电磁场与电磁波第三版答案-西安电子科大出版

11-1zyxezeyexrvvvv++=αβγ),,(zyxαβγ1coscoscos222=++γβαrx=αcosry=βcosrz=γcos222zyxr++=1coscoscos2222222222222=++=++=++rzyxrzryrxγβα1-2zyxeeeAvvvv−−=9zyxeeeBvvvv342+−=(1)BAvv+(2)BAvv−(3)BAvv⋅(4)BAvv×1zyxzyxzyxeeeeeeeeeBAvvvvvvvvvvv2133)342()9(+−=+−+−−=+2zyxzyxzyxeeeeeeeeeBAvvvvvvvvvvv45)342()9(−−−=+−−−−=−3)342()9(zyxzyxeeeeeeBAvvvvvvvv+−⋅−−=⋅3533623)1()4()9(21=−+=×−+−×−+×=4342191−−−=×zyxeeeBAvvvvvzyxeeevvv))9(21)4(())1(231())1()4(39(−×−×−+−×−×−−×−−×−=zyxeeevvv14531+−−=1-3zyxecebeAvvvv++=zyxeeeBvvvv83++−=BAvv⊥BAvv//bcBAvv⊥0=⋅BAvv0831=++−cb0183=−+cbbc2BAvv//0=×BAvv()()()038388311=+++−−=−=×zyxzyxebcecbecbeeeBAvvvvvvvv8,3−=−=cb1-4zyxeeeAvvvv++=912yxebeaBvvv+=ABvv⊥Bv1abABvv⊥0=⋅BAvv0912=+ba1=Bv122=+ba⎪⎩⎪⎨⎧−==5453ba⎪⎩⎪⎨⎧=−=5453ba1-5xyzzxy−+=22ϕ)2,1,1(3πα=4πβ=3πγ=γϕβϕαϕϕcoscoscoszyxl∂∂+∂∂+∂∂=∂∂()()()3cos24cos23cos2πππxyzxzxyyzy−+−+−=()()()()2111222221221212,1,1×−×+×−+−=∂∂xxlϕ121321=×+−=1-6xyz=ϕ)2,1,5()2,1,5()19,4,9(()()()3142191459222=−+−+−=r3314459cos=−=rα314314cos=−=rβ31417219cos=−=rγγβαγϕβϕαϕϕcoscoscoscoscoscosxyxzyzzyxl+⋅+⋅=∂∂+∂∂+∂∂=∂∂()()()314123853083141171532542131412,1,5=++=××+××+××=∂∂lϕ1-7zyxxyzyx62332222−−++++=ϕ)0,0,0()1,1,1(zyxezeyexgradvvv∂∂+∂∂+∂∂==∇ϕϕϕϕϕ()()()zyxezexyeyxvvv662432−+−++++=000zyxeeevvv623)0,0,0(−−=ϕgrad111yxeevv36)1,1,1(+=ϕgrad1-8uvxyzuv(1)()vu+gradvugradgrad+=(2)()uvgraduvvugradgrad+=(3)()uuugradgrad22=1()()()()zyxezvueyvuexvuvuvvv∂+∂+∂+∂+∂+∂=+grad4⎟⎠⎞⎜⎝⎛∂∂+∂∂+∂∂+⎟⎠⎞⎜⎝⎛∂∂+∂∂+∂∂=zyxzyxezveyvexvezueyuexuvvvvvvvugradgrad+=2()()()()zyxezuveyuvexuvuvvvv∂∂+∂∂+∂∂=gradzzyyxxezuvezvueyuveyvuexuvexvuvvvvvv∂∂+∂∂+∂∂+∂∂+∂∂+∂∂=⎟⎠⎞⎜⎝⎛∂∂+∂∂+∂∂+⎟⎠⎞⎜⎝⎛∂∂+∂∂+∂∂=zyxzyxezueyuexuvezveyvexvuvvvvvvuvvugradgrad+=32()uuuuuuugradgradgradgrad22=+=1-9(1)BABAvvvv⋅∇+⋅∇=+⋅∇)((2)ϕϕϕ∇⋅+⋅∇=⋅∇AAAvvv)(1()BAvv+⋅∇()zzyyxxzzyyxxzyxeBeBeBeAeAeAezeyexvvvvvvvvv+++++⋅⎟⎠⎞⎜⎝⎛∂∂+∂∂+∂∂=BAzByBxBzAyAxAzyxzyxvv⋅∇+⋅∇=∂∂+∂∂+∂∂+∂∂+∂∂+∂∂=2()()()zzyyxxzyxeAeAeAezeyexAvvvvvvv++⋅⎟⎠⎞⎜⎝⎛∂∂+∂∂+∂∂=⋅∇ϕϕ()()()zAyAxAzyx∂∂+∂∂+∂∂=ϕϕϕ5zAyAxAzAyAxAzyxzyx∂∂+∂∂+∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂=ϕϕϕϕϕϕ∇⋅+⋅∇=AAvv1-10zyxezeyexrvvvv++=21222)(zyxr++=(1)03=⋅∇rrv(2)()()nnrnrr3+=⋅∇v133311rrrrrr∇⋅+⋅∇=⋅∇vvv⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂=zryrxrezeyexrzzyyxxrvvvv33311131⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛⋅⋅⋅−+⎟⎠⎞⎜⎝⎛⋅⋅⋅−⋅+=−−yrryxrrxr21213212133443⎟⎟⎠⎞⎟⎠⎞⎜⎝⎛⋅⋅⋅−+−zrrz2121340333352352223=⋅−=++−=rrrrzyxr2()nnnrrrrrr∇⋅+⋅∇=⋅∇vvv⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂+=znynxnnezreyrexrrrvvvv3⎟⎠⎞⎜⎝⎛⋅⋅⋅+⋅⋅⋅+⋅⋅⋅+=−−−rzrnzryrnyrxrnxrnnnn1113()nnnrnrzryrxnrr33222222+=⎟⎟⎠⎞⎜⎜⎝⎛+++=1-11()()∫∫⋅++−+=SzyxSezyxyezyxexzIvvvvd]2[23226S0=z21222)(yxaz−−=∫∫⋅=⋅∇VSSAVAvvvdd()()zyxezyxyezyxexzAvvvv23222++−+=()()[]SezyxyezyxexzISzyxvvvvd22322⋅++−+=∫∫()∫∫∫∫∫∫∫∫++=⋅∇=⋅=VSVVyxzVASAddd222vvv∫∫∫∫∫∫==VVrrrVrφθθdddsind22255022202052512ddsindaarrraππθθφππ=⋅⋅==∫∫∫1-121()AcAcvv×∇=×∇(c)(2)()Avϕ×∇AAvv×∇+×∇=ϕϕ(1)()AcAAAzyxeeeccAcAcAzyxeeeAczyxzyxzyxzyxvvvvvvvv×∇=∂∂∂∂∂∂=∂∂∂∂∂∂=×∇2()zyxzyxAAAzyxeeeAϕϕϕϕ∂∂∂∂∂∂=×∇vvvv()()()()()()zxyyzxxyzeyAxAexAzAezAyAvvv⎟⎟⎠⎞⎜⎜⎝⎛∂∂−∂∂+⎟⎠⎞⎜⎝⎛∂∂−∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂−∂∂=ϕϕϕϕϕϕxyzyzeAzAyzAyAv⎥⎦⎤⎢⎣⎡∂∂−∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂−∂∂=ϕϕϕyzzzxeAxAzxAzAv⎥⎦⎤⎢⎣⎡∂∂−∂∂+⎟⎠⎞⎜⎝⎛∂∂−∂∂+ϕϕϕ7zxyxyeAyAxyAxAv⎥⎦⎤⎢⎣⎡∂∂−∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂−∂∂+ϕϕϕzyxzyxzyxzyxAAAzyxeeeAAAzyxeee∂∂∂∂∂∂+∂∂∂∂∂∂=ϕϕϕϕvvvvvvAAvv×∇+×∇=ϕϕ1-13)(BAvv×⋅∇()()BAABvvvv×∇⋅−×∇⋅=()zyxzyxzyxBBBAAAeeeBAvvvvv⋅∇=×⋅∇()()()xyyxzxzxyzzyBABAzBAAByBABAx−∂∂+−∂∂+−∂∂=yBAyABxABxBAxABxBAxzzxzyyzyzzy∂∂+∂∂+∂∂−∂∂−∂∂+∂∂=zABzBAzABzBAyAByBAyxxyxyyxxzzx∂∂−∂∂−∂∂+∂∂+∂∂−∂∂−⎟⎟⎠⎞⎜⎜⎝⎛∂∂−∂∂+⎟⎠⎞⎜⎝⎛∂∂−∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂−∂∂=yAxABxAzABzAyABzyzzxyyzx⎟⎟⎠⎞⎜⎜⎝⎛∂∂−∂∂−⎟⎠⎞⎜⎝⎛∂∂−∂∂−⎟⎟⎠⎞⎜⎜⎝⎛∂∂−∂∂−yBxBAxBzBAzByBAxyzzxyyxx()()BAABvvvv×∇⋅−×∇⋅=1-14zyxezeyexrvvvv++=21222)(zyxr++=(1)0=×∇rv8(2)0=⎟⎠⎞⎜⎝⎛×∇rrv(3)0)(=⎟⎠⎞⎜⎝⎛×∇rfrrv()(rfr)10=⎟⎟⎠⎞⎜⎜⎝⎛∂∂−∂∂+⎟⎠⎞⎜⎝⎛∂∂−∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂−∂∂=∂∂∂∂∂∂=×∇zyxzyxeyxxyexzzxezyyzzyxzyxeeervvvvvvv2rrrrrrrrvvvv×⎟⎠⎞⎜⎝⎛∇=×⎟⎠⎞⎜⎝⎛∇+×∇=⎟⎠⎞⎜⎝⎛×∇111rezreyrexrzyxvvvv×⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛∂∂+∂∂+∂∂=111()rezeyexrzyxvvvv×++−=31()013=×−=rrrvv3()()()rrfrfrr0=⎥⎦⎤⎢⎣⎡×∇v()()()()rrrfrrrfrrrfrfrrvvvv×⎟⎠⎞⎜⎝⎛∇=×⎟⎠⎞⎜⎝⎛∇+×∇=⎥⎦⎤⎢⎣⎡×∇()()()rezrrfeyrrfexrrfzyxvvvv×⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛∂∂+∂∂+∂∂=()rezreyrexrrrrfzyxvvvv×⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂∂∂=9()()rezeyexrrrrfzyxvvvv×++∂∂=1()()01=×∂∂=rrrrrrfvv1-15),,,(tzyxEv),,,(tzyxHv0=⋅∇EvtHcE∂∂−=×∇vv10=⋅∇HvtEcH∂∂−=×∇vv1EvHv22221tAcA∂∂=∇vv(AvEvHv)()AAAvvv2∇−⋅∇∇=×∇×∇()AAAvvv×∇×∇−⋅∇∇=∇2()EEEEvvvv×∇×−∇=×∇×∇−⋅∇∇=∇2()222111111tEctEctcHtctHctHc∂∂⋅=⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂=×∇∂∂=∂∂×∇=⎟⎟⎠⎞⎜⎜⎝⎛∂∂−×−∇=vvvvv22221tHcH∂∂=∇vv1-16()uv2∇vuuvvu∇⋅∇+∇+∇=222()()()()uvzuvyuvxuv2222222∂∂+∂∂+∂∂=∇10⎟⎠⎞⎜⎝⎛∂∂+∂∂∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂∂∂+⎟⎠⎞⎜⎝⎛∂∂+∂∂∂∂=zuvzvuzyuvyvuyxuvxvuxyvyuyvuxuvxvxuxvu∂∂∂∂+∂∂+∂∂+∂∂∂∂+∂∂=222222222222222zuvzvzuzvuyuv∂∂+∂∂∂∂+∂∂+∂∂+vuuvvu∇⋅∇+∇+∇=2221-171zyxzyxγβαϕ−=esinsin),,((222βαγ+=)2φρφρϕnzncos),,(−=3θφθϕcos),,(rr=12222222yyx∂∂+∂∂+∂∂=∇ϕϕϕϕ()zzeyxeyx22sinsinsinsinγγβαββαα−−−+−=()zyex2sinsinγβαγ−−+()0sinsin222=+−−=−γβαβαγzyex222222211z∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂=∇ϕφϕρρϕρρρϕ()()0cos1cos12+−+−∂∂=φρρφρρρnnnnnn()φρρρφnnnnnnncoscos221−−−−−−⋅−=0coscos2222=−=−−−−φρφρnnnnnn11322222222sin1sinsin11φϕθθϕθθθϕϕ∂∂+⎟⎠⎞⎜⎝⎛∂∂∂∂+⎟⎠⎞⎜⎝⎛∂∂∂∂=∇rrrrrr()()0sinsin1cos12222+−∂∂+∂∂=θθθθrrrrr()θθθθcossin2sin12cos22⋅−+⋅=rrrr0cos2cos2=−+=rrθθ1-18Av⋅∇Av×∇(1)zyxeyxezxezxyAvvvv22332++=(2)zeezAvvvφρφρφρρsincos),,(22+=(3)φθθθθφθer