第七章 静电场1.docx

第七章 静电场1.docx

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第七章静电场1

CHAPTER7Electrostatics

Theword"electricity"mayevokeanimageofcomplexmoderntechnology:

computers,lights,motors,electricpower.Buttheelectricforcewouldseemtoplayanevendeeperroleinourlives.Accordingtoatomictheory,electricforcesbetweenatomsandmoleculesholdthemtogethertoformliquidsandsolids,andelectricforcesarealsoinvolvedinthemetabolicprocessesthatoccurwithinourbodies.Manyoftheforceswehavedealtwithsofar,suchaselasticforces,thenormalforce,andothercontactforces（pushesandpulls）arenowconsideredtoresultfromelectricforcesactingattheatomiclevel.Gravity,ontheotherhand,isaseparateforce.

Theearlieststudiesonelectricitydatebacktotheancients,butithasbeenonlyinthepasttwocenturiesthatelectricitywasstudiedindetail.Wewilldiscussthedevelopmentofideasaboutelectricity,includingpracticaldevices,aswellastherelationtomagnetism,inthenextthreechapters.

7.1ElectricChargeandElectricField

7.1.1ElectricChargeandCoulomb'sLaw

ElectricChargeandItsConservation

ThewordelectricitycomesfromtheGreekwordelektron,whichmeans"amber."Amberispetrifiedtreeresin,andtheancientsknewthatifyourubanamberrodwithapieceofcloth,theamberattractssmallpiecesofleavesordust.Apieceofhardrubber,aglassrod,oraplasticrulerrubbedwithaclothwillalsodisplaythis"ambereffect,"orstaticelectricityaswecallittoday.Youcanreadilypickupsmall

piecesofpaperwithaplasticcomborrulerthatyou'vejustvigorouslyrubbedwithevenapapertowel.SeethephotoonthepreviouspageandFig.7-1.Youhaveprobablyexperiencedstaticelectricitywhencombingyourhairorwhentakingasyntheticblouseshirtfromaclothesdryer.Andyoumayhavefeltashockwhenyoutouchedametaldoorknobafterslidingacrossacarseatorwalkingacrossanyloncarpet.Ineachcase,anobjectbecomes"charged"duetoarubbingprocessandissaidtopossessanetelectriccharge.

Isallelectricchargethesame,orisitpossiblethatthereismorethanonetype?

Infact,therearetwotypesofelectriccharge,asthefollowingsimpleexperimentsshow.Aplasticrulerissuspendedbyathreadandrubbedvigorouslywithaclothtochargeit.Whenasecondruler,whichhasalsobeenchargedinthesameway,isbroughtclosetothefirst,itisfoundthattheonerulerrepelsthe

other.ThisisshowninFig.7-2a.Similarly,ifarubbedglassrodisbroughtclosetoasecondchargedglassrod,againarepulsiveforceisseentoact,Fig.7-2b.However,

ifthechargedglassrodisbroughtclosetothechargedplasticruler,itisfoundthattheyattracteachother,Fig.7-2c.Thechargeontheglassmustthereforebedifferentfromthatontheplastic.Indeed,itisfoundexperimentallythatallchargedobjectsfallintooneoftwocategories.Eithertheyareattractedtotheplasticandrepelledbytheglass,justasglassis;ortheyarerepelledbytheplasticandattractedtotheglass,justastheplasticruleris.Thusthereseemtobetwo,andonlytwo,typesofelectriccharge.Eachtypeofchargerepelsthesametypebutattractstheoppositetype.Thatis:

unlikechargesattract;likechargesrepel.

ThetwotypesofelectricchargewerereferredtoaspositiveandnegativebytheAmericanstatesman,philosopher,andscientistBenjaminFranklin（1706-1790）.Thechoiceofwhichnamewentwithwhichtypeofchargewasofcoursearbitrary.Franklin'schoicesetthechargeontherubbedglassrodtobepositivecharge,sothechargeonarubbedplasticruler（oramber）iscallednegativecharge.Westillfollowthisconventiontoday.

Franklinarguedthatwheneveracertainamountofchargeisproducedononebodyinaprocess,anequalamountoftheoppositetypeofchargeisproducedonanotherbody.Thepositiveandnegativearetobetreatedalgebraically,sothatduringanyprocess,thenetchangeintheamountofchargeproducediszero.Forexample,whenaplasticrulerisrubbedwithapapertowel,theplasticacquiresanegativechargeandthetowelanequalamountofpositivecharge.Thechargesareseparated,butthesumofthetwoiszero.Thisisanexampleofalawthatisnowwellestablished:

thelawofconservationofelectriccharge,whichstatesthat

thenetamountofelectricchargeproducedinanyprocessiszero.

Ifoneobjectoroneregionofspaceacquiresapositivecharge,thenanequalamountofnegativechargewillbefoundinneighboringareasorobjects.Noviolationshaveeverbeenfound,andthisconservationlawisasfirmlyestablishedasthoseforenergyandmomentum.

Coulomb'sLaw

Wehaveseenthatanelectricchargeexertsaforceonotherelectriccharges.Whatfactorsaffectthemagnitudeofthisforce?

Toanswerthis,theFrenchphysicistCharlesCoulomb（1736-1806）investigatedelectricforcesinthe1780susingatorsionbalance（Fig.7-3）muchlikethatusedbyCavendishforhisstudiesofthegravitationalforce.

AlthoughpreciseinstrumentsforthemeasurementofelectricchargewerenotavailableinCoulomb'stime,hewasabletopreparesmallsphereswithdifferentmagnitudesofchargeinwhichtheratioofthechargeswasknown.Hereasonedthatifachargedconductingsphereisplacedincontactwithanidenticalunchargedsphere,thechargeonthefirstwouldbesharedequallybythetwoofthembecauseofsymmetry.Hethushadawaytoproducechargesequalto1/2,1/4andsoon,oftheoriginalcharge.Althoughhehadsomedifficultywithinducedcharges,Coulombwasabletoarguethattheforceonetinychargedobjectexertedonasecondtinychargedobjectisdirectlyproportionaltothechargeoneach

ofthem.Thatis,ifthechargeoneitheroneoftheobjectswasdoubled,theforcewasdoubled;andifthechargeonbothoftheobjectswasdoubled,theforceincreasedtofourtimestheoriginalvalue.Thiswasthecasewhenthedistancebetweenthetwochargesremainedthesame.Ifthedistancebetweenthemwasallowedtoincrease,hefoundthattheforcedecreasedwiththesquareofthedistancebetweenthem.Thatis,ifthedistancewasdoubled,theforcefelltoone-fourthof

itsoriginalvalue.Thus,Coulombconcluded,theforceonetinychargedobjectexertsonasecondoneisproportionaltotheproductofthemagnitudeofthechargeonone,Q1,timesthemagnitudeofthechargeontheother,Q2,andinverselyproportionaltothesquareofthedistancerbetweenthem（Fig.7-4）.Asanequation,wecanwriteCoulomb'slawas

（7-1）

wherekisaproportionalityconstant.

Equation7-1givesthemagnitudeoftheelectricforcethateitherobjectexertsontheother.Thedirectionoftheelectricforceisalwaysalongthelinejoiningthetwoobjects.Ifthetwochargeshavethesamesign,theforceoneitherobjectisdirectedawayfromtheother.Ifthetwochargeshaveoppositesigns,theforceononeisdirectedtowardtheother,Fig.7-5.Noticethattheforceonechargeexertsonthesecondisequalbutoppositetothatexertedbythesecondonthefirst,inaccordwithNewton'sthirdlaw.

ThevalidityofCoulomb'slawtodayrestsonprecisionmeasurementsthataremuchmoresophisticatedthan

Coulomb'soriginalexperiment.Theexponent,2,inCoulomb'slawhasbeenshowntobeaccurateto1partin1016[thatis,2±（l×10-16）].

Sincewearedealingherewithanewquantity（electriccharge）,wecouldchooseitsunitsothattheproportionalityconstantkinEq.7-1wouldbeone.Indeed,suchasystemofunitswasoncecommon.1However,themostwidelyusedunitnowisthecoulomb（C）,whichistheSIunit.Theprecisedefinitionofthecoulombtodayisintermsofelectriccurrentandmagneticfield,andwillbediscussedlater.InSIunits,khasthevalue

k=8.988×109N.m2/C29.0×109N.m2/C2.

Thus,1Cisthatamountofchargewhich,ifplacedoneachoftwopointobjectsthatare1.0mapart,willresultineachobjectexertingaforceof（9.0×109N.m2/C2）（1.0C）（1.0C）/（1.0m）2=9.0×109Nontheother.Thiswouldbeanenormousforce,equaltotheweightofalmostamilliontons.Wedon'tnormallyencounterchargesaslargeasacoulomb.

Chargesproducedbyrubbingordinaryobjects（suchasacomborplasticruler）aretypicallyaroundamicrocoulomb（lμC=10-6C）orless.Objectsthatcarryapositivechargehaveadeficitofelectrons,whereasnegativelychargedobjectshaveanexcessofelectrons.Themagnitudeofthechargeononeelectronhasbeendeterminedtobeabout1.602×10-19C,anditssignisnegative.Thisisthesmallestchargefoundinnature,andbecauseofitsfundamentalnature,itisgiventhesymboleandisoftenreferredtoastheelementarycharge:

e=1.602×10-19C.

Notethateisdefinedasapositivenumber,sothechargeontheelectronis-e.（Thechargeonaproton,ontheotherhand,is+e）.Sinceanobjectcannotgainorloseafractionofanelectron,thenetchargeonanyobjectmustbeanintegralmultipleofthischarge.Electricchargeisthussaidtobequantized（existingonlyindiscreteamounts:

\e,2e,3e,etc.）.Becauseeissosmall,however,wenormallydon'tnoticethisdiscretenessinmacroscopiccharges（l/xCrequiresabout1013electrons）,whichthusseemcontinuous.

NotethesimilarityofCoulomb'slawtothelawofuniversalgravitation,Eq.7-1.Bothareinversesquarelaws（F∝1/r2）.Bothalsohaveaproportionalitytoaproductofapropertyofeachbody—massforgravity,electricchargeforelectricity.Amajordifferencebetweenthetwolawsisthatgravityisalwaysanattractiveforce,whereastheelectricforcecanbeeitherattractiveorrepulsive.

TheconstantkinEq.7-1isoftenwrittenintermsofanotherconstant,ε0,calledthepermittivityoffreespace.Itisrelatedtokby