电动力学导论David J Griffith答案ch01.doc

INSTRUCTORSSOLUTIONSMANUALINTRODUCTIONtoELECTRODYNAMICSThirdEditionDavidJ.GriffithsTABLEOFCONTENTSChapter1VectorAnalysis1Chapter2Electrostatics22Chapter3SpecialTechniques42Chapter4ElectrostaticFieldsinMatter73Chapter5Magnetostatics89Chapter6MagnetostaticFieldsinMatter113Chapter7Electrodynamics125Chapter8ConservationLaws146Chapter9ElectromagneticWaves157Chapter10PotentialsandFields179Chapter11Radiation195Chapter12ElectrodynamicsandRelativity219Chapter1VectorAnalysisProblem1.1aFromthediagramIBCICOSO3IBICOSO1ICICOSO2MultiplybyIAI.IAIIBCICOSO3IAIIBICOSO1IAIICICOSO2.So:A.BCA.BA.C.Dotproductisdistributive.Similarly:IBCIsin03IBIsin01ICIsinO2MulitplybyIAIn.IAIIBCIsin03nIAIIBIsin01nIAIICIsinO2n.IfnistheunitvectorpointingoutofthepageitfollowsthatAxBeAxBAxe.Crossproductisdistributive.bForthegeneralcaseseeG.E.HaysVectorandTensorAnalysisChapter1Section7dotproductandSection8crossproduct.Problem1.2Thetriplecross-productisnotingeneralassociative.ForexamplesupposeAandCisperpendiculartoAasinthediagram.ThenBXCpointsout-of-the-pageandAXBXCpointsdownandhasmagnitudeABC.ButAxB0soAxBxC0:f.AxBxC.ICIsin82IBlsin81Ak-hBBxCiAxBxeProblem1.3zA1x1Y-HA/3B1x1YHiB/3.A.B11-11ABcosO/3/3cosogtcosO.10COS-1t70.5288IxyProblem1.4Thecross-productofanytwovectorsintheplanewillgiveavectorperpendiculartotheplane.ForexamplewemightpickthebaseAandtheleftsideB:A-1x2y0zB-1x0Y3z.12CHAPTER1.VECTORANALYSISxyZAxBI-12016x3y2z.-103Thishastherightdirectionbutthewrongmagnitude.length:IAxBIv36947.Problem1.5Tomakeaunitvectoroutofitsimplydividebyitsft-AXB16A32I-IAXBI7X7y7z.xyZAxAyAzByCz-BzCyBzCx-BxCzBxCy-ByCxxAyBxCy-ByCx-AzBzCx-BxCzyOzOIlljustcheckthex-componenttheothersgothesameway.xAyBxCy-AyByCx-AzBzCxAzBxCzyOzOoBA.C-CA.BBxAxCxAyCyAzCz-CxAxBxAyByAzBzx0y0zxAyBxCyAzBxCz-AyByCx-AzBzCxyOzOoTheyagree.AxBxCProblem1.6AxBXCBxCxACxA-xBBA.C-CA.BCA.B-AC.BAB.C-BC.Ao.So:AxBxC-AxBxC-BxCxAAB.C-CA.B.IfthisiszerotheneitherAisparalleltoCincludingthecaseinwhichtheypointinoppositedirectionsoroneiszeroorelseB.CB.A0inwhichcaseBisperpendiculartoAandCincludingthecaseB0.Conclusion:AxBxCAxBxClt:eitherAisparalleltoCorBisperpendiculartoAandC.Problem1.74x6y8z-2x8y7z2x-2yzlyl441J.12A2AlAIZ3x-3Y3z:rroblem1.8aA.yByA.zBzcoscpAysincpAzcoscpBysincpBz-sincpAycoscpAz-sincpBycoscpBzCOS2cpAyBysincpcoscpAyBzAzBysin2cpAzBzsin2cpAyBy-sincpcoscpAyBzAzByCOS2cpAzBzCOS2cpsin2cpAyBysin2cpCOS2cpAzBzAyByAzBz./-2-2-2-3-333.bAxAyAz-ilAiAi-iljlRijAjklRikAkjkiRijRikAjAk.Th.IA2A2A2.ddI3RR-IifjkISequasxyzprovzeLJilijik-0ifjI-kMoreoverifRistopreservelengthsforallvectorsA.thenthisconditionisnotonlysufficientbutalsonecessary.ForsupposeA100.ThenjkiRijRikAjAkiRilRilandthismustequal1sincewe-2-2-2.33.wantAxAyAz1.LIkewIseilRi2Ri2ilRi3Ri31.TocheckthecaseJI-kchooseA110.Thenwewant2jkiRijRikAjAkiRilRiliRi2Ri2iRilRi2iRi2Ril.Butwealreadyknowthatthefirsttwosumsareboth1thethirdandfourthareequalsoiRilRi2iRi2Ril0andsoonforotherunequalcombinationsofjk./Inmatrixnotation:RR1whereRisthetransposeofR.3Problem1.9yxLookingdowntheaxis:xzA1200rotationcarriesthezaxisintotheyzaxisyintoxyandxintozx.AyAxAzAy.001R100010SoAxAzProblem1.10aINochange.AxAxAyAyAzAzbIA-t-AIinthesenseAx-AxAy-AyAz-AzcAxB-t-AX-BAxB.ThatisifCAxBIe-tCI.Nominussignincontrasttobehaviorofanquotordinaryquotvectorasgivenbyb.IfAandBarepseudovectorsthenAXB-tAXBAxB.Sothecrossproductoftwopseudovectorsisagainapseudovector.Inthecrossproductofavectorandapseudovectoronechangessigntheotherdoesntandthereforethecross-productisitselfavector.AngularmomentumLrxpandtorqueNrxFarepseudovectors.dA.BxC-t-A.?-Bx-C-A.BxC.SoifaA.BxCthenIa-t-aIapseudoscalarchangessignunderinversionofcoordinates.Problem1.11aVf2xx3y2y4z3zbVf2xy3z4X3x2y2z4y4x2y3z3ZcVfeXsinylnzxeXcosylnzyexsinyl/zzProblem1.12aVh1O2y-6x-18x2x-8y28y.Vh0atsummitso2y-6x-1802x-8y280gt6x-24y8402y-18-24y84O.22y66gty3gt2x-24280gtx-2.Topis13milesnorth2mileswestofSouthHadley.bPuttinginx-2y3:h10-12-12-363684121720ft.IcPuttinginx1y1:Vh102-6-18x2-828y10-22x22y220-xy.IVhi220v2:J1311ft/mileIdirection:Inorthwest.4CHAPTER1.VECTORANALYSISProblem1.13x-xxy-yyz-ZlZjVex-X2y-y2z-ZI2.aV2txX_X2y_y2Z_Z2JxtyOytzOz2X_XX2y_yy2z_ZZ2.bVktxx-X2y-y2z-ZI2-xtyo-ytzO-z1-J.21-J.21-J.2-22x-xX-22y-yY-22z-ZZ-O-x-xxy-yyz-ZlIi_1/1-3-I/1-2i.c/xnn1-n-lfin1-n-lH21-xn1-n-lixsoIV1-nn1-n-li.1Problem1.14yycosifgtzsinifgtmultiplybysinifgt:ysinifgtysinifgtcosifgtZsin2ifgt.Z-ysinifgtZcosifgtmultiplybycosifgt:zcosifgt-ysinifgtcosifgtZCOS2ifgt.Add:ysinifgtzcosifgtzsin2ifgtcos2if?z.Likewiseycosifgt-zsinifgty.Socosifgt-sinifgtvsinifgtcosifgt.ThereforeVJyUMvcosifgtVJysinifgtVJzVIli.0:-.V/V/SoVItransformsasavector.zozoyozozozsmlycoSlzProblem1.15aV.vatxX2ty3XZ2tz-2xz2x0:-2xO.qedbV.Vbtxxyty2yztz3xzy2x3x.cV.vctxy2ty2xyZ2tz2yz02x2y2xy.Problem1.16V.vtx-tytz-txxx2y2z2-tyyX2y2Z2-tzzx2y2z2-3535350-2x-3/2-22x0-2y-3/20-22y0-2z-3/2O-22z3r-3-3r-5x2y2Z23r-3-3r-3O.Thisconclusionissurprisingbecausefromthediagramthisvectorfieldisobviouslydivergingawayfromtheorigin.HowthencanV.vOTheansweristhatV.v0everywhereexceptattheoriginbutattheoriginourcalculationisnogoodsincer0andtheexpressionforvblowsup.InfactV.visinfiniteatthatonepointandzeroelsewhereasweshallseeinSect.1.5.Problem1.17VycosifgtVysinifgtvzVz-sinifgtvycosifgtvz.-lJL.quot-bl.ozlJL.bl.lJL.8Z.UI.Pb114.8y-8ycoslogsml-oy8y8z8ycosl8y8y8zoysml.seresutmra.COgifgtsinif?COgifgtlJL.cosifgtlJL.sinif?sinifgt.8y8z8yOZ-sinifgtcosifgt-8:.sinifgtlJL.lJL.8:.cosifgt8zOZ8z8y8zOZ8z8y8zoz8z-vsinifgti:cosif?sinifgt-sinifgtcosif?cosifgt.SoCOS2sincosifgtlJL.sincoslJL.sin2ifgtsin2-sincos8y8z8yI8zI8yII8z8yI8zII5-sinifgtcosifgtCOS2ifgtoy8zcOS2ifgtsin2if?sin2A.COS2A.8y8zIfIf8y8zProblem1.18aVxvaxyz0008xoyozx23xz2-2xzxyZ800ox8yozxy2yz3xzxO-6xzyO2zz3z2-01-6xzx2zy3Z2z.1bVXVbxO-2yyO-3zzO-x1-2yx-3zy-xz.xcVxvcIIxy2Problem1.19vyxxyorvyzxxzyxyZjorv3x2z-Z3X3yx3-3xz2zorvsinxcoshyx-cosxsinhyyjetc.Problem1.20iVfg8:x8:yoZtgUxJgUytg?ZzJxyzgUxUy?ZzJVggVI.qedivV.AxBIxAyBz-AzBylyAzBx-AxBzlzAxBy-AyBx-AB-A8Bv-BMLA8BzBML-A-BoA.-yOXz8xz8xy8xz8yx8yxoyzoyABML-A8B.-Bxozy8zy8zx8zBML-BML-MLB-OA.-A-x8yOZy8z8zzoz8yx8y8z-Ay8tz.-Az8v-o.B.VxA-A.VxB.qedvVxfAO:-Ov?x8.-8:?y8:v-8:.?ZJM.A-J-AxJ8A.A-JMAyA8yz8y8zY8zOZzozoxz8xJA-JoAz-Az8zy8x8yx8yJM.-xML-MLYA-8AzZfufu-Ay?Z-AzUxAzU-Az?ZyAxU-AyUzJVxA-AxVI.qedProblem1.21aA.1BAoB.A8B.A8B.XA8BvAoBvA8BvYAxyZfuzyZfuAAAz.x8xyoyz8zbf.xxyyzz.Letsjustdothexcomponent.rylx2y2z2f.1fx-xIxYlyztzylz2y2z2yZ00oyoz2xyz22yzx2z-2zyO-0z2y-2yQJ6CHAPTER1.VECTORANALYSISxJ-X-32Xyx-32YZX-32Z-X3xy2XZ2-X2y2Z2-O.Samegoesfortheothercomponents.Hence:If.Vf0I.cVa.VVbX2tx3XZ2ty-2xztzxyx2yzy3xzzX2yxOy3zz3XZ2xx2zyOz-2xzOx2yy3xzX2y3x2z2X6XZ3-4xyzy3X2z-6X2ZZIX2y3Z2x2xz3z2-2yY-3X2ZZIProblem1.22iiVA.Bx:xAxBxAyByAzBzozBxA:Xza:quotByAy:XquotBzAzAxVxBxAyVxBz-AzVxByAy:Xquot-.:vz-Azfzz-BxVxAB-B-xyaxoyzozaxA.VBxAxtxAytyAztJBxAx:XzAy.:vzAzfzzB.VABoAzBoAzBoAzxxaxyoyzozSoAxVxBBxVxAA.VBB.VAx-AoBquot-AoBz-AoBzAllBoAquot-BoAz-BoAzB-YoxYoyzozzaxYoxyoyZozzaXAoBzAilkAoBzBBoAzBM.xampyZhxampyZhBoAzAoBzBoAquot-AOBquot-xaxxaxyaxTvTvyaxTvTvB-A-llzTzaxTzzTzaxTzVA.BxsameforyandzviVxAxBxtyAXBz-tzAxBytyAxBy-AyBx-tzAzBx-AxBz-oAzBA-B-AoBz-MLB-AlloAzBAll.-oyyxoyoyxYoyOZxzOZozzxozB.VA-A.VBAV.B-BV.Ax-BoAzBoAzBoAz-AoBz-AoBz-AoBzAOBzll-B8Az-xoxYoyzozxaxyoyzozxaxoyoz.xaxoyozBoAzA-oBquotllB-MLyoyxTx-quot7fXquotoyozxTJXTJXoyozA-A-BM.yoyzozzozVxAxBxsameforyandzProblem1.23txU/gxtyU/gytzU/gzgU-fAg-fAgU-fA2Xg2Yg2Z1rg.Lx.LyA.Lz-f?i.x?i.yA?i.zgVf-fVg.Qedloxoyozaxoyozg2VU/gtxAx/gtyAy/gtzAz/gg-Azg-Ag-A.g2g2g21gOAzoAquotML-A?i.A?i.A?i.gV.A-A.VgQedaxoyozxaxyoyzozg2.V.A/g7VxA/g.tyAz/g-tzAy/guAg-Alf119-v2-g2gg?:-?:-Azv-Ay-gVXAquotAXVgquotsameforyandz.Qed-g2Problem1.24xyZaAxBIx2y3zIx6xzy9zyz_2X2-6y23y-2x0V.AxBt.6xzty9zytz-2x2-6y26z9z015zVxAxty3Z-tz2Yytzx-t.3zzt.2Y-tyx0B.VxA0VxBxtyO-tz-2Xytz3y-t.0zt.-2X-ty3Y-5zA.VxB-15zV.AxBB.VxA-A.VxB0-15z15z./bA.B3xy-4xy-xyVA.BV-xyxt.-xyyty-xy-yx-xyxyzAxVxBIx2y3z1x-lOyy5xBxVxA000-5IIA.VBxt.2yty3ztz3yx-2xyx6yy-2xB.VA3yto:-2xtyxx2yy3zzx3yy-4xAxVxBBxVxAA.VBB.VA-10yx5xy6yx-2xy3yx-4xy-yx-xyV.A.B./cVXAxBxty-2X2-6y2-tz9ZYytz6xz-t.-2x2-6y2zt.9zy-ty6xzx-12y-9yy6x4xzO-21yxlOxyV.At.xty2ytz3z1236V.Bt.3yty-2x0.B.VA-A.VBAV.B-BV.A3yx-4xy-6yx2xy-18yx12xy-21yxlOxyVxAxB./Problem1.25a8a28/8a0gtI72Ta2.1827182718271I2.IbWW-ngt7n-3Tb-3smxsmysmz.c8725Tc8c-16Tc8c-9TcgtI72Tc0.1d82quot2aquot80gt72V.282v82v82v2WW0F6xgt7Vy6xa8.810gt72vz0Ivv2X6xy.18CHAPTER1.VECTORANALYSISProblem1.26V.VXv.JL-JL-JL-axayazayazaxazax8y-a2vza2vz82vz82vquota2vquot-0baltfd.t.-ax8y-8y8xayoz-ozoyozax-ax8z-yequ1y0cross-envaIves.FromProb.1.18:VXVb-2yx-3zy-xzV.VXVbz-2y/y-3z/z-xO./Problem1.27xyzIaa8VXVt8xfJyfJzfJtfJtfJtfJxfJyaz0byequalityofcross-derivatives.InProb.1.l1bVf2xy3z4X3x2y2z4y4x2y3z3zsoXyzaaafJxfJyaz2xy3z43x2y3z44x2y3z3x3.4x2y2z3-4.3x2y2z3y4.2xy3z3-2.4xy3z3z2.3xy2z4-3.2xy2Z4O./Problem1.28a000100.x:0-t1yzOJdldxXjv.dlx2dxjJv.dlJX2dxx3/3IA1/3.100110.x1Y:0-t1z0dldyyv.dl2yzdyOJJv.dlO.110111.xY1z:0-tIjdldzZjv.dly2dzdzjJv.dlJo1dzzlA1.Total:Jv.dl1/30114/3.1b000001.xy0z:0-t1dldzzv.dly2dz0Jv.dlO.001011.xOy:0-tlzIjdldyyv.dl2yzdy2ydyJv.dlJ2ydyy21A1.011111.x:0-t1yzIjdldxxv.dlx2dxJv.dlJx2dxx3/3IA1/3.Total:Jv.dl011/314/3.1cxyz:0-tIjdxdydzjv.dlX2dx2yzdyy2dzx2dx2X2dxx2dx4X2dxJv.dlJ4x2dx4X3/3IA14/3.1dfv.dl4/3-4/3Problem1.29xy:0-tlzOjdadxdyzjvdayz2-3dxdy-3ydxdyJv.da-3J:dxJ:ydy-3xlfl-322IInEx.1.7wegot20forthesameboundarylinethesquareinthexy-planesotheanswerisIno:Ithesurfaceintegraldoesnotdependonlyontheboundaryline.Thetotalfluxforthecubeis2012illProblem1.30JTdrJZ2dxdydz.Youcandotheintegralsinanyorder-hereitissimplesttosavezforlast:-fJ2tfJ2t-fJ2tfJ2t-a2ta2txfJyfJz-fJzfJyYfJzfJx-fJxfJzzaxfJy-fJyaxVXVf/Z2/dXdYdz.Theslopingsurfaceisxyz1sothexintegralisJI-y-Zdx1-y-z.Foragivzyrangesfrom0to1-zsotheyintegralisJI-Z1-y-zdy1-zy-y2/2JII-Z1-z2-1-z2/21-z2/291/2-zz2/2.FinallythezintegralisJZ2-ZZdzJo1-Z34dz-4flbi-tfoProblem1.31Tb1427jTa0.gtITb-Ta7.1VT2x4yx4x2Z3y6yz2zVT.dl2x4ydx4x2x3dy6yz2dzaSegment1:x:0-t1yzdydzO.JVT.dlJ2xdxx211.Segment2:y:O-tlxlzOdxdz0.JVT.dlJo14dy4YI4.J:VT.dl7./Segment3:z:0-t1xy1dxdyO.JVT.dlJ6Z2dz2z312.bSegment:z:O-tlxydxdyO.JVT.dlJOdzO.Segment2:y:0-t1x0z1dxdzO.JVT.dlJo12dy2yl2.J.bVT.dl7./Segment3:x:0-t1yz1dydzO.JVT.dlJo2x4dxaX24xl145.cx:O-tlyxzX2dydxdz2xdx.VT.dl2x4xdx4x2X6dx6XX42xdxlOx14x6dx.1:VT.dlJlOx14x6dx5X22X71527./Problem1.32V.vy2z3xJV.vdrJy2z3xdxdydzJJJy2z3xdxdydz-ty2zxX22y2z6JJ2y4z6dYdzty24z6Y424z68z16J8z16dz4z216zl1632148.1NumberingthesurfacesasinFig.1.29:idadydzxx2.v.da2ydydz.Jv.daJJ2ydydz2y218.iida-dydzxxO.v.daO.Jv.daO.iiidadxdzyy2.v.da4zdxdz.Jv.daJJ4zdxdz16.ivda-dxdzyyO.v.daO.Jv.daO.vdadxdyzz2.v.da6xdxdy.Jv.da24.vida-dxdyzzO.v.daO.Jv.daO.gtJv.da8162448./Problem1.33VxvxO-2yyO-3zzO-x-2yx-3zy-xz.dadydzXifweagreethatthepathintegralshallruncounterclockwise.SoVxv.da-2ydydz.10CHAPTER1.VECTORANALYSISJVxv.daJJ:-Z-2YdYdzyy21-Z-2-Z2-J:4-4zZ2dz-4Z-2z2I:-S-sI-1Meanwhilev.dlxydx2yzdy3zxdz.Therearethreesegments.zyz31y1xz0dxdzO.y:0-t2.Jv.dlO.2x0z2-ydx0dzdyy:2-tO.v.dl2yzdy.Jv.dlJ202y2-ydy-J:4y-2y2dy-2y2-y3I-S-.S-i.3xy0dxdy0z:2-tO.v.dlO.Jv.dlO.Sovdl-i./Problem1.34ByCorollary1JVxv.dashouldequal.VXv4z2-2xx2zz.idadydzxx1yz:0-t1.Vxv.da4z2-2dydzJVxv.daJ4z2-2dzz3-2zl-2-.iida-dxdyzzOJxy:0-t1.Vxv.da0JVxv.daO.iiidadxdzyy1xz:0-t1.Vxv.da0fVxv.daO.ivda-dxdzyy0xz:0-t1.Vxv.da0JVxv.daO.vdadxdyzz1xy:0-t1.Vxv.da2dxdyJVxv.da2.gtfVxv.da-2./Problem1.35aUsetheproductruleVXfAfVXA-AxVf:LfVXA.daLVxfA.daLrAXVf.daifA.dlLrAXVfda.qed.IusedStokestheoreminthelaststep.bUsetheproductruleV.AxBB.VxA-A.VxB:IvBVXAdrIvV.AxBdrIvAVXBdrtAXB.da-IvAVXBdr.qed.Iusedthedivergencetheoreminthelaststep.11Problem1.361rjx2y2Z2-I0cosyxyquotqtan-Im.Problem1.37Therearemanywaystodothisone-probablythemostilluminatingwayistoworkitoutbytrigonometryfromFig.1.36.Themostsystematicapproachistostudytheexpression:rxxyYzzrsin0cosqxrsin0sinqyrcos0z.IfIonlyvaryrslightlythendrfj-.rdrisashortvectorpointinginthedirectionofincreaseinr.TomakeitaunitvectorImustdividebyitslength.Thus:8r8rkfIIh11Jgt11?isinOcosqxsinOsinqycosOz1?i12sin20cos2qsin20sin2qcos201.rcosOcosqxrcosOsinqy-rsinOz112r2cos20cos2qr2cos20sin2qr2sin20r2-rsinOsinqxrsinOco