清华大学_电动力学讲义

20103:hi,,,.²:(,,.)²:()1.2.3.4.5.1..(P340):(a)²ijk:3Xk=1²ijk²lmk=±il±jm¡±im±jl(b)±ij²ijk,~a£(~b£~c)=(~a¢~c)~b¡(~a¢~b)~c(~a£~b)£(~c£~d)=[~a¢(~b£~d)]~c¡[~a¢(~b£~c)]~d=[~a¢(~c£~d)]~b¡[~b¢(~c£~d)]~a(c)i.ii.¸±ijiii.¸²ijkiv.¸±ij±kl+®±ik±jl+¯±il±jk12..(P341):1.1,1.3,1.6:r¢~A0,r¢~A=0,r¢~A0?r£~A0,r£~A=0,r£~A0?3..(P345),4.±.(P76):2.14:2.155.Helmholtz1..(1,2):S,HS~j¢d~S0,HS~j¢d~S=0,HS~j¢d~S0?2..(1,2,3):1.5,1.103..(3):(a)(l!0,q!1,lq),Á=14¼²0~r¢~pr3+(b)(S!0,J!1,JS),~A=¹04¼~m£~rr3+(c)±,~f21=k1q1q2R3~Re¡±RR0i.,~E=¡rÁ,,Á.ii.:H~E¢d~S=Q=²0?(d)±,d2~F21=k2J2d~l2£(J1d~l1£~RR3)e¡±RR0i.,~B=r£~A,,~A.ii.:H~B¢d~l=¹0J?:2i.?ii.,,?4..(4):1.7,1.8,1.9,1.11:,,?5..(5):1.12,1.13²!1.6..(6,7):1.14:,.():1.,~S~E~B?2.,(,)?1..(1):(a)2.1(b):,,1²r.(c):,1²r.2..(2):(a),iqi,,,~rq:Á(~r;q1;q2;:::)=Xipi(~r)qipi(~r).(b)Q,,?33.,.(3)11223:2.2,2.3,2.4,2.5,2.6,2.184..(4)12345:2.8,2.9,2.10,2.11,2.125..(5):2.19::Pn(cos£)=nXm=¡n(n¡m)!(n+m)!Pmn(cosµ)Pmn(cosµ0)eim(Á¡Á0)cos£=cosµcosµ0+sinµsinµ0cos(Á¡Á0)Zd­0f(µ0)j~r¡~r0j=Zd­0f(µ0)1Xn=0rnrn+1Pn(cos£)=Z2¼0dÁ0Z¼0dµ0sinµ0f(µ0)1Xn=0nXm=¡nrnrn+1(n¡m)!(n+m)!Pmn(cosµ)Pmn(cosµ0)eim(Á¡Á0)=2¼1Xn=0rnrn+1Pn(cosµ)Z¼0dµ0sinµ0f(µ0)Pn(cosµ0)6.7..(,)12:(a)2.72.13(b)a°,.V,,.(c)a°,b(ba),d(da),(a=b;a=d).(d)p,,p..8..(1):3.1,3.2,3.7:3.89..(2):3.4,3.9,3.10,3.11,3.12,3.13410..(6,3),:(a)3.14,3.15(b)a;bc,Q,,.:x=arsinµcosÁ,y=brsinµsinÁ,z=crcosµ.:~p=0;Dij=0(i6=j);D11=Q5(2a2¡b2¡c2)D22=Q5(2b2¡c2¡a2)D33=Q5(2c2¡a2¡b2)Á=14¼²0·QR+Qa2(3x2¡R2)+b2(3y2¡R2)+c2(3z2¡R2)10R5¸(c),,z,,i.µ=¼2,R;ii.µ=0,R..:i.W=p24¼²0R3~F=3p2~R4¼²0R5()ii.W=¡p24¼²0R3~F=¡3p2~R2¼²0R5()(d)~p,a,.:~F=¡3p264¼²0a4(1+cos2®),®.(e)W=1Xn=0WnWn=1n3Xi1;¢¢¢in~Ji1¢¢¢in¢[@n@xi1¢¢¢@xin~A(~r)],W0=0W1=~m¢~B(f)~F=1Xn=0~Fn~Fn=1n3Xi1;¢¢¢in~Ji1¢¢¢in£[@n@xi1¢¢¢@xin~B(~r)],~F0=0~F1=(r~B)¢~m(g)~L=1Xn=0~Ln:~Ji1¢¢¢in=3Xi~eiJii1¢¢¢in~Ln=1n3Xi;i1;¢¢¢inf~Jii1¢¢¢in[@n@xi1¢¢¢@xinBi(~r)]¡Jiii1¢¢¢in[@n@xi1¢¢¢@xin~B(~r)],~L0=~m£~B51..(1):(a)4.4(b),½=0,~j=0,.2..(1,3):(a)4.1,4.5,4.7(b),(°6=0),.(c),°±!q2!¹°.(d),?:,kI?,??3..(3)»»»»»»:(a)4.2,4.3,4.6(b),,:~E20?=2E10?~ey1+¹1k2z¹2k1cosµ1~E20k=1¡¹1k2z¹2k1cosµ11+¹1k2z¹2k1cosµ1E10?~ey¹1=¹2,°2=0,~E20?=2cosµ1sinµ2sin(µ1+µ2)E10?~ey~E20k=sin(µ2¡µ1)sin(µ2+µ1)E10?~ey(c)Dk=4¹1cosµ1Re(k2k¤2zk¤2)¹2k1jk2zk2+k2¹1k1¹2cosµ1j2Rk=¯¯¯¯k2zk2¡k2¹1k1¹2cosµ1k2zk2+k2¹1k1¹2cosµ1¯¯¯¯Dk+Rk=14..(4):4.9,4.14,4.155..(5)TEMTE:6(a)4.11,4.12,4.13(b),TM(Bz=0),(r2xy+¹²!2¡k2z)Ez=0Ez¯¯¯¯=0~Et=ikz¹²!2¡k2zrxyEz~Bt=¡i¹²!¹²!2¡k2zrxyEz().6..1..(1):5.1,5.2,5.3,5.42..(2)():(a)5.5(b),r¢~A=03.(3,4):(a)5.6,5.7,5.8,5.10,5.11,5.12,5.13(b),p0!,©,.:,xy,dId­=¹0p20!432¼2c(1+cos2µ¡sin2µsin2Ácos©)(c)¸4z(¸),©,p0,!,.:,dId­=¹0p20!416¼2c·1+cos(¼2cosµ+©)¸sin2µ(d)l=¸2,,J=J0cos2¼z¸cos!t¡¸4·z·¸4,.:dId­=c¹0J208¼2cos2(¼2cosµ)sin2µ7(e)~a.(f),,,,,.4..(6)1..(1)»»»2.,.(2,3):(a),Ui,U4xi,ict.(b)q1¡v2c2.(c)'',l=8:64£108,''45c,A'B'.A,(12),:i.B'B?(::tA=tB=tA0=12,tB0=1238:4.:tA=tA0=tB0=12,tB=1238:4)ii.A'B',?(::tA=tB=13,tA0=1236,tB0=1314:4.:tA0=tB0=1236,tA=1221:6,tB=13)3..(3)¢s20¢s20¢s2=0:(a)6.2,6.3,6.4,6.5,6.6,6.7(b):,i..ii.,,.iii.,,.(c),,?.4..(4):(a)6.11(b),nn¡2.85..(5,6,7):(a)6.17,6.18,6.19,6.25,6.26,6.27(b),dp¹d¿=¡K¹K¹=XºeF¹ºUº(c)K4=ic~K¢v,~KK¹.,.6..(6,7)--:(a)6.12,6.13,6.15,6.16(b),,~E0=~E¢~vv2~v+~E¡~E¢~vv2~v+~v£~Bq1¡v2c2~B0=~B¢~vv2~v+~B¡~B¢~vv2~v¡~vc2£~Eq1¡v2c2(c),q,~E=q~R4¼²0S3(1¡v2c2)~B=~vc2£~E:~R=~r¡~vtS2=(~R¢~vv)2+(1¡v2c2)(~R¡~R¢~vv2~v)¢(~R¡~R¢~vv2~v)=(~R¢~v)2c2+(1¡v2c2)R2:6.14,6.23,6.241..(12)-:(a)7.7(b),1J=Rd3x0±µ~r0¡~r0(t¡j~r¡~r0jc)¶:J=1¡~R¤R¤¢~v¤c.J.,ccn(n),J0,?(c),~E(~r;t)=q4¼²0S¤3(~R¤¡R¤~v¤c)(1¡v¤2c2)+q4¼²0c2S¤3~R¤£[(~R¤¡R¤~v¤c)£~a¤]~B(~r;t)=~R¤cR¤£~E(~r;t)(d)7.1,7.2,7.3,7.42..(3)93..(4)4..(5):7.9:6?5.,.(6):7.5,7.6,7.8:1.2.3.4.5.6.10x0.1²~e1,~e2,~e3(~ex,~ey,~ez;~i;~j;~k)~A~A=A1~e1+A2~e2+A3~e3=3Xi=1Ai~eij~Aj=qA21+A22+A33=vuut3Xi=1AiAij~e1j=j~e2j=j~e3j=1j~eij=1²{~A§~B=(A1~e1+A2~e2+A3~e3)§(B1~e1+B2~e2+B3~e3)=(A1§B1)~e1+(A2§B2)~e2+(A3§B3)~e3=3Xi=1(Ai§Bi)~ei{()~e1¢~e1=~e2¢~e2=~e3¢~e3=1~e1¢~e2=~e2¢~e1=~e1¢~e3=~e3¢~e1=~e2¢~e3=~e3¢~e2=0~ei¢~ej=±ij´(1i=j0i6=j~A¢~B=(A1~e1+A2~e2+A3~e3)¢(B1~e1+B2~e2+B3~e3)=A1B1~e1¢~e1+A1B2~e1¢~e2+A1B3~e1¢~e3+A2B1~e2¢~e1+A2B2~e2¢~e2+A2B3~e2¢~e3+A3B1~e3¢~e1+A3B2~e3¢~e2+A3B3~e3¢~e3=A1B1+A2B2+A3B3=3Xi=1AiBi=~B¢~A±ij±ij=±ji3Xj=1Aj±ij=3Xj=1Aj±ji=Ai3Xi=1±ii=310±ij~A¢~B~A¢~B=3Xi=1Ai~ei¢3Xj=1Bj~ej=3Xi;j=1AiBj~ei¢~ej=3Xi;j=1AiBj±ij=3Xi=1AiBi138:():~A¢~A=3Xi=1AiAi=j~Aj2~Aj~Aj1;~A¢~Bj~Ajj~Bj´cosµ~A¢~B=0()~A?~Bcosµi´Aij~AjAi{()~e1£~e1=~e2£~e2=~e3£~e3=0~e1£~e2=¡~e2£~e1=~e3~e2£~e3=¡~e3£~e2=~e1~e3£~e1=¡~e1£~e1=~e2~ei£~ej=3Xk=1²ijk~ek²ijk´8:11;2;3²123;²231;²312¡11;2;3²132;²213;²3210²111;²223;²313~A£~B=(A1~e1+A2~e2+A3~e3)£(B1~e1+B2~e2+B3~e3)=A1B1~e1£~e1+A1B2~e1£~e2+A1B3~e1£~e3+A2B1~e2£~e1+A2B2~e2£~e2+A2B3~e2£~e3+A3B1~e3£~e1+A3B2~e3£~e2+A3B3~e3£~e3=(A1B2¡A2B1)~e3+(A3B1¡A1B3)~e2+(A2B3¡A3B1)~e1=¡~B£~A=3Xi;j;k=1²ijkAiBj~ek²ijk²ijk²ijk=¡²jik=¡²kji=¡²ikj3Xk=1²ijk²lmk=±il±jm¡±im±jl3Xj;k=1²ijk²ljk=±il±jj¡±ij±jl=3±il¡±il=2±il3Xi;j;k=1²ijk²ijk=±ii±jj¡±ij±ji=9¡3=611²ijk~A£~B~A£~B=3Xi=1Ai~ei£3Xj=1Bj~ej=3Xi;j=1AiBj~ei£~ej=3Xi;j;k=1AiBj²ijk~A£~A=0.(~A£~B)¢~C=(3Xi;j;k=1²ijkAiBj~ek)¢3Xn=1Cn~en=3Xi;j;k;n=1²ijkAiBjCn~ek¢