重邮大学物理英文版 (5)ppt课件

2 3ELECTRICFLUX GAUSS SLAW 17 04 2020 1 1 ElectricFieldLines Aconvenientspecializedpictorialrepresentationforvisualizingelectricfieldpatternsiscreatedbydrawinglineswhicharecalledelectricfieldlines Theelectricfieldlinesarerelatedtotheelectricfieldinanyregionofspaceinthefollowingmanner 17 04 2020 2 1 Thetangentdirectionateverypointonanelectricfieldlineisjustthedirectionofthefieldintensityatthatpointorthedirectionoftheforceonthepositivepointchargeatthatpoint 2 Theelectricfieldlinesaredenserintheplacewherethefieldintensityisstronger andtheelectricfieldlinesaresparserintheplacewherethefieldintensityisweaker 3 Theelectricfieldlinesstartonpositivechargesandterminateonnegativecharges andneverintersectedeachother Itisneverinterruptedinregionwithoutcharge thisiscalledthecontinuityofelectricfieldline 4 Keepinmind electricfieldlinesdonotactuallyexist 17 04 2020 3 Forapositivepointcharge thelinesaredirectedradiallyoutward Foranegativepointcharge thelinesaredirectedradiallyinward Theelectricfieldlinesfortwochargesofequalmagnitudeandoppositesign anelectricdipole NOTE thenumberoflinesleavingthepositivechargeequalsthenumberterminatingatthenegativecharge Theelectricfieldlinesfortwopositivepointcharges Theelectricfieldlinesforapointcharge 2qandasecondpointcharge q RingAmount 17 04 2020 8 FluxAmount 2 ElectricFlux e 1 Uniformelectricfield 2 Uniformelectricfield S 3 Nonuniformelectricfield arbitrarysurface UNIT Vm 17 04 2020 10 Where 17 04 2020 11 Aclosedsurfaceisdefinedasonethatcompletelydividesspaceintoaninsideregionandoutsideregion sothatmovementcannottakeplacefromoneregiontotheotherwithoutpenetratingthesurface Foraclosedsurface usuallydefinethenormallineateverypointonthesurfacepointsoutoftheclosedsurface Aclosedsurface Aopensurface 17 04 2020 12 0 0 0 n n Accordingtotheconvention outwardtheclosedsurface inwardtheclosedsurface 17 04 2020 14 Thereisacubesurfaceofedgelengthaintheuniformelectricfield isaconstant asshowninfigure Findtheelectricfluxofeveryplaneandthecubesurface Example 17 04 2020 15 Gaussworkedinawidevarietyoffieldsinbothmathematicsandphysicsincludingnumbertheory analysis differentialgeometry geodesy magnetism astronomyandoptics Hisworkhashadanimmenseinfluenceinmanyareas Sometimesknownas theprinceofmathematicians and greatestmathematiciansinceantiquity JohannCarlFriedrichGauss1777 1855 3 GAUSS SLAW 17 04 2020 16 Thenetelectricfluxofanarbitraryclosedsurfaceinthevacuumisequaltothenetchargeinsidethesurfacedividedby 1 GAUSS SLAW 17 04 2020 17 S Theclosedsurface i e gaussiansurface Itisanimaginarysurfaceandneednotcoincidewithanyrealphysicalsurface isthetotalelectricfieldatanypointonthesurfaceduetoallcharges Surfaceelement Itsorientationisperpendiculartothesurfaceandpointsoutwardfromtheinsideregion Thealgebrasumofchargesintheclosedsurface Theclosesurfaceintegralisoverallgaussiansurface 17 04 2020 18 Itgiveasimplewaytocalculatethedistributionofelectricfieldforagivenchargedistributionwithsufficientsymmetry 17 04 2020 19 r 2 Proving Asphericalgaussiansurfaceofradiusrsurroundingapointchargeqwhichisatthecentreofthesphere Theelectricfieldisnormaltothesurfaceandconstantinmagnitudeeverywhereonthesurface 17 04 2020 20 Asphericalgaussiansurfaceofradiusrsurroundingapointchargeqwhichisnotatthecentreofthesphere r q O 17 04 2020 21 Anarbitrarygaussiansurfacesurroundingapointchargeq Thenetelectricfluxthrougheachsurfaceisthesame 17 04 2020 22 S q1 q2 q3 Therearemanychargesinsidetheguassiansurface 17 04 2020 23 Apointchargelocatedoutsideaclosedsurface Thenumberoflinesenteringthesurfaceequalsthenumberleavingthesurface Thenetelectricfluxthroughaclosedsurfacethatsurroundsnonetchargeiszero Zerofluxisnotzerofield 17 04 2020 24 Conclusion Thenetfluxthroughanyclosedsurfacesurroundingthepointchargeqisgivenby Asystemofcharges Continuousdistributionofcharges 17 04 2020 25 3 PhysicalMeaning Thepositivechargeisthesourceoftheelectrostaticfield Gauss slawisvalidfortheelectricfieldofanysystemofchargesorcontinuousdistributionofcharge Guass slawcanbeusedtoevaluatetheelectricfieldforchargedistributionsthathavespherical cylindrical orplanesymmetry Thetechniqueisusefulonlyinsituationswherethedegreeofsymmetryishigh 17 04 2020 26 QuickQuiz Whyaretheelectrostaticfieldlinesneverinterruptedinregionwithoutcharge 17 04 2020 27 Findthefluxthroughthesquare QuickQuiz 17 04 2020 28 Considerwhetherthefollowingstatementsaretruth ElectricfluxoftheGausssurfaceisrelatedwithchargesinGausssurfaceandisnotrelatedwithchargesoutofGausssurface ThefieldintensityatapointontheGausssurfaceisrelatedwithchargesintheGausssurfaceandisnotrelatedwithchargesoutoftheGausssurface IftheelectricfluxofaGausssurfaceequalszero theremustbenotchargeintheGausssurface 17 04 2020 29 IftheelectricfluxofaGausssurfaceequalszero thenfieldintensityateverypointontheGausssurfaceiszero Gausstheoremistenableonlytotheelectrostaticfieldwhosedistributionissymmetricalinspace 17 04 2020 30 4 ApplicationofGauss sLawtoSymmetricChargeDistribution Thegaussiansurfaceshouldalwaysbechosentotakeadvantageofthesymmetryofthechargedistribution sothatwecanremoveEfromtheintegralandsolveit 17 04 2020 31 Thegaussiansurfacehadbettersatisfiesoneormoreofthefollowingconditions Thevalueoftheelectricfieldisconstant andareparallel andareperpendicular isequaltozeroeverywhereonthesurface Note ThesurfaceintegralinGuass slawistakenovertheentiregaussiansurface 17 04 2020 32 Planesymmetry Sphericalsymmetry Thethreesymmetries Cylindricalsymmetry 17 04 2020 33 Problem solvingstrategy Analysisthesymmetryofthefieldintensitydistribution Selectappropriategaussiansurface Selectappropriatecoordinates applyGauss slaw 17 04 2020 34 Example1 ElectricquantityQdistributesuniformlyonasphericalsurfaceofcenterOandradiusR Findthefieldintensity 1 ASphericallySymmetricChargeDistribution Analysisthesymmetryofthefieldintensitydistribution 17 04 2020 35 Sphericalsymmetry Themagnitudeoftheelectricfieldisconstanteverywhereontheconcentricsphericalsurface andthefieldisnormaltothesurfaceateachpoint 36 Selectappropriategaussiansurface 17 04 2020 37 Note Forauniformlychargedsphericalsurface thefieldintheregionexternaltothesphericalsurfaceisequivalenttothatofapointchargeatthecenterofthespheresurface 17 04 2020 38 QuickQuiz TherearetwoconcentricchargedsphericalshellsofradiusR1andR2 Chargequantitiesdistributeuniformly 17 04 2020 39 Example2 Aninsulatingsolidsphereofradiusrhasauniformvolumechargedensity andcarriesatotalpositivechargeQ Calculatetheelectricfieldintensity Solution Becausethechargedistributionissphericallysymmetric weselectasphericalgaussiansurfaceofradiusr concentricwiththesphere 17 04 2020 40 17 04 2020 41 Theelectricfieldinsidethesphere varieslinearlywithr Theelectricfieldoutsidethesphere isthesameasthatofapointchargeQlocatedatr 0 Theexpressionsoffieldintensitymatchwhenr a 17 04 2020 42 2 ACylindricallySymmetricChargeDistribution aninfiniteuniformchargedstraightline aninfiniteuniformchargedcylinder Example1 Asectionofaninfinitelylongcylindricalplasticrodwithauniform Letusfindanexpressionforthemagnitudeoftheatadistancerfromtheaxisoftherod Infinitelength 17 04 2020 43 Analysisthesymmetryofthefieldintensitydistribution 17 04 2020 44 CylindricalSymmetry Themagnitudeoftheelectricfieldisconstanteverywhereonthecoaxialcylindricalsurface andthefieldisnormaltothesurfaceateachpoint 17 04 2020 45 2 Selectappropriategaussiansurface acylindricalgaussiansurfaceofradiusrandlengthlthatiscoaxialwiththelinecharge 17 04 2020 46 17 04 2020 47 Thefieldintensityofaninfiniteuniformchargedstraightline 17 04 2020 48 Example2 17 04 2020 49 QuickQuiz ThechargedensityofaninfiniteuniformsolidchargedcylinderofradiusRis findtheelectricfieldintensity 17 04 2020 50 3 PlanarSymmetry InfiniteChargedPlaneorSheet 17 04