《电动力学》讲义第03章静磁场

121.1............................21.2.............................21.3.............................21.4...........................31.5.............................31.6.............................41.7..............................41.8............................41.9..........................51.10...................................6262.1..............................62.2...........................72.3............................72.4..........................72.5.....................82.6...................................82.7...........................92.8...................................9393.1............................93.2..........................103.3...........................103.4...........................113.5.....................113.6.......................123.7.....................123.8..........................123.9...................................133{(Aharonov{Bohm)135145.1..........................145.2|.................145.3|.................145.4..........................151x1.1Ir£H=Jr¢B=0B=¹0(H+M)I)r¢B=0)B=r£AAx1.2AAILA¢dl=ZZSr£AdS=ZZSBdSBABEx1.3AzABx=By=0;Bz=B0@Ay@x¡@Ax@y=B0;@Az@y¡@Ay@z=@Ax@z¡@Az@x=0Ay=Az=0;Ax=¡B0yAx=Az=0;Ay=B0xIA+rÃABr£(A+rÃ)=r£AIAAIr¢A=02x1.4AB=r£AAB=r£Ar¢A=0r¢A=u6=0A0=A+rÃr¢A0=r¢A+r2Ã=u+r2ÃÃr2Ã=¡uA0A0r¢A0=0Ax1.5IB=¹HB=r£Ar£H=Jr£(r£A)=¹Jr£(r£A)=r(r¢A)¡r2A(r¢A=0)r2A=¡¹Jr2Ai=¡¹Ji;(i=1;2;3)Ir2'=¡½0A(x)=¹4¼ZZZJ(x0)rdV0Ir¢A=0IABB=r£A=r£·¹4¼ZZZJ(x0)rdV0¸=¹4¼ZZZµr1r¶£J(x0)dV0=¹4¼ZZZJ(x0)£rr3dV0I|3x1.6n¢(B2¡B1)=0n£(H2¡H1)=®fIH=B¹n¢(r£A2¡r£A1)=0(1)n£(1¹2r£A2¡1¹1r£A1)=®fIHA¢dl=RRB¢dS!0(1)A2t=A1tAAnx1.7I!=12(E¢D+H¢B)IW=12ZZZH¢BdVIW=12ZZZH¢BdV=12ZZZ(r£A)¢HdV(2)=12ZZZ[r¢(A£H)+A¢(r£H)]dV=12I(A£H)¢dS+12ZZZA¢JdV=12ZZZA¢JdVx1.8I(2)I12A¢JI12A¢JI(2)AJ4x1.9JAeJeJ+JeA+AeW=12ZZZ(A+Ae)¢(J+Je)dVIJeJW1=12ZZZAe¢JedV;W2=12ZZZA¢JdVIWi=W¡W1¡W2=12ZZZ(A¢Je+Ae¢J)dVP104Ae=¹4¼ZZZJe(x0)rdV0;A=¹4¼ZZZJ(x0)rdV012ZZZ(A¢Je)dV=12ZZZ(Ae¢J)dVJWi=ZZZAe¢JdVIA(x)=¹4¼RRRJ(x0)rdV0I²IE/1r2$'/1rIIE/1r$'/lnr²B/1r$A/lnrIA=¡³¹I2¼lnrR0´ezr!1A!1Ir£A=Br£B=¹0J()()Ir£(eµr)=0(r6=0);r£(reµ)=2ez5aIA(x)=¹04¼H1rIdlI2rasinµ¿r2+a2AA(r;µ)=¹0Ia4·rasinµ(r2+a2)3=2+158r3a3sin3µ(r2+a2)7=2¸eÁ(rÀa)A(r;µ)=¹04¼I¼a2ez£rr3=¹04¼m£rr3(3)I(3)IP82x1.10IAB=r£AIAAIr¢A=0AIIW=12RRRA¢JdVIWi=RRRAe¢JdVPage131:1,2,3,4,7x2.1IAIr£H=0;r¢B=0IIILE¢dl=0orZC1E¢dl=ZC2E¢dl6IILH¢dl=ZZSJ¢dS6=0Ix2.2Ix2LH(x)=0HLH¢dl=0Ix2Lr£H(x)=0HLH¢dl=0Ix2Sr£H(x)=0HLH¢dl=RRSr£H¢dS=0IHr=0Ir£HJHIr¢E½Er¢E?)HE¢dS=0Ir£A=B=0(r6=0)HA¢dl=RRB¢dS?=0BA{Bx2.3ILH¢dl=0VL1r£H=02VVSVx2.4B=¹HBHr£H=0r¢B=0(4)B=¹0(H+M)=f(H)(5)(5)(4)r¢H=¡r¢Mr¢P=¡½p½m=¡¹0r¢M7r¢H=½m¹0r£H=0r¢E=½f+½p0r£E=0'mH=¡r'mEHEBEHx2.5r¢E=½f+½p0r¢H=½m¹0r£E=0r£H=0½p=¡r¢P½m=¡¹0r¢MD=0E+PB=¹0(H+M)E=¡r'H=¡r'mr2'=¡½f+½p0r2'm=¡½m¹0x2.6¹!1H12n¢(B2¡B1)=0;n£(H2¡H1)=0B1=¹1H1B2=¹2H2¹2H2n=¹1H1n;H2t=H1tHnµtanµ1tanµ2=H1tH1nH2tH2n=¹1¹2¹1!1µ2!0H28x2.7I!1=DEE!0I¹!1¹=BHH!0I¹!0B!0I¾!1¾=JEE!0I¾!0¾=JEJ!0I!0x2.8III²r£H=0²Ir¢H=½m¹0;r£H=0H=¡r'm;½m=¡¹0r¢MPage133:8,9,11x3.1J(x0)A(x)A(x)=¹04¼ZZZJ(x0)rdV0(6)(rÀx0)(6)A(x)=¹04¼ZZZJ(x0)241r0¡x0¢r1r0+12!Xi;jx0ix0j@2@xi@xj1r0+¢¢¢35dV0(7)A(0)(x)=¹04¼r0ZZZJ(x0)dV0=09A(1)(x)=¡¹04¼ZZZJ(x0)x0¢r1r0dV0=¹04¼m£r0r30A(2)(x)=¹08¼ZZZJ(x0)x0x0:rr1r0dV0x3.2J(x0)r0¢(Jx0)=(r0¢J)x0+(J¢r0)x0=JA(0)(x)=¹04¼r0ZZZJ(x0)dV0=¹04¼r0ZZZr0¢(Jx0)dV0=¹04¼r0IdS0¢Jx0=0x3.3fg¢r0r30m£r0r30fg¢r0=f(g¢r0)=(g£f)£r0+g(f¢r0)A(1)(x)=¡¹04¼ZZZJ(x0)x0¢r1r0dV0=¹04¼ZZZJ(x0)(x0¢r0r30)dV0=¹04¼r30Xir0iZZZJx0idV0=¹04¼r30Xir0iZZZ·12(Jx0i+Jix0)+12(Jx0i¡Jix0)¸dV0r0¢(Jx0x0i)=x0ir0¢(Jx0)+r0x0i¢Jx0=x0iJ+(e0i¢J)x0=Jx0i+Jix0A(1)(x)=¹04¼r30Xir0iZZZ·12(Jx0i¡Jix0)¸dV0=¹08¼r30ZZZ[J(x0¢r0)¡x0(J¢r0)]dV0A(1)(x)=¹08¼r30ZZZ(x0£J)£r0dV0(8)=¹04¼ZZZ12(x0£J)dV0£r0r30=¹04¼m£r0r3010m=12ZZZx0£J(x0)dV0m(8)r20m=12Ix0£Idl=ISH12x0£dl=H±S=Sx3.4Sm=IZZSdSSZZZrÃdV=IdSÃÃ=1ZZS1dS¡ZZS2dS=IdS=0mx3.5B(1)=r£A(1)=¹04¼r£µm£r0r30¶=¡¹04¼r£µm£r1r0¶=¡¹04¼·mr21r0¡(m¢r)r1r0¸=¡¹04¼(m¢r)r0r30rµm¢r0r30¶=m£µr£r0r30¶+(m¢r)r0r30=¡m£µr£r1r0¶+(m¢r)r0r30=(m¢r)r0r30B(1)=¡¹0r'(1)m'm¡HA¡B'(1)m=m¢r04¼r3011x3.6J(x)Ae(x)W=ZZZJ¢AedV(9)I©eLW=IILAe¢dl=IZZSBe¢dS=I©eBeW=IZZSdS¢[Be(0)+x¢rBe(0)+¢¢¢]¼Be(0)¢IZZSdS=m¢Be(0)x3.7U=¡m¢Be(10)IIIIeI(10)¡p¢EBrmF=¡rU=r(m¢Be)=m£(r£Be)+(m¢r)Be=(m¢r)BeLµ=¡@U@µ=@@µ(mBecosµ)=¡mBesinµL=m£Bex3.8IIeIIeW=12µIIAe¢dl+IeIA¢dl¶=12(I©e+Ie©)IIe±W=12(I±©e+Ie±©)12LLeE=¡d©edt;Ee=¡d©dt±tEI±t+EeIe±t=¡I±©e¡Ie±©±Ws=I±©e+Ie±©=2±W±A=±Ws¡±W=±WU=¡W=¡ZZZJ¢AedV¼¡m¢Bex3.9IIIm=12RRRx0£J(x0)dV0IA(1)(x)=¹04¼m£r0r30'(1)m=m¢r04¼r30B(1)¡¹04¼(m¢r)r0r30IW=m¢Be(0)U=¡m¢BeIIF=(m¢r)BeL=m£BeIPage134:13,14,15{13x5.11.TcTcHc(T)=Hc(0)1¡µTTc¶2#2.(Meissner)B=0(Meissner)x5.2|snn=ns+nn;J=Js+JnJn=¾Em_v=¡eE;Js=¡nsev@Js@t=®E;µ®=nee2m¶@Js@t=0)E=0)Jn=0|x5.3|@Js@t=®E)@@tr£Js=®r£E=¡®@B@tr£Js=¡®B+f(x)f(x)=014r£Js=¡®BB=0r£B=¹JB=0J=0|r£B=¹J=¹Js)r£(r£B)=¹r£Jsr(r¢B)¡r2B=¡®¹Br2B=1¸2LB;¸L=1p¹®r£Js=¡®B)r£(r£Js)=¡®r£Br(r¢Js)¡r2Js=¡®¹Jsr2Js=1¸2LJs;¸L=1p¹®¸Lx5.4r£M=JsB=¹0(H+M)B=0M=¡H{M=MH=¡1¹=¹0(1+{M)=015