetweencurrents,i.e.,betweenmoving charges,itisreasonabletowishforamagneticanalogofCoulomb's law,anequationthatwouldtellusthemagneticforcebetweenany twomovingpointcharges. Suchalaw,unfortunately,doesnotexist.Coulomb'slawdescribesthespecialcaseofelectrostatics:ifasetofchargesissitting aroundandnotmoving,ittellsustheinteractionsamongthem. Coulomb'slawfailsifthechargesareinmotion,sinceitdoesnot incorporateanyallowanceforthetimedelayintheoutwardpropagationofachangeinthelocationsofthecharges. Apairofmovingpointchargeswillcertainlyexertmagnetic forcesononeanother,buttheirmagneticeldsarelikethev-shaped bowwavesleftbyboats.Eachpointchargeexperiencesamagnetic eldthatoriginatedfromtheotherchargewhenitwasatsome previousposition.Thereisnowaytoconstructaforcelawthattells 146 Chapter6Electromagnetism

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i / Somemagneticelds. h / Electricelds,1,havesources andsinks,butmagneticelds,2, don't. ustheforcebetweenthembasedonlyontheircurrentpositionsin space. Thereis,however,ascienceofmagnetostaticsthatcoversagreat manyimportantcases.Magnetostaticsdescribesmagneticforces amongcurrentsinthespecialcasewherethecurrentsaresteady andcontinuous,leadingtomagneticeldsthroughoutspacethat donotchangeovertime. Ifwecannotbuildamagnetostaticsfromaforcelawforpoint charges,thenwheredowestart?Itcanbedone,butthelevel ofmathematicsrequiredvectorcalculusisinappropriateforthis course.Luckilythereisanalternativethatismorewithinourreach. Physicistsofgenerationspasthaveusedthefancymathtoderive simpleequationsfortheeldscreatedbyvariousstaticcurrentdistributions,suchasacoilofwire,acircularloop,orastraightwire. Virtuallyallpracticalsituationscanbetreatedeitherdirectlyusing theseequationsorbydoingvectoraddition,e.g.,foracaselikethe eldoftwocircularloopswhoseeldsaddontooneanother.Figure ishowstheequationsforsomeofthemorecommonlyencountered congurations,withillustrationsoftheireldpatterns. Fieldcreatedbyalong,straightwirecarryingcurrent I : B = o I 2 r Here r isthedistancefromthecenterofthewire.Theeldvectors tracecirclesinplanesperpendiculartothewire,goingclockwisewhen viewedfromalongthedirectionofthecurrent. Fieldcreatedbyasinglecircularloopofcurrent: Theeldvectorsformadipole-likepattern,comingthroughtheloop andbackaroundontheoutside.Eachovalpathtracedoutbytheeld vectorsappearsclockwiseifviewedfromalongthedirectionthecurrent isgoingwhenitpunchesthroughit.Thereisnosimpleequationfora eldatanarbitrarypointinspace,butforapointlying alongthecentral axis perpendiculartotheloop,theeldis B = 1 2 o Ib 2 )]TJ/F107 9.9627 Tf 4.567 -8.07 Td [(b 2 + z 2 )]TJ/F39 6.9738 Tf 6.227 0 Td [(3 = 2 where b istheradiusoftheloopand z isthedistanceofthepointfrom theplaneoftheloop. Fieldcreatedbyasolenoidcylindricalcoil: Theeldpatternissimilartothatofasingleloop,butforalongsolenoid thepathsoftheeldvectorsbecomeverystraightontheinsideofthe coilandontheoutsideimmediatelynexttothecoil.Forasufciently longsolenoid,theinterioreldalsobecomesverynearlyuniform,with amagnitudeof B = o IN =` where N isthenumberofturnsofwireand ` isthelengthofthesolenoid. Theeldnearthemouthsoroutsidethecoilisnotconstant,andis moredifculttocalculate.Foralongsolenoid,theexterioreldismuch smallerthantheinterioreld. Section6.2Calculatingmagneticeldsandforces 147

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j / Example1. Don'tmemorizetheequations!Thesymbol o isanabbreviation fortheconstant4 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(7 T m = A.Itisthemagneticcounterpartof theCoulombforceconstant k .TheCoulombconstanttellsushow muchelectriceldisproducedbyagivenamountofcharge,while o relatescurrentstomagneticelds.Unlike k o hasadenite numericalvaluebecauseofthedesignofthemetricsystem. Forceonachargemovingthroughamagneticeld Wenowknowhowtocalculatemagneticeldsinsometypical situations,butonemightalsoliketobeabletocalculatemagnetic forces,suchastheforceofasolenoidonamovingchargedparticle, ortheforcebetweentwoparallelcurrent-carryingwires. Wewillrestrictourselvestothecaseoftheforceonacharged particlemovingthroughamagneticeld,whichallowsustocalculatetheforcebetweentwoobjectswhenoneisamovingcharged particleandtheotherisonewhosemagneticeldweknowhowto nd.AnexampleistheuseofsolenoidsinsideaTVtubetoguide theelectronbeamasitpaintsapicture. Experimentsshowthatthemagneticforceonamovingcharged particlehasamagnitudegivenby j F j = q j v jj B j sin where v isthevelocityvectoroftheparticle,and istheanglebetweenthe v and B vectors.Unlikeelectricandgravitationalforces, magneticforcesdonotliealongthesamelineastheeldvector. Theforceisalways perpendicular toboth v and B .Giventwovectors,thereisonlyonelineperpendiculartobothofthem,sothe forcevectorpointsinoneofthetwopossibledirectionsalongthis line.Forapositivelychargedparticle,thedirectionoftheforce vectorcanbefoundasfollows.First,positionthe v and B vectors withtheirtailstogether.Thedirectionof F issuchthatifyousight alongit,the B vectorisclockwisefromthe v vector;foranegativelychargedparticlethedirectionoftheforceisreversed.Note thatsincetheforceisperpendiculartotheparticle'smotion,the magneticeldneverdoesworkonit. Magneticlevitationexample1 Ingurej,asmall,disk-shapedpermanentmagnetisstuckonthe sideofabattery,andawireisclaspedlooselyaroundthebattery, shortingit.Alargecurrentowsthroughthewire.Theelectrons movingthroughthewirefeelaforcefromthemagneticeldmade bythepermanentmagnet,andthisforcelevitatesthewire. Fromthephoto,it'spossibletondthedirectionofthemagnetic eldmadebythepermanentmagnet.Theelectronsinthecopper wirearenegativelycharged,sotheyowfromthenegativeat terminalofthebatterytothepositiveterminaltheonewiththe bump,infront.Astheelectronspassbythepermanentmagnet, 148 Chapter6Electromagnetism

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l / MichaelFaraday-1867, thesonofapoorblacksmith,discoveredinductionexperimentally. k / Example2. wecanimaginethattheywouldexperienceaeldeithertoward themagnet,orawayfromit,dependingonwhichwaythemagnet wasippedwhenitwasstuckontothebattery.Imaginesighting alongtheupwardforcevector,whichyoucoulddoifyouwere atinybuglyingonyourbackunderneaththewire.Sincethe electronsarenegativelycharged,the B vectormustbecounterclockwisefromthe v vector,whichmeanstowardthemagnet. Acircularorbitexample2 Magneticforcescauseabeamofelectronstomoveinacircle. Thebeamiscreatedinavacuumtube,inwhichasmallamount ofhydrogengashasbeenleft.Afewoftheelectronsstrikehydrogenmolecules,creatinglightandlettingusseethebeam.A magneticeldisproducedbypassingacurrentmeterthrough thecircularcoilsofwireinfrontofandbehindthetube.Inthe bottomgure,withthemagneticeldturnedon,theforceperpendiculartotheelectrons'directionofmotioncausesthemto moveinacircle. HallucinationsduringanMRIscanexample3 DuringanMRIscanofthehead,thepatient'snervoussystem isexposedtointensemagneticelds.Theaveragevelocitiesof thecharge-carryingionsinthenervecellsisfairlylow,butifthe patientmovesherheadsuddenly,thevelocitycanbehighenough thatthemagneticeldmakessignicantforcesontheions.This canresultinvisualandauditoryhallucinations,e.g.,fryingbacon sounds. 6.3Induction Electromagnetismandrelativemotion Thetheoryofelectricandmagneticeldsconstructedupto thispointcontainsaparadox.Oneofthemostbasicprinciples ofphysics,datingbacktoNewtonandGalileoandstillgoingstrong today,statesthatmotionisrelative,notabsolute.Thusthelawsof physicsshouldnotfunctionanydierentlyinamovingframeofreference,orelsewewouldbeabletotellwhichframeofreferencewas theoneinanabsolutestateofrest.Asanexamplefrommechanics, imaginethatachildistossingaballupanddowninthebackseatof amovingcar.Inthechild'sframeofreference,thecarisatrestand thelandscapeismovingby;inthisframe,theballgoesstraightup anddown,andobeysNewton'slawsofmotionandNewton'slawof gravity.Intheframeofreferenceofanobserverwatchingfromthe sidewalk,thecarismovingandthesidewalkisn't.Inthisframe, theballfollowsaparabolicarc,butitstillobeysNewton'slaws. Whenitcomestoelectricityandmagnetism,however,wehavea problem,whichwasrstclearlyarticulatedbyEinstein:ifwestate thatmagnetismisaninteractionbetweenmovingcharges,wehave Section6.3Induction 149

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n / ObserverAseesapositivelychargedparticlemoves througharegionofupward magneticeld,whichweassume tobeuniform,betweenthepoles oftwomagnets.Theresulting forcealongthe z axiscausesthe particle'spathtocurvetowardus. m / Alineofpositivecharges. apparentlycreatedalawofphysicsthatviolatestheprinciplethat motionisrelative,sincedierentobserversindierentframeswould disagreeabouthowfastthechargesweremoving,orevenwhether theyweremovingatall.TheincorrectsolutionthatEinsteinwas taughtanddisbelievedasastudentaroundtheyear1900wasthat therelativenatureofmotionappliedonlytomechanics,nottoelectricityandmagnetism.ThefullstoryofhowEinsteinrestoredthe principleofrelativemotiontoitsrightfulplaceinphysicsinvolves histheoryofspecialrelativity,whichwewillnottakeupuntilbook6 ofthisseries.However,afewsimpleandqualitativethoughtexperimentswillsucetoshowhow,basedontheprinciplethatmotion isrelative,theremustbesomenewandpreviouslyunsuspectedrelationshipsbetweenelectricityandmagnetism.Theserelationships formthebasisformanypractical,everydaydevices,suchasgeneratorsandtransformers,andtheyalsoleadtoanexplanationoflight itselfasanelectromagneticphenomenon. Let'simagineanelectricalexampleofrelativemotioninthe samespiritasthestoryofthechildinthebackofthecar.Suppose wehavealineofpositivecharges,m.ObserverAisinaframeof referencewhichisatrestwithrespecttothesecharges,andobserves thattheycreateanelectriceldpatternthatpointsoutward,away fromthecharges,inalldirections,likeabottlebrush.Suppose, however,thatobserverBismovingtotherightwithrespecttothe charges.AsfarasBisconcerned,she'stheoneatrest,whilethe chargesandobserverAmovetotheleft.InagreementwithA,she observesanelectriceld,butsincetoherthechargesareinmotion, shemustalsoobserveamagneticeldinthesameregionofspace, exactlylikethemagneticeldmadebyacurrentinalong,straight wire. Who'sright?They'rebothright.InA'sframeofreference, thereisonlyan E ,whileinB'sframethereisbothan E anda B Theprincipleofrelativemotionforcesustoconcludethatdependingonourframeofreferencewewillobserveadierentcombination ofelds.Althoughwewillnotproveittheproofrequiresspecial relativity,whichwegettoinbook6,itistruethateitherframeof referenceprovidesaperfectlyself-consistentdescriptionofthings. Forinstance,ifanelectronpassesthroughthisregionofspace,both AandBwillseeitswerve,speedup,andslowdown.Awillsuccessfullyexplainthisastheresultofanelectriceld,whileBwill ascribetheelectron'sbehaviortoacombinationofelectricandmagneticforces. Thus,ifwebelieveintheprincipleofrelativemotion,thenwe mustacceptthatelectricandmagneticeldsarecloselyrelated phenomena,twosidesofthesamecoin. Nowconsiderguren.ObserverAisatrestwithrespecttothe barmagnets,andseestheparticleswervingointhe z direction,as itshouldaccordingtotherulegiveninsection6.2sightingalong 150 Chapter6Electromagnetism

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o / Thegeometryofinduced elds.Theinducedeldtendsto formawhirlpoolpatternaround thechangeinthevectorproducingit.Notehowtheycirculatein oppositedirections. theforcevector,i.e.,frombehindthepage,the B vectorisclockwise fromthe v vector.SupposeobserverB,ontheotherhand,ismovingtotherightalongthe x axis,initiallyatthesamespeedasthe particle.Bseesthebarmagnetsmovingtotheleftandtheparticle initiallyatrestbutthenacceleratingalongthe z axisinastraight line.Itisnotpossibleforamagneticeldtostartaparticlemoving ifitisinitiallyatrest,sincemagnetismisaninteractionofmoving chargeswithmovingcharges.Bisthusledtotheinescapableconclusionthatthereisanelectriceldinthisregionofspace,which pointsalongthe z axis.Inotherwords,whatAperceivesasapure B eld,Bseesasamixtureof E and B Ingeneral,observerswhoarenotatrestwithrespecttooneanotherwillperceivedierentmixturesofelectricandmagneticelds. Theprincipleofinduction Sofareverythingwe'vebeendoingmightnotseemterriblyuseful,sinceitseemsthatnothingsurprisingwillhappenaslongas westicktoasingleframeofreference,anddon'tworryaboutwhat peopleinotherframesthink.Thatisn'tthewholestory,however, aswasdiscoveredexperimentallybyFaradayin1831andexplored mathematicallybyMaxwelllaterinthesamecentury.Let'sstate Faraday'sidearst,andthenseehowsomethinglikeitmustfollow inevitablyfromtheprinciplethatmotionisrelative: theprincipleofinduction Anyelectriceldthatchangesovertimewillproduceamagneticeldinthespacearoundit. Anymagneticeldthatchangesovertimewillproducean electriceldinthespacearoundit. Theinducedeldtendstohaveawhirlpoolpattern,asshownin gureo,butthewhirlpoolimageisnottobetakentooliterally;the principleofinductionreallyjustrequiresaeldpatternsuchthat, ifoneinsertedapaddlewheelinit,thepaddlewheelwouldspin.All oftheeldpatternsshowningurepareonesthatcouldbecreated byinduction;allhaveacounterclockwisecurl"tothem. p / Threeeldswithcounterclockwisecurls. Section6.3Induction 151

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r / Agenerator q / 1.ObserverAisatrestwithrespecttothebarmagnet,andobserves magneticeldsthathavedifferentstrengthsatdifferentdistancesfromthe magnet.2.ObserverB,hangingoutintheregiontotheleftofthemagnet, seesthemagnetmovingtowardher,anddetectsthatthemagneticeld inthatregionisgettingstrongerastimepasses.Asin1,thereisan electriceldalongthezaxisbecauseshe'sinmotionwithrespecttothe magnet.The B vectorisupward,andtheelectriceldhasacurliness toit:apaddlewheelinsertedintheelectriceldwouldspinclockwiseas seenfromabove,sincetheclockwisetorquemadebythestrongelectric eldontherightisgreaterthanthecounterclockwisetorquemadebythe weakerelectriceldontheleft. Figureqshowsanexampleofthefundamentalreasonwhya changing B eldmustcreatean E eld.Theelectriceldwould beinexplicabletoobserverBifshebelievedonlyinCoulomb'slaw, andthoughtthatallelectriceldsaremadebyelectriccharges.If sheknowsabouttheprincipleofinduction,however,theexistence ofthiseldistobeexpected. Thegeneratorexample4 Agenerator,r,consistsofapermanentmagnetthatrotateswithin acoilofwire.Themagnetisturnedbyamotororcrank,not shown.Asitspins,thenearbymagneticeldchanges.Accordingtotheprincipleofinduction,thischangingmagneticeldresultsinanelectriceld,whichhasawhirlpoolpattern.Thiselectriceldpatterncreatesacurrentthatwhipsaroundthecoilsof wire,andwecantapthiscurrenttolightthelightbulb. self-checkA Whenyou'redrivingacar,theenginerechargesthebatterycontinuouslyusingadevicecalledanalternator,whichisreallyjustageneratorliketheoneshownonthepreviouspage,exceptthatthecoilrotates whilethepermanentmagnetisxedinplace.Whycan'tyouusethe alternatortostarttheengineifyourcar'sbatteryisdead? Answer, p.204 Thetransformerexample5 Insection4.3wediscussedtheadvantagesoftransmittingpower overelectricallinesusinghighvoltagesandlowcurrents.However,wedon'twantourwallsocketstooperateat10000volts! Forthisreason,theelectriccompanyusesadevicecalledatransformer,g,toconverttolowervoltagesandhighercurrentsinside yourhouse.Thecoilontheinputsidecreatesamagneticeld. Transformersworkwithalternatingcurrent,sothemagneticeld surroundingtheinputcoilisalwayschanging.Thisinducesan electriceld,whichdrivesacurrentaroundtheoutputcoil. Ifbothcoilswerethesame,thearrangementwouldbesymmetric, andtheoutputwouldbethesameastheinput,butanoutputcoil withasmallernumberofcoilsgivestheelectricforcesasmaller distancethroughwhichtopushtheelectrons.Lessmechanical workperunitchargemeansalowervoltage.Conservationofen152 Chapter6Electromagnetism

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ergy,however,guaranteesthattheamountofpowerontheoutput sidemustequaltheamountputinoriginally, I in V in = I out V out ,so thisreducedvoltagemustbeaccompaniedbyanincreasedcurrent. Amechanicalanalogyexample6 Figuresshowsanexampleofinductionleftwithamechanical analogyright.Thetwobarmagnetsareinitiallypointinginoppositedirections,1,andtheirmagneticeldscancelout.Ifone magnetisipped,2,theireldsreinforce,butthechangeinthe magneticeldtakestimetospreadthroughspace.Eventually, 3,theeldbecomeswhatyouwouldexpectfromthetheoryof magnetostatics.Inthemechanicalanalogy,thesuddenmotionof thehandproducesaviolentkinkorwavepulseintherope,the pulsetravelsalongtherope,andittakessometimefortherope tosettledown.Anelectriceldisalsoinducedinbythechangingmagneticeld,eventhoughthereisnonetchargeanywhere totoactasasource.Thesesimplieddrawingsarenotmeant tobeaccuraterepresentationsofthecompletethree-dimensional patternofelectricandmagneticelds. s / Example6. DiscussionQuestion A Inguresnandq,observerBismovingtotheright.Whatwould havehappenedifshehadbeenmovingtotheleft? Section6.3Induction 153

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6.4Electromagneticwaves Themostimportantconsequenceofinductionistheexistenceof electromagneticwaves.Whereasagravitationalwavewouldconsist ofnothingmorethanaripplingofgravitationalelds,theprinciple ofinductiontellsusthattherecanbenopurelyelectricalorpurely magneticwaves.Instead,wehavewavesinwhichthereareboth electricandmagneticelds,suchasthesinusoidaloneshowninthe gure.Maxwellprovedthatsuchwaveswereadirectconsequence ofhisequations,andderivedtheirpropertiesmathematically.The derivationwouldbebeyondthemathematicallevelofthisbook,so wewilljuststatetheresults. t / Anelectromagneticwave. Asinusoidalelectromagneticwavehasthegeometryshownin guret.The E and B eldsareperpendiculartothedirectionof motion,andarealsoperpendiculartoeachother.Ifyoulookalong thedirectionofmotionofthewave,the B vectorisalways90degrees clockwisefromthe E vector.Themagnitudesofthetwoeldsare relatedbytheequation j E j = c j B j Howisanelectromagneticwavecreated?Itcouldbeemitted, forexample,byanelectronorbitinganatomorcurrentsgoingback andforthinatransmittingantenna.Ingeneralanyaccelerating chargewillcreateanelectromagneticwave,althoughonlyacurrent thatvariessinusoidallywithtimewillcreateasinusoidalwave.Once created,thewavespreadsoutthroughspacewithoutanyneedfor chargesorcurrentsalongthewaytokeepitgoing.Astheelectric eldoscillatesbackandforth,itinducesthemagneticeld,and theoscillatingmagneticeldinturncreatestheelectriceld.The wholewavepatternpropagatesthroughemptyspaceatavelocity c =3.0 10 8 m/s,whichisrelatedtotheconstants k and o by c = p 4 k= o Polarization Twoelectromagneticwavestravelinginthesamedirectionthrough spacecandierbyhavingtheirelectricandmagneticeldsindifferentdirections,apropertyofthewavecalleditspolarization. 154 Chapter6Electromagnetism

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u / HeinrichHertz-1894. Lightisanelectromagneticwave OnceMaxwellhadderivedtheexistenceofelectromagneticwaves, hebecamecertainthattheywerethesamephenomenonaslight. Botharetransversewavesi.e.,thevibrationisperpendicularto thedirectionthewaveismoving,andthevelocityisthesame. HeinrichHertzforwhomtheunitoffrequencyisnamedveried Maxwell'sideasexperimentally.Hertzwasthersttosucceedin producing,detecting,andstudyingelectromagneticwavesindetail usingantennasandelectriccircuits.Toproducethewaves,hehad tomakeelectriccurrentsoscillateveryrapidlyinacircuit.Infact, therewasreallynohopeofmakingthecurrentreversedirections atthefrequenciesof10 15 Hzpossessedbyvisiblelight.Thefastest electricaloscillationshecouldproducewere10 9 Hz,whichwould giveawavelengthofabout30cm.Hesucceededinshowingthat, justlikelight,thewavesheproducedwerepolarizable,andcouldbe reectedandrefractedi.e.,bent,asbyalens,andhebuiltdevices suchasparabolicmirrorsthatworkedaccordingtothesameoptical principlesasthoseemployinglight.Hertz'sresultswereconvincing evidencethatlightandelectromagneticwaveswereoneandthe same. Theelectromagneticspectrum Today,electromagneticwaveswithfrequenciesintherangeemployedbyHertzareknownasradiowaves.Anyremainingdoubts thattheHertzianwaves,"astheywerethencalled,werethesame typeofwaveaslightwavesweresoondispelledbyexperimentsin thewholerangeoffrequenciesinbetween,aswellasthefrequencies outsidethatrange.Inanalogytothespectrumofvisiblelight,we speakoftheentireelectromagneticspectrum,ofwhichthevisible spectrumisonesegment. Theterminologyforthevariouspartsofthespectrumisworth memorizing,andismosteasilylearnedbyrecognizingthelogicalreSection6.4Electromagneticwaves 155

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lationshipsbetweenthewavelengthsandthepropertiesofthewaves withwhichyouarealreadyfamiliar.Radiowaveshavewavelengths thatarecomparabletothesizeofaradioantenna,i.e.,metersto tensofmeters.Microwaveswerenamedthatbecausetheyhave muchshorterwavelengthsthanradiowaves;whenfoodheatsunevenlyinamicrowaveoven,thesmalldistancesbetweenneighboring hotandcoldspotsishalfofonewavelengthofthestandingwave theovencreates.Theinfrared,visible,andultravioletobviously havemuchshorterwavelengths,becauseotherwisethewavenature oflightwouldhavebeenasobvioustohumansasthewavenatureof oceanwaves.Torememberthatultraviolet,x-rays,andgammarays alllieontheshort-wavelengthsideofvisible,recallthatallthreeof thesecancausecancer.Aswe'lldiscusslaterinthecourse,thereis abasicphysicalreasonwhythecancer-causingdisruptionofDNA canonlybecausedbyveryshort-wavelengthelectromagneticwaves. Contrarytopopularbelief,microwavescannotcausecancer,which iswhywehavemicrowaveovensandnotx-rayovens! Whytheskyisblueexample7 Whensunlightenterstheupperatmosphere,aparticularairmolecule ndsitselfbeingwashedoverbyanelectromagneticwaveoffrequency f .Themolecule'schargedparticlesnucleiandelectrons actlikeoscillatorsbeingdrivenbyanoscillatingforce,andrespondbyvibratingatthesamefrequency f .Energyissucked outoftheincomingbeamofsunlightandconvertedintothekineticenergyoftheoscillatingparticles.However,theseparticles areaccelerating,sotheyactlikelittleradioantennasthatputthe energybackoutassphericalwavesoflightthatspreadoutinall directions.Anobjectoscillatingatafrequency f hasanaccelerationproportionalto f 2 ,andanacceleratingchargedparticle createsanelectromagneticwavewhoseeldsareproportional toitsacceleration,sotheeldofthereradiatedsphericalwave isproportionalto f 2 .Theenergyofaeldisproportionaltothe squareoftheeld,sotheenergyofthereradiatedisproportional to f 4 .Sincebluelighthasabouttwicethefrequencyofredlight, thisprocessisabout2 4 =16timesasstrongforblueasforred, andthat'swhytheskyisblue. 6.5Calculatingenergyinelds Wehaveseenthattheenergystoredinawaveactuallytheenergy densityistypicallyproportionaltothesquareofthewave'samplitude.Fieldsofforcecanmakewavepatterns,forwhichwemight expectthesametobetrue.Thisturnsouttobetruenotonlyfor 156 Chapter6Electromagnetism

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wave-likeeldpatternsbutforallelds: energystoredinthegravitationaleldperm 3 = )]TJ/F15 10.9091 Tf 17.274 7.38 Td [(1 8 G j g j 2 energystoredintheelectriceldperm 3 = 1 8 k j E 2 j energystoredinthemagneticeldperm 3 = 1 2 o j B j 2 Althoughfunnyfactorsof8 andtheplusandminussignsmay haveinitiallycaughtyoureye,theyarenotthemainpoint.The importantideaisthattheenergydensityisproportionaltothe squareoftheeldstrengthinallthreecases.Werstgiveasimple numericalexampleandworkalittleontheconcepts,andthenturn ourattentiontothefactorsoutinfront. Gettingkilledbyasolenoidexample8 Solenoidsareverycommonelectricaldevices,buttheycanbea hazardtosomeonewhoisworkingonthem.Imagineasolenoid thatinitiallyhasaDCcurrentpassingthroughit.Thecurrentcreatesamagneticeldinsideandaroundit,whichcontainsenergy. Nowsupposethatwebreakthecircuit.Sincethereisnolonger acompletecircuit,currentwillquicklystopowing,andthemagneticeldwillcollapseveryquickly.Theeldhadenergystored init,andevenasmallamountofenergycancreateadangerouspowersurgeifreleasedoverashortenoughtimeinterval.It isprudentnottoddlewithasolenoidthathascurrentowing throughit,sincebreakingthecircuitcouldbehazardoustoyour health. Asatypicalnumericalestimate,let'sassumea40cm 40cm 40cmsolenoidwithaninteriormagneticeldof1.0Tquite astrongeld.Forthesakeofthisroughestimate,weignore theexterioreld,whichisweak,andassumethatthesolenoidis cubicalinshape.Theenergystoredintheeldis energyperunitvolumevolume= 1 2 o j B j 2 V =3 10 4 J That'salotofenergy! Inchapter5whenwediscussedtheoriginalreasonforintroducingtheconceptofaeldofforce,aprimemotivationwasthat otherwisetherewasnowaytoaccountfortheenergytransfersinvolvedwhenforcesweredelayedbyaninterveningdistance.We usedtothinkoftheuniverse'senergyasconsistingof Section6.5Calculatingenergyinelds 157

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kineticenergy +gravitationalpotentialenergybasedonthedistancesbetween objectsthatinteractgravitationally +electricpotentialenergybasedonthedistancesbetween objectsthatinteractelectrically +magneticpotentialenergybasedonthedistancesbetween objectsthatinteractmagnetically, butinnonstaticsituationswemustuseadierentmethod: kineticenergy +gravitationalpotentialenergystoredingravitationalelds +electricpotentialenergystoredinelectricelds +magneticpotentialstoredinmagneticelds 158 Chapter6Electromagnetism

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v / Example9. w / Example10. Surprisingly,thenewmethodstillgivesthesameanswersforthe staticcases. Energystoredinacapacitorexample9 Apairofparallelmetalplates,seenfromthesideingurev, canbeusedtostoreelectricalenergybyputtingpositivecharge ononesideandnegativechargeontheother.Suchadeviceis calledacapacitor.Wehaveencounteredsuchanarrangement previously,butthereitspurposewastodeectabeamofelectrons,nottostoreenergy. Intheoldmethodofdescribingpotentialenergy,1,wethinkin termsofthemechanicalworkthathadtobedonetoseparate thepositiveandnegativechargesontothetwoplates,working againsttheirelectricalattraction.Thenewdescription,2,attributesthestorageofenergytothenewlycreatedelectriceld occupyingthevolumebetweentheplates.Sincethisisastatic case,bothmethodsgivethesame,correctanswer. Potentialenergyofapairofoppositechargesexample10 Imaginetakingtwooppositecharges,w,thatwereinitiallyfar apartandallowingthemtocometogetherundertheinuence oftheirelectricalattraction. Accordingtotheoldmethod,potentialenergyislostbecausethe electricforcedidpositiveworkasitbroughtthechargestogether. Thismakessensebecauseastheycometogetherandaccelerateitistheirpotentialenergythatisbeinglostandconvertedto kineticenergy. Bythenewmethod,wemustaskhowtheenergystoredinthe electriceldhaschanged.Intheregionindicatedapproximately bytheshadinginthegure,thesuperposingeldsofthetwo chargesundergopartialcancellationbecausetheyareinopposingdirections.Theenergyintheshadedregionisreducedby thiseffect.Intheunshadedregion,theeldsreinforce,andthe energyisincreased. Itwouldbequiteaprojecttodoanactualnumericalcalculationof theenergygainedandlostinthetworegionsthisisacasewhere theoldmethodofndingenergygivesgreatereaseofcomputation,butitisfairlyeasytoconvinceoneselfthattheenergyis lesswhenthechargesarecloser.Thisisbecausebringingthe chargestogethershrinksthehigh-energyunshadedregionand enlargesthelow-energyshadedregion. Energyinanelectromagneticwaveexample11 Theoldmethodwouldgivezeroenergyforaregionofspace containinganelectromagneticwavebutnocharges.Thatwould bewrong!Wecanonlyusetheoldmethodinstaticcases. Nowlet'sgiveatleastsomejusticationfortheotherfeatures ofthethreeexpressionsforenergydensity, )]TJ/F18 7.9701 Tf 15.554 4.295 Td [(1 8 G j g j 2 1 8 k j E 2 j ,and Section6.5Calculatingenergyinelds 159

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x / DiscussionquestionA. 1 2 o j B j 2 ,besidestheproportionalitytothesquareoftheeldstrength. First,whythedierentplusandminussigns?Thebasicideais thatthesignshavetobeoppositeinthegravitationalandelectric casesbecausethereisanattractionbetweentwopositivemasses whicharetheonlykindthatexist,buttwopositivechargeswould repel.Sincewe'vealreadyseenexampleswherethepositivesignin theelectricenergymakessense,thegravitationalenergyequation mustbetheonewiththeminussign. Itmayalsoseemstrangethattheconstants G k ,and o arein thedenominator.Theytellushowstrongthethreedierentforces are,soshouldn'ttheybeontop?No.Consider,forinstance,an alternativeuniverseinwhichgravityistwiceasstrongasinours. Thenumericalvalueof G isdoubled.Because G isdoubled,allthe gravitationaleldstrengthsaredoubledaswell,whichquadruples thequantity j g j 2 .Intheexpression )]TJ/F18 7.9701 Tf 15.554 4.296 Td [(1 8 G j g j 2 ,wehavequadrupled somethingontopanddoubledsomethingonthebottom,which makestheenergytwiceasbig.Thatmakesperfectsense. DiscussionQuestions A Thegureshowsapositivechargeinthegapbetweentwocapacitor plates.Firstmakealargedrawingoftheeldpatternthatwouldbeformed bythecapacitoritself,withouttheextrachargeinthemiddle.Next,show howtheeldpatternchangeswhenyouaddtheparticleatthesetwopositions.Comparetheenergyoftheelectriceldsinthetwocases.Does thisagreewithwhatyouwouldhaveexpectedbasedonyourknowledge ofelectricalforces? B Criticizethefollowingstatement:Asolenoidmakesachargeinthe spacesurroundingit,whichdissipateswhenyoureleasetheenergy. C Inexample10,Iarguedthattheeldssurroundingapositive andnegativechargecontainlessenergywhenthechargesarecloser together.Perhapsasimplerapproachistoconsiderthetwoextremepossibilities:thecasewherethechargesareinnitelyfarapart,andtheone inwhichtheyareatzerodistancefromeachother,i.e.,rightontopof eachother.Carryoutthisreasoningforthecaseofapositivecharge andanegativechargeofequalmagnitude,twopositivechargesof equalmagnitude,thegravitationalenergyoftwoequalmasses. 6.6 ? Symmetryandhandedness ThephysicistRichardFeynmanhelpedtogetmehookedonphysics withaneducationallmcontainingthefollowingpuzzle.Imagine thatyouestablishradiocontactwithanalienonanotherplanet. Neitherofyouevenknowswheretheotherone'splanetis,andyou aren'tabletoestablishanylandmarksthatyoubothrecognize.You managetolearnquiteabitofeachother'slanguages,butyou're stumpedwhenyoutrytoestablishthedenitionsofleftandright or,equivalently,clockwiseandcounterclockwise.Isthereanyway todoit? 160 Chapter6Electromagnetism

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Iftherewasanywaytodoitwithoutreferencetoexternallandmarks,thenitwouldimplythatthelawsofphysicsthemselveswere asymmetric,whichwouldbestrange.Whyshouldtheydistinguish leftfromright?Thegravitationaleldpatternsurroundingastar orplanetlooksthesameinamirror,andthesamegoesforelectricelds.However,theeldpatternsshowninsection6.2seem toviolatethisprinciple,butdotheyreally?Couldyouusethese patternstoexplainleftandrighttothealien?Infact,theansweris no.Ifyoulookbackatthedenitionofthemagneticeldinsection 6.1,italsocontainsareferencetohandedness:thecounterclockwise directionoftheloop'scurrentasviewedalongthemagneticeld. Thealiensmighthavereversedtheirdenitionofthemagneticeld, inwhichcasetheirdrawingsofeldpatternswouldlooklikemirror imagesofours. Untilthemiddleofthetwentiethcentury,physicistsassumed thatanyreasonablesetofphysicallawswouldhavetohavethis kindofsymmetrybetweenleftandright.Anasymmetrywould begrotesque.Whatevertheiraestheticfeelings,theyhadtochange theiropinionsaboutrealitywhenexperimentsshowedthattheweak nuclearforcesection6.5violatesright-leftsymmetry!Itisstill amysterywhyright-leftsymmetryisobservedsoscrupulouslyin general,butisviolatedbyoneparticulartypeofphysicalprocess. Section6.6 ? Symmetryandhandedness 161

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Summary SelectedVocabulary magneticeld..aeldofforce,denedintermsofthetorque exertedonatestdipole magneticdipole.anobject,suchasacurrentloop,anatom, orabarmagnet,thatexperiencestorquesdue tomagneticforces;thestrengthofmagnetic dipolesismeasuredbycomparisonwithastandarddipoleconsistingofasquareloopofwire ofagivensizeandcarryingagivenamountof current induction.....theproductionofanelectriceldbyachangingmagneticeld,orvice-versa Notation B .........themagneticeld D m ........magneticdipolemoment Summary Magnetismisaninteractionofmovingchargeswithothermoving charges.Themagneticeldisdenedintermsofthetorqueona magnetictestdipole.Ithasnosourcesorsinks;magneticeld patternsneverconvergeonordivergefromapoint. Themagneticandelectriceldsareintimatelyrelated.The principleofinductionstatesthatanychangingelectriceldproduces amagneticeldinthesurroundingspace,andvice-versa.These inducedeldstendtoformwhirlpoolpatterns. Themostimportantconsequenceoftheprincipleofinduction isthattherearenopurelymagneticorpurelyelectricwaves.Disturbancesintheelectricalandmagneticeldspropagateoutward ascombinedmagneticandelectricwaves,withawell-denedrelationshipbetweentheirmagnitudesanddirections.Theseelectromagneticwavesarewhatlightismadeof,butotherformsofelectromagneticwavesexistbesidesvisiblelight,includingradiowaves, x-rays,andgammarays. Fieldsofforcecontainenergy.Thedensityofenergyisproportionaltothesquareofthemagnitudeoftheeld.Inthecase ofstaticelds,wecancalculatepotentialenergyeitherusingthe previousdenitionintermsofmechanicalworkorbycalculating theenergystoredintheelds.Iftheeldsarenotstatic,theold methodgivesincorrectresultsandthenewonemustbeused. 162 Chapter6Electromagnetism

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Problem2. Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 Inanelectricalstorm,thecloudandthegroundactlikea parallel-platecapacitor,whichtypicallychargesupduetofrictional electricityincollisionsoficeparticlesinthecoldupperatmosphere. Lightningoccurswhenthemagnitudeoftheelectriceldbuildsup toacriticalvalue, E c ,atwhichairisionized. aTreatthecloudasaatsquarewithsidesoflength L .Ifitisat aheight h abovetheground,ndtheamountofenergyreleasedin thelightningstrike. p bBasedonyouranswerfromparta,whichismoredangerous,a lightningstrikefromahigh-altitudecloudoralow-altitudeone? cMakeanorder-of-magnitudeestimateoftheenergyreleasedby atypicallightningbolt,assumingreasonablevaluesforitssizeand altitude. E c isabout10 6 V/m. Seeproblem21foranoteonhowrecentresearchaectsthisestimate. 2 Theneuroninthegurehasbeendrawnfairlyshort,butsome neuronsinyourspinalcordhavetailsaxonsuptoameterlong. Theinnerandoutersurfacesofthemembraneactastheplates" ofacapacitor.Thefactthatithasbeenrolledupintoacylinder hasverylittleeect.Inordertofunction,theneuronmustcreate avoltagedierence V betweentheinnerandoutersurfacesofthe membrane.Letthemembrane'sthickness,radius,andlengthbe t r ,and L .aCalculatetheenergythatmustbestoredintheelectric eldfortheneurontodoitsjob.Inreallife,themembraneismade outofasubstancecalledadielectric,whoseelectricalproperties increasetheamountofenergythatmustbestored.Forthesakeof thisanalysis,ignorethisfact.[Hint:Thevolumeofthemembrane isessentiallythesameasifitwasunrolledandattenedout.] p bAnorganism'sevolutionarytnessshouldbebetterifitneeds lessenergytooperateitsnervoussystem.Basedonyouranswerto parta,whatwouldyouexpectevolutiontodotothedimensions t and r ?Whatotherconstraintswouldkeeptheseevolutionarytrends fromgoingtoofar? 3 Considertwosolenoids,oneofwhichissmallersothatitcan beputinsidetheother.Assumetheyarelongenoughsothateach oneonlycontributessignicantlytotheeldinsideitself,andthe interioreldsarenearlyuniform.Considerthecongurationwhere thesmalloneisinsidethebigonewiththeircurrentscirculatingin thesamedirection,andasecondcongurationinwhichthecurrents circulateinoppositedirections.Comparetheenergiesofthesecongurationswiththeenergywhenthesolenoidsarefarapart.Based Problems 163

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Problem4. onthisreasoning,whichcongurationisstable,andinwhichcongurationwillthelittlesolenoidtendtogettwistedaroundorspit out?[Hint:Astablesystemhaslowenergy;energywouldhaveto beaddedtochangeitsconguration.] 4 Thegureshowsanestedpairofcircularwireloopsused tocreatemagneticelds.Thetwistingoftheleadsisapractical trickforreducingthemagneticeldstheycontribute,sotheelds areverynearlywhatwewouldexpectforanidealcircularcurrent loop.Thecoordinatesystembelowistomakeiteasiertodiscuss directionsinspace.Oneloopisinthe y )]TJ/F20 10.9091 Tf 10.833 0 Td [(z plane,theotherinthe x )]TJ/F20 10.9091 Tf 11.2 0 Td [(y plane.Eachoftheloopshasaradiusof1.0cm,andcarries 1.0Ainthedirectionindicatedbythearrow. aUsingtheequationinoptionalsection6.2,calculatethemagnetic eldthatwouldbeproducedby one suchloop,atitscenter. p bDescribethedirectionofthemagneticeldthatwouldbeproduced,atitscenter,bytheloopinthe x )]TJ/F20 10.9091 Tf 10.909 0 Td [(y planealone. cDothesamefortheotherloop. dCalculatethemagnitudeofthemagneticeldproducedbythe twoloopsincombination,attheircommoncenter.Describeits direction. p 5 aShowthatthequantity p 4 k= o hasunitsofvelocity. bCalculateitnumericallyandshowthatitequalsthespeedof light. cProvethatinanelectromagneticwave,halftheenergyisinthe electriceldandhalfinthemagneticeld. 6 Onemodelofthehydrogenatomhastheelectroncircling aroundtheprotonataspeedof2.2 10 6 m/s,inanorbitwitha radiusof0.05nm.Althoughtheelectronandprotonreallyorbit aroundtheircommoncenterofmass,thecenterofmassisveryclose totheproton,sinceitis2000timesmoremassive.Forthisproblem, assumetheprotonisstationary.Inhomeworkproblem9onpage 102,youcalculatedtheelectriccurrentcreated. aNowestimatethemagneticeldcreatedatthecenterofthe atombytheelectron.Wearetreatingthecirclingelectronasacurrentloop,eventhoughit'sonlyasingleparticle. p bDoestheprotonexperienceanonzeroforcefromtheelectron's magneticeld?Explain. cDoestheelectronexperienceamagneticeldfromtheproton? Explain. dDoestheelectronexperienceamagneticeldcreatedbyitsown current?Explain. eIsthereanelectricforceactingbetweentheprotonandelectron? Ifso,calculateit. p fIsthereagravitationalforceactingbetweentheprotonandelectron?Ifso,calculateit. gAninwardforceisrequiredtokeeptheelectroninitsorbit{ 164 Chapter6Electromagnetism

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otherwiseitwouldobeyNewton'srstlawandgostraight,leaving theatom.Basedonyouranswerstothepreviousparts,whichforce orforceselectric,magneticandgravitationalcontributessignicantlytothisinwardforce? 7 [Youneedtohavereadoptionalsection6.2todothisproblem.]Supposeachargedparticleismovingthrougharegionofspace inwhichthereisanelectriceldperpendiculartoitsvelocityvector,andalsoamagneticeldperpendiculartoboththeparticle's velocityvectorandtheelectriceld.Showthattherewillbeone particularvelocityatwhichtheparticlecanbemovingthatresults inatotalforceofzeroonit.Relatethisvelocitytothemagnitudes oftheelectricandmagneticelds.Suchanarrangement,calleda velocitylter,isonewayofdeterminingthespeedofanunknown particle. 8 Ifyouputfourtimesmorecurrentthroughasolenoid,how manytimesmoreenergyisstoredinitsmagneticeld? p 9 Supposewearegivenapermanentmagnetwithacomplicated, asymmetricshape.Describehowaseriesofmeasurementswith amagneticcompasscouldbeusedtodeterminethestrengthand directionofitsmagneticeldatsomepointofinterest.Assumethat youareonlyabletoseethedirectiontowhichthecompassneedle settles;youcannotmeasurethetorqueactingonit. ? 10 Considertwosolenoids,oneofwhichissmallersothatit canbeputinsidetheother.Assumetheyarelongenoughtoact likeidealsolenoids,sothateachoneonlycontributessignicantly totheeldinsideitself,andtheinterioreldsarenearlyuniform. Considerthecongurationwherethesmalloneispartlyinsideand partlyhangingoutofthebigone,withtheircurrentscirculatingin thesamedirection.Theiraxesareconstrainedtocoincide. aFindthemagneticpotentialenergyasafunctionofthelength x ofthepartofthesmallsolenoidthatisinsidethebigone.Your equationwillincludeotherrelevantvariablesdescribingthetwo solenoids. bBasedonyouranswertoparta,ndtheforceactingbetween thesolenoids. Problems 165

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Problem11. 11 Fourlongwiresarearranged,asshown,sothattheircrosssectionformsasquare,withconnectionsattheendssothatcurrent owsthroughallfourbeforeexiting.Notethatthecurrentistothe rightinthetwobackwires,buttotheleftinthefrontwires.Ifthe dimensionsofthecross-sectionalsquareheightandfront-to-back are b ,ndthemagneticeldmagnitudeanddirectionalongthe longcentralaxis. p 12 Todothisproblem,youneedtounderstandhowtodo volumeintegralsincylindricalandsphericalcoordinates.aShow thatifyoutrytointegratetheenergystoredintheeldofalong, straightwire,theresultingenergyperunitlengthdivergesbothat r 0and r !1 .Takenatfacevalue,thiswouldimplythata certainreal-lifeprocess,theinitiationofacurrentinawire,would beimpossible,becauseitwouldrequirechangingfromastateof zeromagneticenergytoastateofinnitemagneticenergy.b Explainwhytheinnitiesat r 0and r !1 don'treallyhappen inarealisticsituation.cShowthattheelectricenergyofapoint chargedivergesat r 0,butnotat r !1 Aremarkregardingpartc:Naturedoesseemtosupplyuswith particlesthatarechargedandpointlike,e.g.,theelectron,butone couldarguethattheinniteenergyisnotreallyaproblem,because anelectrontravelingaroundanddoingthingsneithergainsnorloses inniteenergy;onlyaninnite change inpotentialenergywouldbe physicallytroublesome.However,therearereal-lifeprocessesthat createanddestroypointlikechargedparticles,e.g.,theannihilation ofanelectronandantielectronwiththeemissionoftwogamma rays.Physicistshave,infact,beenstrugglingwithinnitieslike thissinceabout1950,andtheissueisfarfromresolved.Some theoristsproposethatapparentlypointlikeparticlesareactuallynot pointlike:closeup,anelectronmightbelikealittlecircularloopof string. R ? 13 Thepurposeofthisproblemistondtheforceexperiencedby astraight,current-carryingwirerunningperpendiculartoauniform magneticeld.aLet A bethecross-sectionalareaofthewire, n thenumberoffreechargedparticlesperunitvolume, q thecharge perparticle,and v theaveragevelocityoftheparticles.Showthat thecurrentis I = Avnq .bShowthatthemagneticforceperunit lengthis AvnqB .cCombiningtheseresults,showthattheforce 166 Chapter6Electromagnetism

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Problem15. onthewireperunitlengthisequalto IB Solution,p.206 14 Supposetwolong,parallelwiresarecarryingcurrent I 1 and I 2 .Thecurrentsmaybeeitherinthesamedirectionorinoppositedirections.aUsingtheinformationfromsection6.2,determineunderwhatconditionstheforceisattractive,andunder whatconditionsitisrepulsive.Notethat,becauseofthedicultiesexploredinproblem12,it'spossibletogetyourselftiedupin knotsifyouusetheenergyapproachofsection6.5.bStarting fromtheresultofproblem13,calculatetheforceperunitlength. Solution,p.206 15 Thegureshowscross-sectionalviewsoftwocubicalcapacitors,andacross-sectionalviewofthesametwocapacitorsput togethersothattheirinteriorscoincide.Acapacitorwiththeplates closetogetherhasanearlyuniformelectriceldbetweentheplates, andalmostzeroeldoutside;thesecapacitorsdon'thavetheirplates veryclosetogethercomparedtothedimensionsoftheplates,but forthepurposesofthisproblem,assumethattheystillhaveapproximatelythekindofidealizedeldpatternshowninthegure. Eachcapacitorhasaninteriorvolumeof1.00m 3 ,andischargedup tothepointwhereitsinternaleldis1.00V/m.aCalculatethe energystoredintheelectriceldofeachcapacitorwhentheyare separate.bCalculatethemagnitudeoftheinterioreldwhenthe twocapacitorsareputtogetherinthemannershown.Ignoreeects arisingfromtheredistributionofeachcapacitor'schargeunderthe inuenceoftheothercapacitor.cCalculatetheenergyofthe put-togetherconguration.Doesassemblingthemlikethisrelease energy,consumeenergy,orneither? 16 Section6.2statesthefollowingrule: Forapositivelychargedparticle,thedirectionofthe F vectoristhe onesuchthatifyousightalongit,the B vectorisclockwisefrom the v vector. Makeathree-dimensionalmodelofthethreevectorsusingpencils orrolled-uppiecesofpapertorepresentthevectorsassembledwith theirtailstogether.Nowwritedowneverypossiblewayinwhich therulecouldberewrittenbyscramblingupthethreesymbols F B ,and v .Referringtoyourmodel,whicharecorrectandwhichare incorrect? 17 Provethatanytwoplanarcurrentloopswiththesamevalue of IA willexperiencethesametorqueinamagneticeld,regardless oftheirshapes.Inotherwords,thedipolemomentofacurrentloop canbedenedas IA ,regardlessofwhetheritsshapeisasquare. ? Problems 167

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Problem18. 18 AHelmholtzcoilisdenedasapairofidenticalcircular coilsseparatedbyadistance, h ,equaltotheirradius, b .Eachcoil mayhavemorethanoneturnofwire.Currentcirculatesinthe samedirectionineachcoil,sotheeldstendtoreinforceeachother intheinteriorregion.Thiscongurationhastheadvantageofbeing fairlyopen,sothatotherapparatuscanbeeasilyplacedinsideand subjectedtotheeldwhileremainingvisiblefromtheoutside.The choiceof h = b resultsinthemostuniformpossibleeldnearthe center.aFindthepercentagedropintheeldatthecenterof onecoil,comparedtothefullstrengthatthecenterofthewhole apparatus.bWhatvalueof h notequalto b wouldmakethis percentagedierenceequaltozero? 19 aInthephotoofthevacuumtubeapparatusinsection 6.2,inferthedirectionofthemagneticeldfromthemotionofthe electronbeam.bBasedonyouranswertoa,ndthedirectionof thecurrentsinthecoils.cWhatdirectionaretheelectronsinthe coilsgoing?dArethecurrentsinthecoilsrepellingorattracting thecurrentsconsistingofthebeaminsidethetube?Comparewith partaofproblem14. 20 Inthephotoofthevacuumtubeapparatusinsection6.2, anapproximatelyuniformmagneticeldcausedcircularmotion.Is thereanyotherpossibilitybesidesacircle?Whatcanhappenin general? ? 21 Inproblem1,youestimatedtheenergyreleasedinabolt oflightning,basedontheenergystoredintheelectriceldimmediatelybeforethelightningoccurs.Theassumptionwasthatthe eldwouldbuilduptoacertainvalue,whichiswhatisnecessary toionizeair.However,real-lifemeasurementsalwaysseemedto showelectriceldsstrengthsroughtly10timessmallerthanthose requiredinthatmodel.Foralongtime,itwasn'tclearwhetherthe eldmeasurementswerewrong,orthemodelwaswrong.Research carriedoutin2003seemstoshowthatthemodelwaswrong.Itis nowbelievedthatthenaltriggeringoftheboltoflightningcomes fromcosmicraysthatentertheatmosphereandionizesomeofthe air.Iftheeldis10timessmallerthanthevalueassumedinproblem1,whateectdoesthishaveonthenalresultofproblem1? 22 Insection6.2Igaveanequationforthemagneticeldin theinteriorofasolenoid,butthatequationdoesn'tgivetheright answernearthemouthsorontheoutside.Althoughingeneralthe computationoftheeldintheseotherregionsiscomplicated,itis possibletondaprecise,simpleresultfortheeldatthecenterof oneofthemouths,usingonlysymmetryandvectoraddition.What isit? Solution,p.207 ? 168 Chapter6Electromagnetism

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ChapterA Capacitanceand Inductance Thischapterisoptional. Thelongroadleadingfromthelightbulbtothecomputerstarted withoneveryimportantstep:theintroductionoffeedbackintoelectroniccircuits.Althoughtheprincipleoffeedbackhasbeenunderstoodandandappliedtomechanicalsystemsforcenturies,andto electricalonessincetheearlytwentiethcentury,formostofusthe wordevokesanimageofJimiHendrixorsomemorerecentguitar herointentionallycreatingearsplittingscreeches,oroftheschool principaldoingthesameinadvertentlyintheauditorium.Inthe guitarexample,themusicianstandsinfrontoftheampandturns itupsohighthatthesoundwavescomingfromthespeakercome backtotheguitarstringandmakeitshakeharder.Thisisanexampleof positive feedback:theharderthestringvibrates,thestronger thesoundwaves,andthestrongerthesoundwaves,theharderthe stringvibrates.Theonlylimitisthepower-handlingabilityofthe amplier. Negativefeedbackisequallyimportant.Yourthermostat,for example,providesnegativefeedbackbykickingtheheaterowhen thehousegetswarmenough,andbyringitupagainwhenit getstoocold.Thiscausesthehouse'stemperaturetooscillateback andforthwithinacertainrange.Justasout-of-controlexponential freak-outsareacharacteristicbehaviorofpositive-feedbacksystems, oscillationistypicalincasesofnegativefeedback.Youhavealready studiednegativefeedbackextensivelyin VibrationsandWaves in thecaseofamechanicalsystem,althoughwedidn'tcallitthat. A.1Capacitanceandinductance Inamechanicaloscillation,energyisexchangedrepetitivelybetween potentialandkineticforms,andmayalsobesiphonedointhe formofheatdissipatedbyfriction.Inanelectricalcircuit,resistors arethecircuitelementsthatdissipateheat.Whataretheelectrical analogsofstoringandreleasingthepotentialandkineticenergyofa vibratingobject?Whenyouthinkofenergystorageinanelectrical circuit,youarelikelytoimagineabattery,butevenrechargeable batteriescanonlygothrough10or100cyclesbeforetheywearout. 169

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a / Thesymbolforacapacitor. b / Somecapacitors. c / Twocommongeometries forinductors.Thecylindrical shapeontheleftiscalleda solenoid. d / Thesymbolforaninductor. e / Someinductors. Inaddition,batteriesarenotabletoexchangeenergyonashort enoughtimescaleformostapplications.Thecircuitinamusical synthesizermaybecalledupontooscillatethousandsoftimesa second,andyourmicrowaveovenoperatesatgigahertzfrequencies. Insteadofbatteries,wegenerallyusecapacitorsandinductorsto storeenergyinoscillatingcircuits.Capacitors,whichyou'vealready encountered,storeenergyinelectricelds.Aninductordoesthe samewithmagneticelds. Capacitors Acapacitor'senergyexistsinitssurroundingelectricelds.Itis proportionaltothesquareoftheeldstrength,whichisproportional tothechargesontheplates.Ifweassumetheplatescarrycharges thatarethesameinmagnitude,+ q and )]TJ/F20 10.9091 Tf 8.485 0 Td [(q ,thentheenergystored inthecapacitormustbeproportionalto q 2 .Forhistoricalreasons, wewritetheconstantofproportionalityas1 = 2 C E C = 1 2 C q 2 Theconstant C isageometricalpropertyofthecapacitor,calledits capacitance. Basedonthisdenition,theunitsofcapacitancemustbecoulombs squaredperjoule,andthiscombinationismoreconvenientlyabbreviatedasthefarad,1F=1C 2 = J.Condenser"isalessformal termforacapacitor.Notethatthelabelsprintedoncapacitors oftenuseMFtomean F,eventhoughMFshouldreallybethe symbolformegafarads,notmicrofarads.Confusiondoesn'tresult fromthisnonstandardnotation,sincepicofaradandmicrofaradvaluesarethemostcommon,anditwasn'tuntilthe1990'sthateven millifaradandfaradvaluesbecameavailableinpracticalphysical sizes.Figureashowsthesymbolusedinschematicstorepresenta capacitor. Inductors Anycurrentwillcreateamagneticeld,soinfacteverycurrentcarryingwireinacircuitactsasaninductor!However,thistype ofstray"inductanceistypicallynegligible,justaswecanusually ignorethestrayresistanceofourwiresandonlytakeintoaccount theactualresistors.Tostoreanyappreciableamountofmagnetic energy,oneusuallyusesacoilofwiredesignedspecicallytobe aninductor.Alltheloops'contributiontothemagneticeldadd togethertomakeastrongereld.Unlikecapacitorsandresistors, practicalinductorsareeasytomakebyhand.Onecanforinstance spoolsomewirearoundashortwoodendowel,putthespoolinside aplasticaspirinbottlewiththeleadshangingout,andllthebottle withepoxytomakethewholethingrugged.Aninductorlikethis, intheformcylindricalcoilofwire,iscalledasolenoid,c,anda stylizedsolenoid,d,isthesymbolusedtorepresentaninductorin acircuitregardlessofitsactualgeometry. 170 ChapterACapacitanceandInductance

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h / Avariablecapacitor. f / Inductancesinseriesadd. g / Capacitancesinparallel add. Howmuchenergydoesaninductorstore?Theenergydensityis proportionaltothesquareofthemagneticeldstrength,whichis inturnproportionaltothecurrentowingthroughthecoiledwire, sotheenergystoredintheinductormustbeproportionalto I 2 .We write L= 2fortheconstantofproportionality,giving E L = L 2 I 2 Asinthedenitionofcapacitance,wehaveafactorof1/2, whichispurelyamatterofdenition.Thequantity L iscalledthe inductance oftheinductor,andweseethatitsunitsmustbejoules peramperesquared.Thisclumsycombinationofunitsismore commonlyabbreviatedasthehenry,1henry=1J = A 2 .Rather thanmemorizingthisdenition,itmakesmoresensetoderiveit whenneededfromthedenitionofinductance.Manypeopleknow inductorssimplyascoils,"orchokes,"andwillnotunderstand youifyourefertoaninductor,"buttheywillstillreferto L asthe inductance,"notthecoilance"orchokeance!" Identicalinductancesinseriesexample1 Iftwoinductorsareplacedinseries,anycurrentthatpasses throughthecombineddoubleinductormustpassthroughboth itsparts.Thusbythedenitionofinductance,theinductanceis doubledaswell.Ingeneral,inductancesinseriesadd,justlike resistances.Thesamekindofreasoningalsoshowsthattheinductanceofasolenoidisapproximatelyproportionaltoitslength, assumingthenumberofturnsperunitlengthiskeptconstant. Identicalcapacitancesinparallelexample2 Whentwoidenticalcapacitancesareplacedinparallel,anycharge depositedattheterminalsofthecombineddoublecapacitorwill divideitselfevenlybetweenthetwoparts.Theelectriceldssurroundingeachcapacitorwillbehalftheintensity,andtherefore storeonequartertheenergy.Twocapacitors,eachstoringone quartertheenergy,givehalfthetotalenergystorage.Sincecapacitanceisinverselyrelatedtoenergystorage,thisimpliesthat identicalcapacitancesinparallelgivedoublethecapacitance.In general,capacitancesinparalleladd.Thisisunlikethebehaviorofinductorsandresistors,forwhichseriescongurationsgive addition. Thisisconsistentwiththefactthatthecapacitanceofasingle parallel-platecapacitorproportionaltotheareaoftheplates.If wehavetwoparallel-platecapacitors,andwecombinethemin parallelandbringthemveryclosetogethersidebyside,wehave producedasinglecapacitorwithplatesofdoublethearea,andit hasapproximatelydoublethecapacitance. Inductancesinparallelandcapacitancesinseriesareexplored inhomeworkproblems4and6. SectionA.1Capacitanceandinductance 171

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j / AseriesLRCcircuit. k / Amechanicalanalogyfor theLRCcircuit. i / DiscussionquestionB. Avariablecapacitorexample3 Figureh/1showstheconstructionofavariablecapacitoroutof twoparallelsemicirclesofmetal.Oneplateisxed,whilethe othercanberotatedabouttheircommonaxiswithaknob.The oppositechargesonthetwoplatesareattractedtooneanother, andthereforetendtogatherintheoverlappingarea.Thisoverlappingarea,then,istheonlyareathateffectivelycontributesto thecapacitance,andturningtheknobchangesthecapacitance. Thesimpledesigncanonlyprovideverysmallcapacitancevalues,soinpracticeoneusuallyusesabankofcapacitors,wired inparallel,withallthemovingpartsonthesameshaft. DiscussionQuestions A Supposethattwoparallel-platecapacitorsarewiredinparallel,and areplacedveryclosetogether,sidebyside,sothattheireldsoverlap. Willtheresultingcapacitancebetoosmall,ortoobig?Couldyoutwist thecircuitintoadifferentshapeandmaketheeffectbetheotherway around,ormaketheeffectvanish?Howaboutthecaseoftwoinductors inseries? B Mostpracticalcapacitorsdonothaveanairgaporvacuumgap betweentheplates;instead,theyhaveaninsulatingsubstancecalleda dielectric.Wecanthinkofthemoleculesinthissubstanceasdipolesthat arefreetorotateatleastalittle,butthatarenotfreetomovearound, sinceitisasolid.Thegureshowsahighlystylizedandunrealisticway ofvisualizingthis.Weimaginethatallthedipolesareintiallyturnedsideways,,andthatasthecapacitorischarged,theyallrespondbyturning throughacertainangle,.Inreality,thescenemightbemuchmore random,andthealignmenteffectmuchweaker. Forsimplicity,imagineinsertingjustoneelectricdipoleintothevacuum gap.Foragivenamountofchargeontheplates,howdoesthisaffect theamountofenergystoredintheelectriceld?Howdoesthisaffectthe capacitance? Nowredotheanalysisintermsofthemechanicalworkneededinorder tochargeuptheplates. A.2Oscillations Figurejshowsthesimplestpossibleoscillatingcircuit.Foranyusefulapplicationitwouldactuallyneedtoincludemorecomponents. Forexample,ifitwasaradiotuner,itwouldneedtobeconnectedto anantennaandanamplier.Nevertheless,alltheessentialphysics isthere. Wecananalyzeitwithoutanysweatortearswhatsoever,simplybyconstructingananalogywithamechanicalsystem.Ina mechanicaloscillator,k,wehavetwoformsofstoredenergy, E spring = 1 2 kx 2 K = 1 2 mv 2 172 ChapterACapacitanceandInductance

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Inthecaseofamechanicaloscillator,wehaveusuallyassumed africtionforceoftheformthatturnsouttogivethenicestmathematicalresults, F = )]TJ/F20 10.9091 Tf 8.485 0 Td [(bv .Inthecircuit,thedissipationofenergy intoheatoccursviatheresistor,withnomechanicalforceinvolved, soinordertomaketheanalogy,weneedtorestatetheroleofthe frictionforceintermsofenergy.Thepowerdissipatedbyfriction equalsthemechanicalworkitdoesinatimeinterval t ,dividedby t P = W= t = F x= t = Fv = )]TJ/F20 10.9091 Tf 8.485 0 Td [(bv 2 ,so rateofheatdissipation= )]TJ/F20 10.9091 Tf 8.485 0 Td [(bv 2 self-checkA Equationhas x squared,andequationsandhave v squared. Becausethey'resquared,theresultsdon'tdependonwhetherthese variablesarepositiveornegative.Doesthismakephysicalsense? Answer,p.204 Inthecircuit,thestoredformsofenergyare E C = 1 2 C q 2 0 E L = 1 2 LI 2 0 andtherateofheatdissipationintheresistoris rateofheatdissipation= )]TJ/F20 10.9091 Tf 8.485 0 Td [(RI 2 0 Comparingthetwosetsofequations,werstformanalogiesbetween quantitiesthatrepresentthestateofthesystematsomemoment intime: x $ q v $ I self-checkB Howis v relatedmathematicallyto x ?Howis I connectedto q ?Arethe tworelationshipsanalogous? Answer,p.204 Nextwerelatetheonesthatdescribethesystem'spermanent characteristics: k $ 1 =C m $ L b $ R Sincethemechanicalsystemnaturallyoscillateswithaperiod T =2 p m=k ,wecanimmediatelysolvetheelectricalversionby analogy,giving T =2 p LC SectionA.2Oscillations 173

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Ratherthanperiod, T ,andfrequency, f ,itturnsouttobemore convenientifweworkwiththequantity =2 f ,whichcanbe interpretedasthenumberofradianspersecond.Then = 1 p LC Sincetheresistance R isanalogousto b inthemechanicalcase, wendthatthe Q qualityfactor,notchargeoftheresonance isinverselyproportionalto R ,andthewidthoftheresonanceis directlyproportionalto R Tuningaradioreceiverexample4 Aradioreceiverusesthiskindofcircuittopickoutthedesired station.Sincethereceiverresonatesataparticularfrequency, stationswhosefrequenciesarefaroffwillnotexciteanyresponse inthecircuit.Thevalueof R hastobesmallenoughsothatonly onestationatatimeispickedup,butbigenoughsothatthe tunerisn'ttootouchy.Theresonantfrequencycanbetunedby adjustingeither L or C ,butvariablecapacitorsareeasiertobuild thanvariableinductors. Anumericalcalculationexample5 Thephonecompanysendsmorethanoneconversationatatime overthesamewire,whichisaccomplishedbyshiftingeachvoice signalintodifferentrangeoffrequenciesduringtransmission.The numberofsignalsperwirecanbemaximizedbymakingeach rangeoffrequenciesknownasabandwidthassmallaspossible.Itturnsoutthatonlyarelativelynarrowrangeoffrequencies isnecessaryinordertomakeahumanvoiceintelligible,sothe phonecompanyltersoutalltheextremehighsandlows.Thisis whyyourphonevoicesoundsdifferentfromyournormalvoice. IfthelterconsistsofanLRCcircuitwithabroadresonance centeredaround1.0kHz,andthecapacitoris1 Fmicrofarad, whatinductancevaluemustbeused? Solvingfor L ,wehave L = 1 C 2 = 1 )]TJ/F39 7.9701 Tf 6.587 0 Td [(6 F 10 3 s )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 2 =2.5 10 )]TJ/F39 7.9701 Tf 6.586 0 Td [(3 F )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 s 2 Checkingthatthesereallyarethesameunitsashenriesisalittle tedious,butitbuildscharacter: F )]TJ/F39 7.9701 Tf 6.586 0 Td [(1 s 2 =C 2 = J )]TJ/F39 7.9701 Tf 6.587 0 Td [(1 s 2 =J C )]TJ/F39 7.9701 Tf 6.586 0 Td [(2 s 2 =J = A 2 =H 174 ChapterACapacitanceandInductance

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Theresultis25mHmillihenries. Thisisactuallyquitealargeinductancevalue,andwouldrequire abig,heavy,expensivecoil.Infact,thereisatrickformaking thiskindofcircuitsmallandcheap.Thereisakindofsilicon chipcalledanop-amp,which,amongotherthings,canbeused tosimulatethebehaviorofaninductor.Themainlimitationofthe op-ampisthatitisrestrictedtolow-powerapplications. A.3VoltageandCurrent Whatisphysicallyhappeninginoneoftheseoscillatingcircuits? Let'srstlookatthemechanicalcase,andthendrawtheanalogy tothecircuit.Forsimplicity,let'signoretheexistenceofdamping, sothereisnofrictioninthemechanicaloscillator,andnoresistance intheelectricalone. Supposewetakethemechanicaloscillatorandpullthemass awayfromequilibrium,thenreleaseit.Sincefrictiontendstoresist thespring'sforce,wemightnaivelyexpectthathavingzerofriction wouldallowthemasstoleapinstantaneouslytotheequilibrium position.Thiscan'thappen,however,becausethemasswouldhave tohaveinnitevelocityinordertomakesuchaninstantaneousleap. Innitevelocitywouldrequireinnitekineticenergy,buttheonly kindofenergythatisavailableforconversiontokineticistheenergy storedinthespring,andthatisnite,notinnite.Ateachstepon itswaybacktoequilibrium,themass'svelocityiscontrolledexactly bytheamountofthespring'senergythathassofarbeenconverted intokineticenergy.Afterthemassreachesequilibrium,itovershoots duetoitsownmomentum.Itperformsidenticaloscillationsonboth sidesofequilibrium,anditneverlosesamplitudebecausefrictionis notavailabletoconvertmechanicalenergyintoheat. Nowwiththeelectricaloscillator,theanalogofpositionischarge. Pullingthemassawayfromequilibriumislikedepositingcharges + q and )]TJ/F20 10.9091 Tf 8.484 0 Td [(q ontheplatesofthecapacitor.Sinceresistancetends toresisttheowofcharge,wemightimaginethatwithnofrictionpresent,thechargewouldinstantlyowthroughtheinductor whichis,afterall,justapieceofwire,andthecapacitorwould dischargeinstantly.However,suchaninstantdischargeisimpossible,becauseitwouldrequireinnitecurrentforoneinstant.Innite currentwouldcreateinnitemagneticeldssurroundingtheinductor,andtheseeldswouldhaveinniteenergy.Instead,therate ofowofcurrentiscontrolledateachinstantbytherelationship betweentheamountofenergystoredinthemagneticeldandthe amountofcurrentthatmustexistinordertohavethatstronga eld.Afterthecapacitorreaches q =0,itovershoots.Thecircuit hasitsownkindofelectricalinertia,"becauseifchargewastostop owing,therewouldhavetobezerocurrentthroughtheinductor. Butthecurrentintheinductormustberelatedtotheamountof SectionA.3VoltageandCurrent 175

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l / Theinductorreleasesenergyandgivesittotheblackbox. energystoredinitsmagneticelds.Whenthecapacitorisat q =0, allthecircuit'senergyisintheinductor,soitmustthereforehave strongmagneticeldssurroundingitandquiteabitofcurrentgoing throughit. Theonlythingthatmightseemspookyhereisthatweusedto speakasifthecurrentintheinductorcausedthemagneticeld, butnowitsoundsasiftheeldcausesthecurrent.Actuallythisis symptomaticoftheelusivenatureofcauseandeectinphysics.It's equallyvalidtothinkofthecauseandeectrelationshipineither way.Thismayseemunsatisfying,however,andforexampledoesnot reallygetatthequestionofwhatbringsaboutavoltagedierence acrosstheresistorinthecasewheretheresistanceisnite;there mustbesuchavoltagedierence,becausewithoutone,Ohm'slaw wouldpredictzerocurrentthroughtheresistor. Voltage,then,iswhatisreallymissingfromourstorysofar. Let'sstartbystudyingthevoltageacrossacapacitor.Voltageis electricalpotentialenergyperunitcharge,sothevoltagedierence betweenthetwoplatesofthecapacitorisrelatedtotheamountby whichitsenergywouldincreaseifweincreasedtheabsolutevalues ofthechargesontheplatesfrom q to q + q : V C = E q + q )]TJ/F20 10.9091 Tf 10.909 0 Td [(E q = q = E C q = q 1 2 C q 2 = q C Manybooksusethisasthedenitionofcapacitance.Thisequation, bytheway,probablyexplainsthehistoricalreasonwhy C wasdenedsothattheenergywas inversely proportionalto C foragiven valueof C :thepeoplewhoinventedthedenitionwerethinkingofa capacitorasadeviceforstoringchargeratherthanenergy,andthe amountofchargestoredforaxedvoltagethechargecapacity" isproportionalto C Inthecaseofaninductor,weknowthatifthereisasteady,constantcurrentowingthroughit,thenthemagneticeldisconstant, andsoistheamountofenergystored;noenergyisbeingexchanged betweentheinductorandanyothercircuitelement.Butwhatif thecurrentischanging?Themagneticeldisproportionaltothe current,soachangeinoneimpliesachangeintheother.Forconcreteness,let'simaginethatthemagneticeldandthecurrentare bothdecreasing.Theenergystoredinthemagneticeldisthereforedecreasing,andbyconservationofenergy,thisenergycan'tjust goaway|someothercircuitelementmustbetakingenergyfrom theinductor.Thesimplestexample,showningurel,isaseries circuitconsistingoftheinductorplusoneothercircuitelement.It 176 ChapterACapacitanceandInductance

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doesn'tmatterwhatthisothercircuitelementis,sowejustcallita blackbox,butifyoulike,wecanthinkofitasaresistor,inwhich casetheenergylostbytheinductorisbeingturnedintoheatby theresistor.Thejunctionruletellsusthatbothcircuitelements havethesamecurrentthroughthem,so I couldrefertoeitherone, andlikewisetheloopruletellsus V inductor + V blackbox =0,sothe twovoltagedropshavethesameabsolutevalue,whichwecanrefer toas V .Whatevertheblackboxis,therateatwhichitistaking energyfromtheinductorisgivenby j P j = j IV j ,so j IV j = E L t = t 1 2 LI 2 = LI I t or j V j = L I t whichinmanybooksistakentobethedenitionofinductance. Thedirectionofthevoltagedropplusorminussignissuchthat theinductorresiststhechangeincurrent. There'soneveryintriguingthingaboutthisresult.Suppose, forconcreteness,thattheblackboxingurelisaresistor,and thattheinductor'senergyisdecreasing,andbeingconvertedinto heatintheresistor.Thevoltagedropacrosstheresistorindicates thatithasanelectriceldacrossit,whichisdrivingthecurrent. Butwhereisthiselectriceldcomingfrom?Therearenocharges anywherethatcouldbecreatingit!Whatwe'vediscoveredisone specialcaseofamoregeneralprinciple,theprincipleofinduction:a changingmagneticeldcreatesanelectriceld,whichisinaddition toanyelectriceldcreatedbycharges.Thereverseisalsotrue: anyelectriceldthatchangesovertimecreatesamagneticeld. Inductionformsthebasisforsuchtechnologiesasthegeneratorand thetransformer,andultimatelyitleadstotheexistenceoflight, whichisawavepatternintheelectricandmagneticelds.These arealltopicsforchapter6,butit'strulyremarkablethatwecould cometothisconclusionwithoutyethavinglearnedanydetailsabout magnetism. Thecartoonsinguremcompareselectriceldsmadebycharges, 1,toelectriceldsmadebychangingmagneticelds,2-3.Inm/1, twophysicistsareinaroomwhoseceilingispositivelychargedand SectionA.3VoltageandCurrent 177

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m / Electriceldsmadebycharges,1,andbychangingmagneticelds,2and3. whoseoorisnegativelycharged.Thephysicistonthebottom throwsapositivelychargedbowlingballintothecurvedpipe.The physicistatthetopusesaradarguntomeasurethespeedofthe ballasitcomesoutofthepipe.Theyndthattheballhasslowed downbythetimeitgetstothetop.Bymeasuringthechangeinthe ball'skineticenergy,thetwophysicistsareactingjustlikeavoltmeter.Theyconcludethatthetopofthetubeisatahighervoltage thanthebottomofthepipe.Adierenceinvoltageindicatesan electriceld,andthiseldisclearlybeingcausedbythechargesin theoorandceiling. Inm/2,therearenochargesanywhereintheroomexceptfor thechargedbowlingball.Movingchargesmakemagneticelds,so thereisamagneticeldsurroundingthehelicalpipewhiletheball ismovingthroughit.Amagneticeldhasbeencreatedwherethere wasnonebefore,andthateldhasenergy.Wherecouldtheenergy havecomefrom?Itcanonlyhavecomefromtheballitself,so theballmustbelosingkineticenergy.Thetwophysicistsworking togetherareagainactingasavoltmeter,andagaintheyconclude thatthereisavoltagedierencebetweenthetopandbottomof thepipe.Thisindicatesanelectriceld,butthiselectriceldcan't havebeencreatedbyanycharges,becausetherearen'tanyinthe room.Thiselectriceldwascreatedbythechangeinthemagnetic eld. Thebottomphysicistkeepsonthrowingballsintothepipe,until thepipeisfullofballs,m/3,andnallyasteadycurrentisestablished.Whilethepipewasllingupwithballs,theenergyinthe magneticeldwassteadilyincreasing,andthatenergywasbeing stolenfromtheballs'kineticenergy.Butonceasteadycurrentis established,theenergyinthemagneticeldisnolongerchanging. Theballsnolongerhavetogiveupenergyinordertobuildupthe eld,andthephysicistatthetopndsthattheballsareexitingthe 178 ChapterACapacitanceandInductance

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n / AnRCcircuit. pipeatfullspeedagain.Thereisnovoltagedierenceanymore. Althoughthereisacurrent, I= t iszero. DiscussionQuestions A Whathappenswhenthephysicistatthebottomingurem/3starts gettingtired,anddecreasesthecurrent? A.4Decay UpuntilnowI'vesoft-pedaledthefactthatbychangingthecharacteristicsofanoscillator,itispossibletoproducenon-oscillatory behavior.Forexample,imaginetakingthemass-on-a-springsystem andmakingthespringweakerandweaker.Inthelimitofsmall k ,it'sasthoughtherewasnospringwhatsoever,andthebehavior ofthesystemisthatifyoukickthemass,itsimplystartsslowing down.Forfrictionproportionalto v ,aswe'vebeenassuming,theresultisthatthevelocityapproacheszero,butneveractuallyreaches zero.Thisisunrealisticforthemechanicaloscillator,whichwillnot havevanishingfrictionatlowvelocities,butitisquiterealisticin thecaseofanelectricalcircuit,forwhichthevoltagedropacrossthe resistorreallydoesapproachzeroasthecurrentapproacheszero. Electricalcircuitscanexhibitallthesamebehavior.ForsimplicitywewillanalyzeonlythecasesofLRCcircuitswith L =0or C =0. Therccircuit WerstanalyzetheRCcircuit,n.Inrealityonewouldhave tokick"thecircuit,forexamplebybrieyinsertingabattery,in ordertogetanyinterestingbehavior.WestartwithOhm'slawand theequationforthevoltageacrossacapacitor: V R = IR V C = q=C Theloopruletellsus V R + V C =0, andcombiningthethreeequationsresultsinarelationshipbetween q and I : I = )]TJ/F15 10.9091 Tf 15.426 7.38 Td [(1 RC q Thenegativesigntellsusthatthecurrenttendstoreducethecharge onthecapacitor,i.e.todischargeit.Itmakessensethatthecurrent isproportionalto q :if q islarge,thentheattractiveforcesbetween the+ q and )]TJ/F20 10.9091 Tf 8.485 0 Td [(q chargesontheplatesofthecapacitorarelarge, andchargeswillowmorequicklythroughtheresistorinorderto reunite.Iftherewaszerochargeonthecapacitorplates,therewould benoreasonforcurrenttoow.Sinceamperes,theunitofcurrent, SectionA.4Decay 179

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p / AnRLcircuit. o / Overatimeinterval RC thechargeonthecapacitoris reducedbyafactorof e arethesameascoulombspersecond,itappearsthatthequantity RC musthaveunitsofseconds,andyoucancheckforyourselfthat thisiscorrect. RC isthereforereferredtoasthetimeconstantof thecircuit. Howexactlydo I and q varywithtime?Rewriting I as q= t wehave q t = )]TJ/F15 10.9091 Tf 15.425 7.38 Td [(1 RC q Thisequationdescribesafunction q t thatalwaysgetssmallerover time,andwhoserateofdecreaseisbigatrst,when q isbig,but getssmallerandsmalleras q approacheszero.Asanexampleof thistypeofmathematicalbehavior,wecouldimagineamanwho has1024weedsinhisbackyard,andresolvestopullouthalfof themeveryday.Ontherstday,hepullsouthalf,andhas512 left.Thenextday,hepullsouthalfoftheremainingones,leaving 256.Thesequencecontinuesexponentially:128,64,32,16,8,4,2, 1.Returningtoourelectricalexample,thefunction q t apparently needstobeanexponential,whichwecanwriteintheform ae bt where e =2.718...isthebaseofnaturallogarithms.Wecouldhave writtenitwithbase2,asinthestoryoftheweeds,ratherthan base e ,butthemathlateronturnsoutsimplerifweuse e .It doesn'tmakesensetopluganumberthathasunitsintoafunction likeanexponential,so bt mustbeunitless,and b musttherefore haveunitsofinverseseconds.Thenumber b quantieshowfastthe exponentialdecayis.Theonlyphysicalparametersofthecircuit onwhich b couldpossiblydependare R and C ,andtheonlyway toputunitsofohmsandfaradstogethertomakeunitsofinverse secondsisbycomputing1 =RC .Well,actuallywecoulduse7 =RC or3 =RC ,oranyotherunitlessnumberdividedby RC ,butthis iswheretheuseofbase e comesinhandy:forbase e ,itturnsout thatthecorrectunitlessconstant is 1.Thusoursolutionis q = q o exp )]TJ/F20 10.9091 Tf 16.183 7.38 Td [(t RC Thenumber RC ,withunitsofseconds,iscalledtheRCtimeconstantofthecircuit,andittellsushowlongwehavetowaitifwe wantthechargetofallobyafactorof1 =e Therlcircuit TheRLcircuit,p,canbeattackedbysimilarmethods,andit caneasilybeshownthatitgives I = I o exp )]TJ/F20 10.9091 Tf 9.68 7.38 Td [(R L t TheRLtimeconstantequals L=R 180 ChapterACapacitanceandInductance

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Deathbysolenoid;sparkplugsexample6 WhenwesuddenlybreakanRLcircuit,whatwillhappen?Itmight seemthatwe'refacedwithaparadox,sinceweonlyhavetwo formsofenergy,magneticenergyandheat,andifthecurrent stopssuddenly,themagneticeldmustcollapsesuddenly.But wheredoesthelostmagneticenergygo?Itcan'tgointoresistive heatingoftheresistor,becausethecircuithasnowbeenbroken, andcurrentcan'tow! Thewayoutofthisconundrumistorecognizethattheopengap inthecircuithasaresistancewhichislarge,butnotinnite.This largeresistancecausestheRLtimeconstant L = R tobevery small.Thecurrentthuscontinuestoowforaverybrieftime, andowsstraightacrosstheairgapwherethecircuithasbeen opened.Inotherwords,thereisaspark! Wecandeterminebasedonseveraldifferentlinesofreasoning thatthevoltagedropfromoneendofthesparktotheothermust beverylarge.First,theair'sresistanceislarge,so V = IR requiresalargevoltage.Wecanalsoreasonthatalltheenergy inthemagneticeldisbeingdissipatedinashorttime,sothe powerdissipatedinthespark, P = IV ,islarge,andthisrequires alargevalueof V I isn'tlargeitisdecreasingfromitsinitial value.Yetathirdwaytoreachthesameresultistoconsiderthe equation V L = I = t :sincethetimeconstantisshort,thetime derivative I = t islarge. Thisisexactlyhowacar'ssparkplugswork.Anotherapplication istoelectricalsafety:itcanbedangeroustobreakaninductive circuitsuddenly,becausesomuchenergyisreleasedinashort time.Thereisalsonoguaranteethatthesparkwilldischarge acrosstheairgap;itmightgothroughyourbodyinstead,since yourbodymighthavealowerresistance. DiscussionQuestions A AgophergnawsthroughoneofthewiresintheDClightingsystem inyourfrontyard,andthelightsturnoff.Attheinstantwhenthecircuit becomesopen,wecanconsiderthebareendsofthewiretobelikethe platesofacapacitor,withanairgaporgophergapbetweenthem.What kindofcapacitancevaluearewetalkingabouthere?Whatwouldthistell youaboutthe RC timeconstant? SectionA.4Decay 181

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q / Inacapacitor,thecurrent is90 aheadofthevoltagein phase. A.5Impedance Sofarwehavebeenthinkingintermsofthefreeoscillationsofa circuit.Thisislikeamechanicaloscillatorthathasbeenkickedbut thenlefttooscillateonitsownwithoutanyexternalforcetokeep thevibrationsfromdyingout.SupposeanLRCcircuitisdriven withasinusoidallyvaryingvoltage,suchaswilloccurwhenaradio tunerishookeduptoareceivingantenna.Weknowthatacurrent willowinthecircuit,andweknowthattherewillberesonant behavior,butitisnotnecessarilysimpletorelatecurrenttovoltage inthemostgeneralcase.Let'sstartinsteadwiththespecialcases ofLRCcircuitsconsistingofonlyaresistance,onlyacapacitance, oronlyaninductance.Weareinterestedonlyinthesteady-state response. Thepurelyresistivecaseiseasy.Ohm'slawgives I = V R Inthepurelycapacitivecase,therelation V = q=C letsuscalculate I = q t = C V t Ifthevoltagevariesas,forexample, V t = ~ V sin !t ,thenthe currentwillbe I t = !C ~ V cos !t ,sothemaximumcurrentis ~ I = !C ~ V .ByanalogywithOhm'slaw,wecanthenwrite ~ I = ~ V Z C wherethequantity Z C = 1 !C ,[impedanceofacapacitor] havingunitsofohms,iscalledthe impedance ofthecapacitorat thisfrequency.Notethatitisonlythe maximum current, ~ I ,that isproportionaltothe maximum voltage, ~ V ,sothecapacitorisnot behavinglikearesistor.Themaximaof V and I occuratdierenttimes,asshowningureq.Itmakessensethattheimpedance becomesinniteatzerofrequency.Zerofrequencymeansthatit wouldtakeaninnitetimebeforethevoltagewouldchangebyany amount.Inotherwords,thisislikeasituationwherethecapacitorhasbeenconnectedacrosstheterminalsofabatteryandbeen allowedtosettledowntoastatewherethereisconstantcharge onbothterminals.Sincetheelectriceldsbetweentheplatesare constant,thereisnoenergybeingaddedtoortakenoutofthe 182 ChapterACapacitanceandInductance

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r / Thecurrentthroughaninductorlagsbehindthevoltageby aphaseangleof90 eld.Acapacitorthatcan'texchangeenergywithanyothercircuit componentisnothingmorethanabrokenopencircuit. self-checkC Whycan'tacapacitorhaveitsimpedanceprintedonitalongwithits capacitance? Answer,p.204 Similarmathgives Z L = !L [impedanceofaninductor] foraninductor.Itmakessensethattheinductorhaslowerimpedance atlowerfrequencies,sinceatzerofrequencythereisnochangein themagneticeldovertime.Noenergyisaddedtoorreleased fromthemagneticeld,sotherearenoinductioneects,andthe inductoractsjustlikeapieceofwirewithnegligibleresistance.The termchoke"foraninductorreferstoitsabilitytochokeout"high frequencies. Thephaserelationshipsshowninguresqandrcanberememberedusingmyownmnemonic,eVIL,"whichshowsthatthevoltageVleadsthecurrentIinaninductivecircuit,whiletheoppositeistrueinacapacitiveone.Amoretraditionalmnemonicis ELItheICEman,"whichusesthenotationEforemf,aconcept closelyrelatedtovoltage. Low-passandhigh-passltersexample7 AnLRCcircuitonlyrespondstoacertainrangebandoffrequenciescenteredarounditsresonantfrequency.Asalter,this isknownasabandpasslter.Ifyouturndownboththebassand thetrebleonyourstereo,youhavecreatedabandpasslter. Tocreateahigh-passorlow-passlter,weonlyneedtoinsert acapacitororinductor,respectively,inseries.Forinstance,a verybasicsurgeprotectorforacomputercouldbeconstructed byinsertinganinductorinserieswiththecomputer.Thedesired 60Hzpowerfromthewallisrelativelylowinfrequency,whilethe surgesthatcandamageyourcomputershowmuchmorerapid timevariation.Evenifthesurgesarenotsinusoidalsignals,we canthinkofarapidspikequalitativelyasifitwasveryhighin frequencylikeahigh-frequencysinewave,itchangesvery rapidly. Inductorstendtobebig,heavy,expensivecircuitelements,soa simplesurgeprotectorwouldbemorelikelytoconsistofacapacitorin parallel withthecomputer.Infactonewouldnormallyjust connectonesideofthepowercircuittogroundviaacapacitor. Thecapacitorhasaveryhighimpedanceatthelowfrequencyof thedesired60Hzsignal,soitsiphonsoffverylittleofthecurrent. Butforahigh-frequencysignal,thecapacitor'simpedanceisvery small,anditactslikeazero-impedance,easypathintowhichthe currentisdiverted. SectionA.5Impedance 183

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Themainthingstobecarefulaboutwithimpedancearethat theconceptonlyappliestoacircuitthatisbeingdrivensinusoidally,theimpedanceofaninductororcapacitorisfrequencydependent,andimpedancesinparallelandseriesdon'tcombine accordingtothesamerulesasresistances.Itispossible,however, togetgetaroundthethirdlimitation,asdiscussedinsubsection. DiscussionQuestion A Figureqonpage182showsthevoltageandcurrentforacapacitor. Sketchthe q t graph,anduseittogiveaphysicalexplanationofthe phaserelationshipbetweenthevoltageandcurrent.Forexample,whyis thecurrentzerowhenthevoltageisatamaximumorminimum? B Relatethefeaturesofthegraphingureronpage183tothestory toldincartoonsingurem/2-3onpage178. 184 ChapterACapacitanceandInductance

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Problems Key p Acomputerizedanswercheckisavailableonline. R Aproblemthatrequirescalculus. ? Adicultproblem. 1 IfanFMradiotunerconsistingofanLRCcircuitcontains a1.0 Hinductor,whatrangeofcapacitancesshouldthevariable capacitorbeabletoprovide? p 2 aShowthattheequation V L = L I= t hastherightunits. bVerifythat RC hasunitsoftime. cVerifythat L=R hasunitsoftime. 3 Findtheenergystoredinacapacitorintermsofitscapacitance andthevoltagedierenceacrossit. p 4 Findtheinductanceoftwoidenticalinductorsinparallel. 5 Thewiresthemselvesinacircuitcanhaveresistance,inductance,andcapacitance.Wouldstray"inductanceandcapacitance bemostimportantforlow-frequencyorforhigh-frequencycircuits? Forsimplicity,assumethatthewiresactlikethey'rein series with aninductororcapacitor. 6 aFindthecapacitanceoftwoidenticalcapacitorsinseries. bBasedonthis,howwouldyouexpectthecapacitanceofa parallel-platecapacitortodependonthedistancebetweentheplates? 7 Findthecapacitanceofthesurfaceoftheearth,assuming thereisanoutersphericalplate"atinnity.Inreality,thisouter platewouldjustrepresentsomedistantpartoftheuniversetowhich wecarriedawaysomeoftheearth'schargeinordertochargeupthe earth. p 8 Startingfromtherelation V = L I= t forthevoltagedifferenceacrossaninductor,showthataninductorhasanimpedance equalto L! Problems 185

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Appendix1:Exercises Exercise1A:ElectricandMagneticForces Apparatus: Inthisexercise,youaregoingtoinvestigatetheforcesthatcanoccuramongthefollowing objects: nails magnets smallbitsofpaper speciallypreparedpiecesofscotchtape Tomakethespeciallypreparedpiecesoftape,takeapieceoftape,bendoneendovertoforma handlethatwon'tsticktoyourhand,andstickitonadesk.Makeahandleonasecondpiece, andlayitrightontopoftherstone.Nowpullthetwopiecesothedeskandseparatethem. Yourgoalistoaddressthefollowingquestionsexperimentally: 1.Dotheforcesgetweakerwithdistance?Dotheyhavesomemaximumrange?Istheresome rangeatwhichtheyabruptlycuto? 2.Cantheforcesbeblockedorshieldedagainstbyputtingyourhandoryourcalculatorinthe way?Trythiswithbothelectricandmagneticforces,andwithbothrepulsionandattraction. 3.Aretheforcesamongtheseobjectsgravitational? 4.Ofthemanyforcesthatcanbeobservedbetweendierentpairsofobjects,isthereany naturalwaytoclassifythemintogeneraltypesofforces? 5.DotheforcesobeyNewton'sthirdlaw? 6.Doordinarymaterialslikewoodorpaperparticipateintheseforces?

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Exercise3A:VoltageandCurrent 1.Howmanydierentcurrentscouldyoumeasureinthiscircuit?Makeaprediction,andthen tryit. Whatdoyounotice?Howdoesthismakesenseintermsoftherollercoastermetaphorintroducedindiscussionquestion3.3A? Whatisbeing usedup intheresistor? 2.Byconnectingprobestothesepoints,howmanywayscouldyoumeasureavoltage?How manyofthemwouldbedierentnumbers?Makeaprediction,andthendoit. Whatdoyounotice?Interpretthisusingtherollercoastermetaphor,andcolorinpartsofthe circuitthatrepresentconstantvoltages. 3.Theresistorsareunequal.Howmany dierent voltagesandcurrentscanyoumeasure?Make aprediction,andthentryit. Whatdoyounotice?Interpretthisusingtherollercoastermetaphor,andcolorinpartsofthe circuitthatrepresentconstantvoltages. 187

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Exercise3B:AnalyzingVoltageandCurrent ThisexerciseisbasedononecreatedbyVirginiaRoundy. Apparatus: DCpowersupply 1.5voltbatteries lightbulbsandholders wire highlightingpens,3colors Whenyourstglanceatthisexercise,itmay lookscaryandintimidating|allthosecircuits!However,allthosewild-lookingcircuits canbeanalyzedusingthefollowingfourguides tothinking: 1. Acircuithastobecomplete ,i.e.,itmust bepossibleforchargetogetrecycledasitgoes aroundthecircuit.Ifit'snotcomplete,then chargewillbuildupatadeadend.Thisbuiltupchargewillrepelanyotherchargethattries togetin,andeverythingwillrapidlygrindto astop. 2. Thereisconstantvoltageeverywherealong apieceofwire. Toapplythisruleduringthis lab,Isuggestyouusethecoloredhighlightingpenstomarkthecircuit.Forinstance,if there'sonewholepieceofthecircuitthat'sall atthesamevoltage,youcouldhighlightitin yellow.Asecondpieceofthecircuit,atsome othervoltage,couldbehighlightedinblue. 3. Chargeisconserved, sochargecan'tget usedup." 4.Youcandrawa rollercoasterdiagram ,like theoneshownbelow.Onthiskindofdiagram, heightcorrespondstovoltage|that'swhy thewiresaredrawnashorizontaltracks. ABulbandaSwitch Lookatcircuit1,andtrytopredictwhatwill happenwhentheswitchisopen,andwhatwill happenwhenit'sclosed.Writebothyourpredictionsinthetableonthefollowingpagebeforeyoubuildthecircuit.Whenyoubuildthe circuit,youdon'tneedanactualswitchlikea lightswitch;justconnectanddisconnectthe bananaplugs.Useoneofthe1.5voltbatteries asyourvoltagesource. Circuit1 188 Appendix1:Exercises

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switchopen prediction explanation observation explanation ifdierent switchclosed prediction explanation observation explanation ifdierent Diditworkthewayyouexpected?Ifnot,try togureitoutwiththebenetofhindsight, andwriteyourexplanationinthetableabove. Circuit2Don'tleavetheswitchedclosedfor alongtime! switchopen prediction explanation observation explanation ifdierent switchclosed prediction explanation observation explanation ifdierent 189

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Circuit3 switchopen prediction explanation observation explanation ifdierent switchclosed prediction explanation observation explanation ifdierent Circuit4 switchopen prediction explanation observation explanation ifdierent switchclosed prediction explanation observation explanation ifdierent 190 Appendix1:Exercises

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TwoBulbs Analyzethisonebothbyhighlightingandby drawingarollercoasterdiagram.Insteadofa battery,usetheDCpowersupply,setto2.4 volts. Circuit5 bulba prediction explanation observation explanation ifdierent bulbb prediction explanation observation explanation ifdierent Circuit6 bulba prediction explanation observation explanation ifdierent bulbb prediction explanation observation explanation ifdierent 191

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TwoBatteries Circuits7and8arebothgoodcandidatesfor rollercoasterdiagrams. Circuit7 prediction explanation observation explanation ifdierent Circuit8 prediction explanation observation explanation ifdierent AFinalChallenge Circuit9 bulba prediction explanation observation explanation ifdierent bulbb prediction explanation observation explanation ifdierent 192 Appendix1:Exercises

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Exercise4A:TheLoopandJunctionRules Apparatus: DCpowersupply multimeter resistors 1.Thejunctionrule Constructacircuitlikethisone,usingthepowersupplyasyourvoltagesource.Tomakethings moreinteresting,don'tuseequalresistors.Usenicebigresistorssay100kto1M| thiswillensurethatyoudon'tburnuptheresistors,andthatthemultimeter'ssmallinternal resistancewhenusedasanammeterisnegligibleincomparison. Insertyourmultimeterinthecircuittomeasureallthreecurrentsthatyouneedinordertotest thejunctionrule. 2.Thelooprule Nowcomeupwithacircuittotestthelooprule.Sincetheloopruleisalwayssupposedtobe true,it'shardtogowronghere!Makesureyouhaveatleastthreeresistorsinaloop,andmake sureyouhookinthepowersupplyinawaythatcreatesnon-zerovoltagedierencesacrossall theresistors.Measurethevoltagedierencesyouneedtomeasuretotestthelooprule.Here itisbesttousefairlysmallresistances,sothatthemultimeter'slargeinternalresistancewhen usedinparallelasavoltmeterwillnotsignicantlyreducetheresistanceofthecircuit.Donot useresistancesoflessthanabout100,however,oryoumayblowafuseorburnuparesistor. 193

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Exercise4B:ReasoningAboutCircuits Thequestionsinthisexercisecanallbesolvedusingsomecombinationofthefollowingapproaches: aThereisconstantvoltagethroughoutanyconductor. bOhm'slawcanbeappliedtoany part ofacircuit. cApplythelooprule. dApplythejunctionrule. Ineachcase,discussthequestion,decidewhatyouthinkistherightanswer,andthentrythe experiment. 1.Awireisaddedinparallelwithonebulb. Whichreasoningiscorrect? Eachbulbstillhas1.2Vacrossit,sobothbulbsarestilllitup. Allpartsofawireareatthesamevoltage,andthereisnowawireconnectionfromone sideoftheright-handbulbtotheother.Theright-handbulbhasnovoltagedierence acrossit,soitgoesout. 2.Theseriescircuitischangedasshown. Whichreasoningiscorrect? Eachbulbnowhasitssidesconnectedtothetwoterminalsofthebattery,soeachnowhas 2.4Vacrossitinsteadof1.2V.Theygetbrighter. Justasintheoriginalcircuit,thecurrentgoesthroughonebulb,thentheother.It'sjust thatnowthecurrentgoesinagure-8pattern.Thebulbsglowthesameasbefore. 194 Appendix1:Exercises

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3.Awireisaddedasshowntotheoriginalcircuit. Whatiswrongwiththefollowingreasoning? Thetoprightbulbwillgoout,becauseitstwosidesarenowconnectedwithwire,sotherewill benovoltagedierenceacrossit.Theotherthreebulbswillnotbeaected. 4.Awireisaddedasshowntotheoriginalcircuit. Whatiswrongwiththefollowingreasoning? Thecurrentowsoutoftherightsideofthebattery.Whenithitstherstjunction,someof itwillgoleftandsomewillkeepgoingupThepartthatgoesuplightsthetoprightbulb.The partthatturnsleftthenfollowsthepathofleastresistance,goingthroughthenewwireinstead ofthebottombulb.Thetopbulbstayslit,thebottomonegoesout,andothersstaythesame. 5.Whathappenswhenonebulbisunscrewed,leavinganairgap? 195

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Exercise5A-FieldVectors Apparatus: 3solenoids DCpowersupply compass ruler cut-oplasticcup Atthispointyou'vestudiedthegravitationaleld, g ,andtheelectriceld, E ,butnotthe magneticeld, B .However,theyallhavesomeofthesamemathematicalbehavior:theyact likevectors.Furthermore,magneticeldsaretheeasiesttomanipulateinthelab.Manipulating gravitationaleldsdirectlywouldrequirefuturistictechnologycapableofmovingplanet-sized massesaround!Playingwithelectriceldsisnotasridiculouslydicult,butstaticelectric chargestendtoleakothroughyourbodytoground,andstaticelectricityeectsarehardto measurenumerically.Magneticelds,ontheotherhand,areeasytomakeandcontrol.Any movingcharge,i.e.anycurrent,makesamagneticeld. Apracticaldeviceformakingastrongmagneticeldissimplyacoilofwire,formallyknown asasolenoid.Theeldpatternsurroundingthesolenoidgetsstrongerorweakerinproportion totheamountofcurrentpassingthroughthewire. 1.Withasinglesolenoidconnectedtothepowersupplyandlaidwithitsaxishorizontal,usea magneticcompasstoexploretheeldpatterninsideandoutsideit.Thecompassshowsyouthe eldvector'sdirection,butnotitsmagnitude,atanypointyouchoose.Notethattheeldthe compassexperiencesisacombinationvectorsumofthesolenoid'seldandtheearth'seld. 2.Whathappenswhenyoubringthecompassextremelyfarawayfromthesolenoid? Whatdoesthistellyouaboutthewaythesolenoid'seldvarieswithdistance? Thusalthoughthecompassdoesn'ttellyoutheeldvector'smagnitudenumerically,youcan getatleastsomegeneralfeelforhowitdependsondistance. 196 Appendix1:Exercises

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3.Thegurebelowisacross-sectionofthesolenoidintheplanecontainingitsaxis.Makea sea-of-arrowssketchofthemagneticeldinthisplane.Thelengthofeacharrowshouldatleast approximatelyreectthestrengthofthemagneticeldatthatpoint. Doestheeldseemtohavesourcesorsinks? 4.Whatdoyouthinkwouldhappentoyoursketchifyoureversedthewires? Tryit. 197

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5.Nowhookupthetwosolenoidsinparallel.Youaregoingtomeasurewhathappenswhen theirtwoeldscombineintheatacertainpointinspace.Asyou'veseenalready,thesolenoids' nearbyeldsaremuchstrongerthantheearth'seld;soalthoughwenowtheoreticallyhave threeeldsinvolvedtheearth'splusthetwosolenoids',itwillbesafetoignoretheearth's eld.Thebasicideahereistoplacethesolenoidswiththeiraxesatsomeangletoeachother, andputthecompassattheintersectionoftheiraxes,sothatitisthesamedistancefromeach solenoid.Sincethegeometrydoesn'tfavoreithersolenoid,theonlyfactorthatwouldmakeone solenoidinuencethecompassmorethantheotheriscurrent.Youcanusethecut-oplastic cupasalittleplatformtobringthecompassuptothesamelevelasthesolenoids'axes. aWhatdoyouthinkwillhappenwiththesolenoids'axesat90degreestoeachother,andequal currents?Tryit.Nowrepresentthevectoradditionofthetwomagneticeldswithadiagram. Checkyourdiagramwithyourinstructortomakesureyou'reontherighttrack. bNowtrytomakeasimilardiagramofwhatwouldhappenifyouswitchedthewiresonone ofthesolenoids. Afterpredictingwhatthecompasswilldo,tryitandseeifyouwereright. cNowsupposeyouweretogobacktothearrangementyouhadinparta,butyouchangedone ofthecurrentstohalfitsformervalue.Makeavectoradditiondiagram,andusetrigtopredict theangle. Tryit.Tocutthecurrenttooneofthesolenoidsinhalf,aneasyandaccuratemethodis simplytoputthethirdsolenoidinserieswithit,andputthatthirdsolenoidsofarawaythat itsmagneticelddoesn'thaveanysignicanteectonthecompass. 198 Appendix1:Exercises

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Exercise6A-Polarization Apparatus: calciteIcelandsparcrystal polaroidlm 1.Laythecrystalonapieceofpaperthathasprintonit.Youwillobserveadoubleimage. Seewhathappensifyourotatethecrystal. Evidentlythecrystaldoessomethingtothelightthatpassesthroughitonthewayfromthe pagetoyoureye.Onebeamoflightentersthecrystalfromunderneath,buttwoemergefrom thetop;byconservationofenergytheenergyoftheoriginalbeammustbesharedbetween them.Considerthefollowingthreepossibleinterpretationsofwhatyouhaveobserved: aThetwonewbeamsdierfromeachother,andfromtheoriginalbeam,onlyinenergy. Theirotherpropertiesarethesame. bThecrystaladdstothelightsomemysteriousnewpropertynotenergy,whichcomesin twoavors,XandY.Ordinarylightdoesn'thaveanyofeither.Onebeamthatemergesfrom thecrystalhassomeXaddedtoit,andtheotherbeamhasY. cThereissomemysteriousnewpropertythatispossessedbyalllight.Itcomesintwoavors, XandY,andmostordinarylightsourcesmakeanequalmixtureoftypeXandtypeYlight. Theoriginalbeamisanevenmixtureofbothtypes,andthismixtureisthensplitupbythe crystalintothetwopuriedforms. Inparts2and3you'llmakeobservationsthatwillallowyoutogureoutwhichoftheseis correct. 2.Nowplaceapolaroidlmoverthecrystalandseewhatyouobserve.Whathappenswhen yourotatethelminthehorizontalplane?Doesthisobservationallowyoutoruleoutanyof thethreeinterpretations? 3.Nowputthepolaroidlmunderthecrystalandtrythesamething.Puttingtogetherall yourobservations,whichinterpretationdoyouthinkiscorrect? 4.Lookatanoverheadlightxturethroughthepolaroid,andtryrotatingit.Whatdoyou observe?Whatdoesthistellyouaboutthelightemittedbythelightbulb? 5.Nowpositionyourselfwithyourheadunderalightxtureanddirectlyoverashinysurface, 199

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suchasaglossytabletop.You'llseethelamp'sreection,andthelightcomingfromthelamp toyoureyewillhaveundergoneareectionthroughroughlya180-degreeanglei.e.itvery nearlyreverseditsdirection.Observethisreectionthroughthepolaroid,andtryrotatingit. Finally,positionyourselfsothatyouareseeingglancingreections,andtrythesamething. SummarizewhathappenstolightwithpropertiesXandYwhenitisreected.Thisisthe principlebehindpolarizingsunglasses. 200 Appendix1:Exercises

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Appendix2:PhotoCredits Exceptasspecicallynotedbeloworinaparentheticalcreditinthecaptionofagure,alltheillustrationsin thisbookareundermyowncopyright,andarecopyleftlicensedunderthesamelicenseastherestofthebook. Insomecasesit'sclearfromthedatethatthegureispublicdomain,butIdon'tknowthenameoftheartist orphotographer;Iwouldbegratefultoanyonewhocouldhelpmetogivepropercredit.Ihaveassumedthat imagesthatcomefromU.S.governmentwebpagesarecopyright-free,sinceproductsoffederalagenciesfallinto thepublicdomain.I'veincludedsomepublic-domainpaintings;photographicreproductionsofthemarenot copyrightableintheU.S.BridgemanArtLibrary,Ltd.v.CorelCorp.,36F.Supp.2d191,S.D.N.Y.1999. WhenPSSCPhysics"isgivenasacredit,itindicatesthatthegureisfromthersteditionofthetextbook entitledPhysics,bythePhysicalScienceStudyCommittee.Theearlyeditionsofthesebooksneverhadtheir copyrightsrenewed,andarenowthereforeinthepublicdomain.Thereisalsoablanketpermissiongivenin thelaterPSSCCollegePhysicsedition,whichstatesonthecopyrightpagethatThematerialstakenfromthe originalandsecondeditionsandtheAdvancedTopicsofPSSCPHYSICSincludedinthistextwillbeavailable toallpublishersforuseinEnglishafterDecember31,1970,andintranslationsafterDecember31,1975." CreditstoMillikanandGalerefertothetextbooksPracticalPhysicsandElementsofPhysics. Botharepublicdomain.The1927versiondidnothaveitscopyrightrenewed.Sinceispossiblethatsomeof theillustrationsinthe1927versionhadtheircopyrightsrenewedandarestillundercopyright,Ihaveonlyused themwhenitwasclearthattheywereoriginallytakenfrompublicdomainsources. Inafewcases,Ihavemadeuseofimagesunderthefairusedoctrine.However,Iamnotalawyer,andthelaws onfairusearevague,soyoushouldnotassumethatit'slegalforyoutousetheseimages.Inparticular,fairuse lawmaygiveyoulessleewaythanitgivesme,becauseI'musingtheimagesforeducationalpurposes,andgiving thebookawayforfree.Likewise,ifthephotocreditsayscourtesyof...,"thatmeansthecopyrightownergave mepermissiontouseit,butthatdoesn'tmeanyouhavepermissiontouseit. Cover Catneuron: CourtesyofRichardJ.Harris,UniversityofWesternOntario.. Cover Accelerator: CourtesyofA.Zachau,GSI.. Contents Lightning: C.Clark/NOAAphotolibrary,publicdomain. Contents Electriceel: WikipediauserVsion,duallicensedunderGFDLandCC-BY. Contents CPUdie: Wikipedia userMattGibbs,GFDLlicensed. Contents MarsReconnaissanceOrbiterantenna: NASA,publicdomain. 13 Lightning: C.Clark/NOAAphotolibrary,publicdomain. 26 RobertMillikan: ClarkMillikan,1891, publicdomain. 31 J.J.Thomson: MillikanandGale,1920.. 41 Curies: Harper'sMonthly,1904. 42 Becquerel: MillikanandGale,1920. 42 Becquerel'sphotographicplate: publicdomain. 45 Rutherford: publicdomain. 44 Nuclearfuelpellets: USDOE,publicdomain. 57 Nuclearpowerplant: Wikipedia userStefanKuhn,GFDLlicensed. 62 Raftinneutrinodetector: KamiokaObservatory,ICRRInstitute forCosmicRayResearch,TheUniversityofTokyo. 63 IceCube: JohnJacobsen/NSF.Thismaterialis baseduponworksupportedbytheNationalScienceFoundationunderGrantNos.OPP-9980474AMANDA andOPP-0236449IceCube,UniversityofWisconsin-Madison.". 64 GAMMASPHERE: courtesyofC.J. Lister/R.V.F.Janssens. 64 Hbombtest: publicdomainproductofUSDOE,IvyMiketest. 64 FatuHiva Rainforest: WikipediauserMakemake,GFDLlicensed. 64 fusionreactor: Theseimagesmaybeusedfreeof chargeforeducationalpurposesbutpleaseusetheacknowledgement`photographcourtesyofEFDA-JET"'. 64 Sun: SOHOESA&NASA. 68 Chernobylmap: CIAHandbookofInternationalEconomicStatistics,1996, publicdomain. 69 Horses: c2004ElenaFilatova. 69 Polarbear: U.S.FishandWildlifeService,public domain. 70 CrabNebula: ESOPressPhotosmaybereproduced,ifcreditisgiventotheEuropeanSouthern Observatory.". 77 Transmissionlines: WikipediauserNixdorf,GFDLlicensed. 78 Knifesh: Courtesy ofGregDeGreef. 79 Ampere: MillikanandGale,1920. 83 Volta: MillikanandGale,1920. 87 Ohm: MillikanandGale,1920. 89 Superconductingacceleratorsegment: CourtesyofArgonneNationalLaboratory, managedandoperatedbytheUniversityofChicagofortheU.S.DepartmentofEnergyundercontractNo. W-31-109-ENG-38.. 104 Printedcircuitboard: BillBertram,WikipediauserPixel8,CC-BYlicensed. 105 CPUdie: WikipediauserMattGibbs,GFDLlicensed. 128 LIGO: WikipediauserUmptanum:Thisimage iscopyrighted.Thecopyrightholderhasirrevocablyreleasedallrightstoit.". 133 Hammerheadshark: WikimediaCommonsuserLittlegreenman,publicdomain. 136 Topographicalmaps: FlatmapsbyUSGS, publicdomain;perspectivemapbyWikipediauserKbh3rd,GFDLlicensed. 143 Maxwellandhiswife: public

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domaincontemporaryphotograph. 155 Hertz: Publicdomaincontemporaryphotograph. 149 Faraday: PaintingbyThomasPhillips,1842. 170 Capacitors: Wikipediauserde:Benutzer:Honina,GFDLlicensed. 170 Inductors: Wikipediauserde:Benutzer:Honina,GFDLlicensed. 202 Appendix2:PhotoCredits

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Appendix3:HintsandSolutions AnswerstoSelf-Checks AnswerstoSelf-ChecksforChapter1 Page17,self-checkA: Eithertypecanbeinvolvedineitheranattractionorarepulsion.A positivechargecouldbeinvolvedineitheranattractionwithanegativechargeorarepulsion withanotherpositive,andanegativecouldparticipateineitheranattractionwithapositive orarepulsionwithanegative. Page18,self-checkB: Itwouldn'tmakeanydierence.Therolesofthepositiveandnegative chargesinthepaperwouldbereversed,buttherewouldstillbeanetattraction. Page28,self-checkC: Yes.InU.S.currency,thequantumofmoneyisthepenny. Page55,self-checkA: Thomsonwasacceleratingelectrons,whicharenegativelycharged. Thisapparatusissupposedtoacceleratedatomswithoneelectronstrippedo,whichhave positivenetcharge.Inbothcases,aparticlethatisbetweentheplatesshouldbeattractedby theforwardplateandrepelledbytheplatebehindit. Page65,self-checkB: Thehydrogen-1nucleusissimpleaproton.Thebindingenergyisthe energyrequiredtotearanucleusapart,butforanucleusthissimplethereisnothingtotear apart. AnswerstoSelf-ChecksforChapter3 Page91,self-checkA: Thelargeamountofpowermeansahighrateofconversionofthe battery'schemicalenergyintoheat.Thebatterywillquicklyuseupallitsenergy,i.e.,burn out." AnswerstoSelf-ChecksforChapter5 Page129,self-checkA: Thereasoningisexactlyanalogoustothatusedinexample1on page126toderiveanequationforthegravitationaleldoftheearth.Theeldis F=q t = kQq t =r 2 =q t = kQ=r 2 Page135,self-checkB:

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E x = )]TJ/F15 10.9091 Tf 9.681 7.38 Td [(d V d x = )]TJ/F15 10.9091 Tf 12.798 7.38 Td [(d d x kQ r = kQ r 2 Page137,self-checkC: aThevoltageheightincreasesasyoumovetotheeastornorth. Ifweletthepositive x directionbeeast,andchoosepositive y tobenorth,thend V= d x and d V= d y arebothpositive.Thismeansthat E x and E y arebothnegative,whichmakessense, sincethewaterisowinginthenegative x and y directionssouthandwest. bTheelectriceldsareallpointingawayfromthehigherground.Ifthiswasanelectrical map,therewouldhavetobealargeconcentrationofchargeallalongthetopoftheridge,and especiallyatthemountainpeaknearthesouthend. AnswerstoSelf-ChecksforChapter6 Page152,self-checkA: Aninducedelectriceldcanonlybecreatedbya changing magnetic eld.Nothingischangingifyourcarisjustsittingthere.Apointonthecoilwon'texperience achangingmagneticeldunlessthecoilisalreadyspinning,i.e.,theenginehasalreadyturned over. AnswerstoSelf-ChecksforChapterA Page173,self-checkA: Yes.Themasshasthesamekineticenergyregardlessofwhich directionit'smoving.Frictioncovertsmechanicalenergyintoheatatthesameratewhetherthe massisslidingtotherightortotheleft.Thespringhasanequilibriumlength,andenergycan bestorediniteitherbycompressingit x< 0orstretchingit x> 0. Page173,self-checkB: Velocity, v ,istherateofchangeofposition, x ,withrespecttotime. Thisisexactlyanalogousto I = q= t Page183,self-checkC: Theimpedancedependsonthefrequencyatwhichthecapacitoris beingdriven.Itisn'tjustasinglevalueforaparticularcapacitor. SolutionstoSelectedHomeworkProblems SolutionsforChapter2 Page75,problem6: aInthereactionp+e )]TJ/F23 10.9091 Tf 10.115 -3.959 Td [(! n+ ,thechargesontheleftare e + )]TJ/F20 10.9091 Tf 8.484 0 Td [(e =0, andbothchargesontherightarezero.bTheneutrinohasnegligiblemass.Themasseson theleftadduptolessthanthemassoftheneutrinoontheright,soenergywouldberequired fromanexternalsourceinordertomakethisreactionhappen. SolutionsforChapter3 Page103,problem12: t=Dq/I= e=I =0.160 s. 204 Appendix3:HintsandSolutions

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Page103,problem13: aThechangeinPEis e V,sotheKEgainedis = 2 mv 2 = eV Solvingfor v andplugginginnumbers,weget5.9 10 7 m/s.Thisisabout20%ofthespeed oflight.Sinceit'snotthatclosetothespeedoflight,we'llgetareasonablyaccurateanswer withouttakingintoaccountEinstein'stheoryofrelativity. Page104,problem16: It'smuchmorepracticaltomeasurevoltagedierences.Tomeasure acurrent,youhavetobreakthecircuitsomewhereandinsertthemeterthere,butit'snot possibletodisconnectthecircuitssealedinsidetheboard. SolutionsforChapter4 Page120,problem11: Inseries,theygive11k.Inparallel,theygive = 1k+1 = 10k )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 = 0.9k. Page121,problem12: Theactualshapeisirrelevant;allwecareaboutitwhat'sconnectedto what.Therefore,wecandrawthecircuitattenedintoaplane.Everyvertexofthetetrahedron isadjacenttoeveryothervertex,soanytwoverticestowhichweconnectwillgivethesame resistance.Pickingtwoarbitrarily,wehavethis: Thisisunfortunatelyacircuitthatcannotbeconvertedintoparallelandseriesparts,andthat's whatmakesthisahardproblem!However,wecanrecognizethatbysymmetry,thereiszero currentintheresistormarkedwithanasterisk.Eliminatingthisone,werecognizethewhole arrangementasatripleparallelcircuitconsistingofresistances R ,2 R ,and2 R .Theresulting resistanceis R= 2. SolutionsforChapter5 Page142,problem9: Proceedingassuggestedinthehint,weformconcentricrings,each oneextendingfromradius b toradius b +d b .Theareaofsucharingequalsitscircumference multipledbyd b ,whichis b d b .Itschargeisthus2 b d b .Pluggingthisintotheexpression fromproblem8givesacontributiontotheeldd E =2 bka a 2 + b 2 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 = 2 d b .Thetotaleldis foundbyintegratingthisexpression.Therelevantintegralcanbefoundinatable. E = Z 1 0 d E =2 bka a 2 + b 2 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 = 2 d b =2 ka Z 1 0 b a 2 + b 2 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 = 2 d b =2 ka h )]TJ/F26 10.9091 Tf 10.303 8.837 Td [()]TJ/F20 10.9091 Tf 5 -8.837 Td [(a 2 + b 2 )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 = 2 i 1 b =0 =2 k Page142,problem11: Letthesquare'ssidesbeoflength a .Theeldatthecenteristhe vectorsumoftheeldsthatwouldhavebeenproducedindividuallybythethreecharges.Each oftheseindividualeldsis kq=r 2 ,where r 1 = a= p 2forthetwocharges q 1 ,and r 2 = a= 2for q 2 Vectoradditioncanbedonebyaddingcomponents.Let x behorizontaland y vertical.The y 205

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componentscancelbysymmetry.Thesumofthe x componentsis E x = kq 1 r 2 1 cos45 + kq 1 r 2 1 cos45 )]TJ/F20 10.9091 Tf 12.105 7.38 Td [(kq 2 r 2 2 Substitutingcos45 =1 = p 2andsettingthiswholeexpressionequaltozero,wend q 2 =q 1 = 1 = p 2. SolutionsforChapter6 Page166,problem13: aCurrentmeanshowmuchchargepassesbyagivenpointperunit time.Duringatimeintervalt,allthechargecarriersinacertainregionbehindthepointwill passby.Thisregionhaslength v t andcross-sectionalarea A ,soitsvolumeis Av t,andthe amountofchargeinitis Avnq t.Tondthecurrent,wedividethisamountofchargebyt, giving I = Avnq .bAsegmentofthewireoflength L hasaforce QvB actingonit,where Q = ALnq isthetotalchargeofthemovingchargecarriersinthatpartofthewire.Theforce perunitlengthis ALnqvB =L = AnqvB .cDividingthetworesultsgives F=L = IB Page167,problem14: aThegureshowsthecasewherethecurrentsareinopposite directions. Theeldvectorshownisonemadebywire1,whichcausesaneectonwire2.Itpointsup becausewire1'seldpatternisclockwiseasviewfromalongthedirectionofcurrent I 1 .For simplicity,let'sassumethatthecurrent I 2 ismadebypositivelychargedparticlesmovingin thedirectionofthecurrent.Youcancheckthatthenalresultwouldbethesameiftheywere negativelycharged,aswouldactuallybethecaseinametalwire.Theforceononeofthese positivelychargedparticlesinwire2issupposedtohaveadirectionsuchthatwhenyousight alongit,the B vectorisclockwisefromthe v vector.Thiscanonlybeaccomplishediftheforce ontheparticleinwire2isinthedirectionshown.Wire2isrepelledbywire1. Toverifythatwire1isalsorepelledbywire2,wecaneithergothroughthesametypeof argumentagain,orwecansimplyapplyNewton'sthirdlaw. Simialarargumentsshowthattheforceisattractiveifthecurrentsareinthesamedirection. bTheforceonwire2is F=L = I 2 B ,where B = o I 1 = 2 r istheeldmadebywire1and r is thedistancebetweenthewires.Theresultis F=L = o I 1 I 2 = 2 r Page168,problem19: aBasedonourknowledgeoftheeldpatternofacurrent-carrying loop,weknowthatthemagneticeldmustbeeitherintooroutofthepage.Thismakessense, sincethatwouldmeantheeldisalwaysperpendiculartotheplaneoftheelectrons'motion;if itwasintheirplaneofmotion,thentheanglebetweenthe v and B vectorswouldbechanging allthetime,butweseenoevidenceofsuchbehavior.Withtheeldturnedon,theforcevector isapparentlytowardthecenterofthecircle.Let'sanalyzetheforceatthemomentwhenthe 206 Appendix3:HintsandSolutions

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electronshavestartedmoving,whichisattherightsideofthecircle.Theforceistotheleft. Sincetheelectronsarenegativelychargedparticles,weknowthatifwesightalongtheforce vector,the B vectormustbecounterclockwisefromthe v vector.Themagneticeldmust beoutofthepage.bLookingatgureionpage147,wecantellthatthecurrentinthe coilsmustbecounterclockwiseasviewedfromtheperspectiveofthecamera.cElectronsare negativelycharged,sotoproduceacounterclockwisecurrent,theelectronsinthecoilsmust begoingclockwise.dThecurrentinthecoilsiskeeptheelectronsinthebeamfromgoing straight,i.e.theforceisarepulsion.Thismakessensebycomparisonwithproblem14:thecoil currentsandvacuumtubecurrentsarecounterrotating,whichcausesarepulsion. Page168,problem20: Yes.Forexample,theforcevanishesiftheparticle'svelocityis paralleltotheeld,soifthebeamhadbeenlaunchedparalleltotheeld,itwouldhavegone inastraightlineratherthanacircle.Ingeneral,anycomponentofthevelocityvectorthatis outoftheplaneperpendiculartotheeldwillremainconstant,sothemotioncanbehelical. Page168,problem22: Thetrickistoimagineputtingtogethertwoidenticalsolenoidsto makeonedouble-lengthsolenoid.Theeldofthedoubledsolenoidisgivenbythevectorsum ofthetwosolenoids'individualelds.Atpointsontheaxis,symmetryguaranteesthatthe individualeldsliealongtheaxis,andsimilarlyforthetotaleld.Atthecenterofoneofthe mouths,wethushavetwoparalleleldvectorsofequalstrength,whosesumequalstheinterior eld.Buttheinterioreldofthedoubledsolenoidisthesameasthatoftheindividualones, sincetheequationfortheeldonlydependsonthenumberofturnsperunitlength.Therefore theeldatthecenterofasolenoid'smouthequalsexactlyhalftheinterioreld. 207

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Index alchemy,14 alphadecay,57 alphaparticle,44 ammeter,82 ampereunit,79 antielectron,60 antimatter,60 atom,20 raisin-cookiemodelof,34 atomicnumber dened,48 Atomism,20 betadecay,60 betaparticle,44 bindingenergy nuclear,65 Brownianmotion,25 capacitor,159,169 capacitance,169 cathoderays,30 chainreaction,58 charge,15 conservationof,17 quantizationof,26 Chernobyl,68 circuit,81 complete,81 open,81 parallel,93 series,93 short,91 completecircuit,81 conductor dened,88 conservationofmass,20 coulombunit,16 Coulomb'slaw,17 Crookes,William,24 current dened,79 dipole electric,130 dipolemoment,131 DNA,68 Einstein,Albert andBrownianmotion,26 electriccurrent dened,79 electricdipole,130 electriceld,129 relatedtovoltage,132 electricforces,15 electrolytes,95 electromagneticspectrum,155 electromagneticwaves,154 electron,33 electroncapture,60 electrondecay,60 elements,chemical,23 energy storedinelds,156 equivalentresistance ofresistorsinparallel,108 farad dened,170 Faraday,Michael,77,149,151 typesofelectricity,78 Feynman,Richard,160 eld electric,129 gravitational,125 magnetic,144 elds superpositionof,127 eldsofforce,123 force eldsof,123 Franklin,Benjamin denitionofsignsofcharge,16 Galileo,149 gammaray,44 generator,152,177 gravitationaleld,125 gravitationalwaves,128

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handedness,160 Hertz,Heinrich,155 Hiroshima,69 Hooke,14 Hugo,Victor,13 Hulse,R.A.,128 impedance,182 ofaninductor,183 inductance dened,171 induction,149,177 inductor,169 inductance,169 insulator dened,88 isotopes,55 junctionrule,107 Keynes,JohnMaynard,14 light dened,23 LIGO,128 looprule,113 magneticeld,144 dened,146 magnetostatics,146,147 mass conservationof,20 matter dened,22 Maxwell,JamesClerk,143,151 Mendeleev,Dmitri,24 Millikan,Robert,26 milliremunit,68 moment dipole,131 monopoles magnetic,144 neutralelectrically,17 Newton,149 Newton,Isaac,13 nuclearforces,56,161 nucleus discovery,46 Ohm'slaw,88 ohmic dened,88 op-amp,175 opencircuit,81 operationalamplierop-amp,175 parallelcircuit dened,93 periodictable,24,48 polarization,154 positron,60 positrondecay,60 quantization,26 raisincookiemodel,34 RCcircuit,179 RCtimeconstant,180 remunit,68 resistance dened,87 inparallel,107 inseries,112 resistivity dened,114 resistor,91 resistors inparallel,108 RLcircuit,180 schematic,106 schematics,106 sea-of-arrowsrepresentation,127 seriescircuit dened,93 shortcircuit dened,91 sinksinelds,127 solenoid,170 sourcesofelds,127 sparkplug,181 specialrelativity,66 spectrum electromagnetic,155 strongnuclearforce,56 strongnuclearforce,56 superpositionofelds,127 symmetry,160 Index 209

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Taylor,J.H.,128 teslaunit,146 Thomson,J.J. cathoderayexperiments,31 timeconstant RC,180 transformer,152,177 voltunit dened,83 voltage dened,84 relatedtoelectriceld,132 waves electromagnetic,154 gravitational,128 weaknuclearforce,59,161 210 Index

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Index 211

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212 Index

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Index 213

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214 Index

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Index 215

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216 Index

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Index 217

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UsefulData MetricPrexes M-mega-10 6 k-kilo-10 3 m-milli-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(3 -Greekmumicro-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(6 n-nano-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(9 p-pico-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(12 f-femto-10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(15 Centi-,10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(2 ,isusedonlyinthecentimeter. NotationandUnits quantityunitsymbol distancemeter,m x x timesecond,s t t masskilogram,kg m densitykg = m 3 velocitym/s v accelerationm = s 2 a forceN=kg m = s 2 F pressurePa=1N = m 2 P energyJ=kg m 2 = s 2 E powerW=1J = s P momentumkg m = s p periods T wavelengthm frequencys )]TJ/F19 5.9776 Tf 5.756 0 Td [(1 orHz f chargecoulomb,C q voltagevolt,1V=1J/C V currentampere,1A=1C/s I resistanceohm,1=1V/A R capacitancefarad,1F=1C/V C inductancehenry,1H=1V s = A L electriceldV/morN/C E magneticeldtesla,1T=1N s = C m B FundamentalConstants gravitationalconstant G =6.67 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(11 N m 2 = kg 2 Coulombconstant k =8.99 10 9 N m 2 = C 2 quantumofcharge e =1.60 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(19 C speedoflight c =3.00 10 8 m/s Conversions Nonmetricunitsintermsofmetricones: 1inch=25.4mmbydenition 1pound-force=4.5newtonsofforce kg g =2.2pounds-force 1scienticcalorie=4.18J 1kcal=4.18 10 3 J 1gallon=3.78 10 3 cm 3 1horsepower=746W Whenspeakingoffoodenergy,thewordCalorie"isused tomean1kcal,i.e.,1000calories.Inwriting,thecapitalC maybeusedtoindicate1Calorie=1000calories. RelationshipsamongU.S.units: 1footft=12inches 1yardyd=3feet 1milemi=5280feet Earth,Moon,andSun bodymasskgradiuskmradiusoforbitkm earth5.97 10 24 6.4 10 3 1.49 10 8 moon7.35 10 22 1.7 10 3 3.84 10 5 sun1.99 10 30 7.0 10 5 | SubatomicParticles particlemasskgradiusfm electron9.109 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(31 0.01 proton1.673 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(27 1.1 neutron1.675 10 )]TJ/F19 5.9776 Tf 5.756 0 Td [(27 1.1 Theradiiofprotonsandneutronscanonlybegivenapproximately,sincetheyhavefuzzysurfaces.Forcomparison,a typicalatomisaboutamillionfminradius. 218 Index