习题2

132-1qR334Rqπρ=0ερ=⋅∇ERrv()21002231rCrEErrrErr+=⇒=∂∂=⋅∇ερερv0→rrE01=C30043RqrrErπεερ==Rr0=⋅∇Ev()222201rCEErrrErr=⇒=∂∂=⋅∇vRr=024πεqC=204rqErπε=⎪⎪⎩⎪⎪⎨⎧≤==203044rqRrRqrRrEErπεπε2-2ab(ba)c(bac−)ρ2-1bloov=′rrlvvv′−=a14ρ+bρ−rrEEE′−+=+=−+vvvvv0033ερερρρ()lellrrvvvv000333ερερερ==′−=levoo′2-3a()ρz,φ()zEErrErrEzr∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂=⋅∇ϕϕ11v0ερ=()01ερ=∂∂rrErrrCrEr102+=ερ0→rrE01=C02ερrEr=()01=∂∂rrErrrCEr2=ar=0222ερaC=raEr022ερ=aboo'lrr'2-115⎪⎪⎩⎪⎪⎨⎧≤==arraarrEEr02022ερερ2-4a0Sρ2-2()()()SdrrrrrrESS′′−′−′=∫3041vvvvvvvρπεzezrvv=yxererrvvvsincosφφ′+′=′rrsd′′=′ddφ()2122rzrr′+=′−vvyxzererezrrvvvvvsincosφφ′−′−=′−()()rrrzererezrEyxzS′′′+′−′−=∫∫ddsincos4232200φφφπερvvvvv()()0dcosd42002322200=′+′′−=∫∫πφφπερaSxrzrrE()0dsind42002322200=′′+′−=∫∫πφφπερrrzrEaSy()⎟⎟⎠⎞⎜⎜⎝⎛+−=′′+=∫∫220020023220012dd4azzrrzzESaSzερφπερπ0z()zSzzeazzeErEvvvv⎟⎟⎠⎞⎜⎜⎝⎛+−==220012ερ0z()zSzzeazzeErEvvvv⎟⎟⎠⎞⎜⎜⎝⎛+−−==220||12ερ2-5axyzzrRr'2-216⎪⎪⎩⎪⎪⎨⎧⎟⎠⎞⎜⎝⎛⎟⎠⎞⎜⎝⎛=arearEareraEErr,,020vvvar0ερ=⋅∇Ev()0112202222=⎟⎟⎠⎞⎜⎜⎝⎛⋅∂∂=⋅∂∂=⋅∇raErrrErrrErv00=ερ0=ρar0ερ=⋅∇Ev()aEarErrrErrrEr002222311=⎟⎠⎞⎜⎝⎛⋅∂∂=⋅∂∂=⋅∇vaE003=ερaE003ερ=2-62-12-1⎪⎪⎩⎪⎪⎨⎧≤==203044rqRrRqrRrEErπεπεRr≤()∫∫∫∞∞+=⋅=RRrrrrqrRqrlErd4d4d2030πεπεϕvv17()22300223038424rRRqRqrRRq−=+⎟⎟⎠⎞⎜⎜⎝⎛−=πεπεπεRr()rqrrqlErrr0204d4dπεπεϕ==⋅=∫∫∞∞vv2-72-3lr40223rqlEπε=q2−()()2020204424lreqreqlreqExxx−⋅+⋅−++⋅=vvvvπεπεπε()()⎥⎦⎤⎢⎣⎡−+−+⋅=22201214lrrlrqexπεv⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛−+−⎟⎠⎞⎜⎝⎛+⋅=2220112114rlrlrqexπεv⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛+−+−⋅Lv33222043214rlrlrlrqexπε⎥⎦⎤⎟⎟⎠⎞⎜⎜⎝⎛+++++−L332243212rlrlrlxxxerqlerqlrlrlrqevvLv40240243222023465342πεπεπε=≈⎟⎟⎠⎞⎜⎜⎝⎛++⋅⋅=xllrq2qq2-3182-8q−2q)0,0,(a),,(zyx()22200441zyxqrrqrq++−=′−−⋅=−πεπεϕvvv()22200242241zyxqrrqrq++=′−⋅=πεπεϕvvv()()⎟⎟⎠⎞⎜⎜⎝⎛++−+++−=22222202/114zyxazyxqrπεϕv()0=rvϕ()222222][4zyxzyxa++=++−()2222222232340483⎟⎠⎞⎜⎝⎛=⎟⎠⎞⎜⎝⎛−++⇒=+−++aaxzyaaxzyx04cossin8322=+−aarrφθ⎟⎠⎞⎜⎝⎛0034a32a2-9LazzePPvv0=2-40=⋅−∇=Pvρrenvv=z0=⋅=rSPePvvρP2-419zenvv=0PePzSP=⋅=vvρzenvv−=0)(PePzSP−=−⋅=vvρ2-100x0x03ε)/(5431mVeeeEzyxvvvv++=2Ev()⎪⎩⎪⎨⎧=−×====⇒=⇒=013131312211212201021EEnyozEEEEEEEEDDttxxnnnnnnvvvεε21,EEvvzyzzyyzzyyeeeEeEeEeEvvvvvv542122+=+=+zyxeeeEvvvv542++=∴2-11aab−εq⎪⎪⎪⎩⎪⎪⎪⎨⎧=brrqbrarqarE440202πεπεar20drrqdrrqldldEbbaarr∫∫∫∫∞∞+++=⋅=202440πεπεϕvvv⎟⎟⎠⎞⎜⎜⎝⎛+−=01114εεεπbbaqbra⎟⎠⎞⎜⎝⎛+−=+=⋅=∫∫∫∞∞0202111444εεεππεπεϕbbrqdrrqdrrqldEbbrrvvbrrqdrrqldErr02044πεπεϕ==⋅=∫∫∞∞vv⎪⎪⎪⎪⎩⎪⎪⎪⎪⎨⎧⎟⎟⎠⎞⎜⎜⎝⎛+−⎟⎟⎠⎞⎜⎜⎝⎛+−=brrqbrabbrqarbbaq000411141114πεεεεπεεεπϕr1ϕ2ϕbr0dddd112212=⎟⎠⎞⎜⎝⎛=∇rrrrϕϕbra0dddd122222=⎟⎠⎞⎜⎝⎛=∇rrrrϕϕ211CrC+=ϕ432CrC+=ϕ02=C21(ar=)24aqSπρ=arnSrD=∂∂−==2ϕερεπ43qC=br=21ϕϕ=rr∂∂=∂∂210ϕεϕε431CbCbC+=23210bCbCεε=επ43qC=014επqC=bqbqC4404επεπ−=br40rqπεbra⎟⎟⎠⎞⎜⎜⎝⎛+−01114εεεπbbrq2-12ρεεερ0p−−=()⎪⎩⎪⎨⎧=⋅∇+⋅∇⇒=+⋅∇=⋅∇⇒=⋅∇ρερεερρPEPEEDvvvvvv00ρρερε=−⋅⇒p0ρεεερεεερ00p−−=−=⇒2-13ab22abc=rEπελ2=22121EDEweε=⋅=vvx()bxa()()axLrLrrVEVwxWxaVeln4d2421d21d222222πελπεπλεε=⋅===∫∫∫acLabLln42ln422πελπελ⋅=abc=2-14a0U()q000044aUqUaqπεπε=⇒=q2bab32=03300004244242UaaUbqU=⋅==πεπεπε0587.1U≈2-15adadadaPP0221141πε==dPP0211241πε==23adaddaaC−=−+=000024241411πεπεπεπε2-162-51234411411241148ppppqq−−=1411241411128pppppqq−+−=44332211pppp===14342312pppp===1324pp=1211q22q2121111qpqp+=ϕ2111122221212qpqpqpqp+=+=ϕ21ϕϕ=21qqq+=21qq=22qq=1311q′33q′3242141113132121111qpqpqpqpqpqp++′=++′=′ϕ3332141243332321313qpqpqpqpqpqp++′=++′=ϕ31ϕϕ=′312qqq+=22qq=41qq=′43qq=1411q′′44q13242-5244143242141114143132121111qpqpqpqpqpqpqpqp+++′′=+++′′=′′ϕ4113142241144443432421414qpqpqpqpqpqpqpqp+++′′=+++′′=ϕ41ϕϕ=′′414qqq+′′=22qq=43qq=1411241148ppppqq−−=1411241411128pppppqq−+−=′′2-17dεU20)(21⎟⎠⎞⎜⎝⎛−=dUghεερρgHhl2-6dlhHdhlC)(0−+=εε])([2121022elhHhlUdCUW−+==εεhlUdhWF)(2102eεε−=∂∂=hgdhlFρ=20)(21⎟⎠⎞⎜⎝⎛−=dUghεερ2-6dhH