重邮大学物理英文版PPT (5)

2.3ELECTRICFLUX,GAUSSSLAW,2020/5/14,1,1.ElectricFieldLines,Aconvenientspecializedpictorialrepresentationforvisualizingelectricfieldpatternsiscreatedbydrawinglineswhicharecalledelectricfieldlines.,Theelectricfieldlinesarerelatedtotheelectricfieldinanyregionofspaceinthefollowingmanner:,2020/5/14,2,(1)Thetangentdirectionateverypointonanelectricfieldlineisjustthedirectionofthefieldintensityatthatpointorthedirectionoftheforceonthepositivepointchargeatthatpoint.(2)Theelectricfieldlinesaredenserintheplacewherethefieldintensityisstronger,andtheelectricfieldlinesaresparserintheplacewherethefieldintensityisweaker.(3)Theelectricfieldlinesstartonpositivechargesandterminateonnegativecharges,andneverintersectedeachother.Itisneverinterruptedinregionwithoutcharge;thisiscalledthecontinuityofelectricfieldline.,(4)Keepinmind:electricfieldlinesdonotactuallyexist.,2020/5/14,3,+,+,Forapositivepointcharge,thelinesaredirectedradiallyoutward.,Foranegativepointcharge,thelinesaredirectedradiallyinward.,Theelectricfieldlinesfortwochargesofequalmagnitudeandoppositesign(anelectricdipole),NOTE:thenumberoflinesleavingthepositivechargeequalsthenumberterminatingatthenegativecharge.,Theelectricfieldlinesfortwopositivepointcharges.,Theelectricfieldlinesforapointcharge+2qandasecondpointchargeq.,+,+,+,+,+,+,+,+,+,RingAmount,2020/5/14,8,FluxAmount,2.ElectricFluxe,1.Uniformelectricfield,2.Uniformelectricfield,=,S,3.Nonuniformelectricfield,arbitrarysurface,UNIT：Vm,2020/5/14,10,Where,2020/5/14,11,Aclosedsurfaceisdefinedasonethatcompletelydividesspaceintoaninsideregionandoutsideregion,sothatmovementcannottakeplacefromoneregiontotheotherwithoutpenetratingthesurface.,Foraclosedsurface,usuallydefinethenormallineateverypointonthesurfacepointsoutoftheclosedsurface,Aclosedsurface,Aopensurface,2020/5/14,12,=0,0,0,n,n,Accordingtotheconvention,outwardtheclosedsurface;,inwardtheclosedsurface.,2020/5/14,14,Thereisacubesurfaceofedgelengthaintheuniformelectricfield（isaconstant）asshowninfigure.Findtheelectricfluxofeveryplaneandthecubesurface.,Example,2020/5/14,15,Gaussworkedinawidevarietyoffieldsinbothmathematicsandphysicsincludingnumbertheory,analysis,differentialgeometry,geodesy,magnetism,astronomyandoptics.Hisworkhashadanimmenseinfluenceinmanyareas.Sometimesknownastheprinceofmathematiciansandgreatestmathematiciansinceantiquity,JohannCarlFriedrichGauss1777-1855,3.GAUSSSLAW,2020/5/14,16,Thenetelectricfluxofanarbitraryclosedsurfaceinthevacuumisequaltothenetchargeinsidethesurfacedividedby.,(1)GAUSSSLAW,2020/5/14,17,S：Theclosedsurface,i.e.gaussiansurface.Itisanimaginarysurfaceandneednotcoincidewithanyrealphysicalsurface.,isthetotalelectricfieldatanypointonthesurfaceduetoallcharges.,Surfaceelement.Itsorientationisperpendiculartothesurfaceandpointsoutwardfromtheinsideregion.,Thealgebrasumofchargesintheclosedsurface.,Theclosesurfaceintegralisoverallgaussiansurface.,2020/5/14,18,Itgiveasimplewaytocalculatethedistributionofelectricfieldforagivenchargedistributionwithsufficientsymmetry.,2020/5/14,19,r,(2)Proving,Asphericalgaussiansurfaceofradiusrsurroundingapointchargeqwhichisatthecentreofthesphere.,Theelectricfieldisnormaltothesurfaceandconstantinmagnitudeeverywhereonthesurface.,2020/5/14,20,Asphericalgaussiansurfaceofradiusrsurroundingapointchargeqwhichisnotatthecentreofthesphere.,r,q,O,2020/5/14,21,Anarbitrarygaussiansurfacesurroundingapointchargeq.,Thenetelectricfluxthrougheachsurfaceisthesame.,2020/5/14,22,S,q1,q2,q3,Therearemanychargesinsidetheguassiansurface.,2020/5/14,23,Apointchargelocatedoutsideaclosedsurface.Thenumberoflinesenteringthesurfaceequalsthenumberleavingthesurface.,Thenetelectricfluxthroughaclosedsurfacethatsurroundsnonetchargeiszero.,Zerofluxisnotzerofield.,2020/5/14,24,Conclusion,Thenetfluxthroughanyclosedsurfacesurroundingthepointchargeqisgivenby,Asystemofcharges,Continuousdistributionofcharges,2020/5/14,25,(3)PhysicalMeaning,Thepositivechargeisthesourceoftheelectrostaticfield.,Gaussslawisvalidfortheelectricfieldofanysystemofchargesorcontinuousdistributionofcharge.,Guassslawcanbeusedtoevaluatetheelectricfieldforchargedistributionsthathavespherical,cylindrical,orplanesymmetry.Thetechniqueisusefulonlyinsituationswherethedegreeofsymmetryishigh.,2020/5/14,26,QuickQuiz,Whyaretheelectrostaticfieldlinesneverinterruptedinregionwithoutcharge?,2020/5/14,27,Findthefluxthroughthesquare.,QuickQuiz,2020/5/14,28,Considerwhetherthefollowingstatementsaretruth.,ElectricfluxoftheGausssurfaceisrelatedwithchargesinGausssurfaceandisnotrelatedwithchargesoutofGausssurface.,ThefieldintensityatapointontheGausssurfaceisrelatedwithchargesintheGausssurfaceandisnotrelatedwithchargesoutoftheGausssurface.,IftheelectricfluxofaGausssurfaceequalszero,theremustbenotchargeintheGausssurface.,2020/5/14,29,IftheelectricfluxofaGausssurfaceequalszero,thenfieldintensityateverypointontheGausssurfaceiszero.,Gausstheoremistenableonlytotheelectrostaticfieldwhosedistributionissymmetricalinspace.,2020/5/14,30,(4)ApplicationofGausssLawtoSymmetricChargeDistribution,Thegaussiansurfaceshouldalwaysbechosentotakeadvantageofthesymmetryofthechargedistribution,sothatwecanremoveEfromtheintegralandsolveit.,2020/5/14,31,Thegaussiansurfacehadbettersatisfiesoneormoreofthefollowingconditions:,Thevalueoftheelectricfieldisconstant.,andareparallel.,andareperpendicular.,isequaltozeroeverywhereonthesurface.,Note:ThesurfaceintegralinGuassslawistakenovertheentiregaussiansurface.,2020/5/14,32,Planesymmetry,Sphericalsymmetry,Thethreesymmetries:,Cylindricalsymmetry,2020/5/14,33,Problem-solvingstrategy,Analysisthesymmetryofthefieldintensitydistribution.,Selectappropriategaussiansurface.,Selectappropriatecoordinates,applyGaussslaw.,2020/5/14,34,Example1.ElectricquantityQdistributesuniformlyonasphericalsurfaceofcenterOandradiusR.Findthefieldintensity.,(1)ASphericallySymmetricChargeDistribution,Analysisthesymmetryofthefieldintensitydistribution,2020/5/14,35,Sphericalsymmetry,Themagnitudeoftheelectricfieldisconstanteverywhereontheconcentricsphericalsurface,andthefieldisnormaltothesurfaceateachpoint.,36,Selectappropriategaussiansurface.,2020/5/14,37,Note,Forauniformlychargedsphericalsurface,thefieldintheregionexternaltothesphericalsurfaceisequivalenttothatofapointchargeatthecenterofthespheresurface.,2020/5/14,38,QuickQuiz,TherearetwoconcentricchargedsphericalshellsofradiusR1andR2.Chargequantitiesdistributeuniformly.,2020/5/14,39,Example2.,AninsulatingsolidsphereofradiusrhasauniformvolumechargedensityandcarriesatotalpositivechargeQ.Calculatetheelectricfieldintensity.,Solution,Becausethechargedistributionissphericallysymmetric,weselectasphericalgaussiansurfaceofradiusr,concentricwiththesphere.,2020/5/14,40,2020/5/14,41,Theelectricfieldinsidethesphere()varieslinearlywithr.Theelectricfieldoutsidethesphere()isthesameasthatofapointchargeQlocatedatr=0.,Theexpressionsoffieldintensitymatchwhenr=a.,2020/5/14,42,(2)ACylindricallySymmetricChargeDistribution,aninfiniteuniformchargedstraightline,aninfiniteuniformchargedcylinder,Example1,Asectionofaninfinitelylongcylindricalplasticrodwithauniform+.Letusfindanexpressionforthemagnitudeoftheatadistancerfromtheaxisoftherod.,Infinitelength,2020/5/14,43,Analysisthesymmetryofthefieldintensitydistribution,2020/5/14,44,CylindricalSymmetry,Themagnitudeoftheelectricfieldisconstanteverywhereonthecoaxialcylindricalsurface,andthefieldisnormaltothesurfaceateachpoint.,2020/5/14,45,2.Selectappropriategaussiansurface:acylindricalgaussiansurfaceofradiusrandlengthlthatiscoaxialwiththelinecharge.,2020/5/14,46,？,2020/5/14,47,Thefieldintensityofaninfiniteuniformchargedstraightline:,2020/5/14,48,Example2,2020/5/14,49,QuickQuiz,ThechargedensityofaninfiniteuniformsolidchargedcylinderofradiusRis,findtheelectricfieldintensity.,2020/5/14,50,3.PlanarSy