高等电磁理论-重要定理和原理

第4章重要定理和原理惟一性定边界面上nEnH结论：区域V JE,V

nE 证明：假设有两组解（E1和（E2H2）H JE1,2 m

， J

H1,2

E1,2E0E1E2H0H1H2

E0

H

，E00，H0 且nH0n

SnSn

SnSn

SS t0

t0

t00，

t0

t0

t0 2

H2)dV E2dV

(E

)dtV(2

S S(nE0)H0dS S(H0n)E0dS 2

H2)dV

E2dVdtV(2

(E2H2)dV（t=0） (E2H2)dV0 E2H2 E1E2,H1H2S(EH)dSj[(j)HHV

(j)EE]dV EEVjV[(j)HH(j)EE]dVVEE

dV(0 0

E2

2)dV

E00(0

E2

H2)dV

H0(附加条件：所有源位于有限区域内。

(E0H0)ndSn

Sn

Sn

Sn

Sn

Sn

S2k21）边界S上的给定或2）边界S上的n则区域中的是由源唯一确定2k2

u2vdV

uvdVSuSuv*Sk22

2dVV得 Im(k2)V得

S

影响边值问题转换为 镜像电流元Il

RIl

EE0 dRrIldRr E

sine0sine0

R Il在边界面z0上:0

EE0EIldRdIlrIldRdIlrREE，1R1R1kRkrkdIlrEdIlrEdxdIl

j0Ilsinejkrejkd

sinejkrejkd 故E j0Ilsincos(kdcos)edRrdRr镜像电流元IlIlIlE0

j0040

sinejkRIl

E

4R

sin

ejkR合成波电场：EE0 在边界面y=0上 eyEIlIlRdIlE0dIldRrdIlER0

π1R1R1kRk[x2(yd)2z2]1krkdsinsinkRk[x2(yd)2z2]1krkdsinsinIlRdIlRdIlE0d

j0Ilsinejkrejkdsinsin4πE

j0Il4πr

sinejkrejkdsinsin E0

02πr

sinsin(kdsinsin)e

EE0E

j0

sinsin(kdsinsin)e。E理导J。E理导J电 磁流元（电流元）

JmE

；JJmEa

a,b

V(Ja

JmHdV——源ab a,bb,EEa,Ha；Eb,SJ,JVaaSaJ,JSbbb(EH)(EH)(JEJE)(JmHJmH ) m)mmb[(J b[(J VS(EaHbEbHa)dab Ha)]dVb(J aab Ha)]dVb(J aab

HEHJmjH

EbHaEbJajEbEJm

HEHJmjH EaHbEaJbj HEEHj(HHEE)HJmE HE

HEbEa)EbJHJ

S(EaHbEbHa)dV E)(JmbaabJmH)]dVba源在体积V[(JaEbJE)

JmH)]dVSS(EaHbEbHa)dS

mm

JaE,HnSaaVE,JbbbJbS为无限大球面时，面SE0HS(EaHbEbHa)dS0SEaEb)dS 源在体积V

S(EaHbEbHa)dSV[(V[(J bba)(J mab Ha)]dVmab

S(EaHbEbHa)dS (Ea

EbHa)dSEa,HaJaJanSSEa,HaJaJanSSVVJbJbEb,H dV dV 若J J 0，则Jm dV

Jm SenEaenEb0S(EaHbEbHa)dS

EJE)(Jm JmH)]dV JS导Eb表面的电流源JS不产生辐射场。证明：设JS产生的场为EHIbl

,

SJSEbdSVJbEdVIbl在电壁上，EbJSEb0IblE0，由于IblE0。JEJE,H2J2E,nSJ1J11J2J2nE,HE,HSJm(ESV1Jn(ΗV2S n(ΗΗ (EE)mSS

J,J JE,JE,H2J2E,nSJ1J11J2J2nSJmS(EE21VJSn(Η22将区域VJJm用分界面SJ

E JSnH来替代，且将区域V1内的场设为零，则区域V2讨论：若区域V2内无源，则其中的场由分界面S JmE 和 nH nnE,HJmEnEHSSJnSH(r)1

nH(r)ejkRdS

E(r)

1

nE(r)ejkRd

E(r) H

H(r) E

SchelknoffS区域内V2的场由2Jm产生SE(r)1nE(r)ejkRdJmJmESJSnnE0HV2 H(r)

E(r)?SSH(r)1nH(r)eE(r)

JmJmESJSnnE0HV2? 的均匀平面波E=exE0e-jkz在z=0面S的等效源为JS－exE0/Z0JS

－eyE0，证明此等效源在0

EZ0H

H1kZ考虑

2Ek2E

zS考虑JS

(ˆHˆH)J

zs 2Hk2Hs

ˆE)J

z(ˆHˆH)

z

z

ˆ

ejkz zE

H

ejkz z

ejkz z

SJ S

ejkz z

ˆ

ejkz zE

H xˆE0ejkz z

0ejkz z

jkz

0ˆE0ejkz0

zE z

H z

zSS

=（2）磁壁，紧贴磁壁的电流源为JS－exE0Z0Z0为波阻

JJ

考虑

Sn(ˆH)

z

z EeEe

H

E0ey0E (ab) ln(bJmEnee

zln(b

ln(b bImSezπ db SzS

πU(b2a2

终端开路同轴

2ln(bEe

ImSk

kU(basinejkre kU(ba 4rln(bEEiES、HHiH

Ei,HES,H

nE, 散射SJ n(HSH)Jmn(ESE SiJSJ nHiJS

mnE

EiEi,HnE,散射SES,HJmnEiSJHinSE,nS散射ES,HESES,HJmESnSnE H电 nHSV2：E=H=表面：n(ESEiSS等效源：Jm nEi，JSS

nH

由式★：E=H=n(HSHi

nH

，mnE

由式EES,HnE HJmESnS磁 nHSV2JSJS

JSn

JmE

JS Hi

JmnEiSSSS

EiH为入射场（已知设定，可应用空间的

能应用空间的公式计算例：设理想导电小球的半径为a，且ai ezE0ejkxiEkEka由于a JmnEiee 0对于球面上宽度为dlad0d(I

mS)

madπa2

sin2

πa3Esin3S总等效磁流环：ImSS

d(ImS)

πsin3d00

πa3E

IljImSj4πa3

IlIlj4πa3 ESe2

Il

sine

2k2a3Esine HSeES散射总功率：PS

Re(EH)dS16πk4a6E2S J,J,JnEd,Hd,0,VSEdE0Ep,HdH0J,JJ,JeeS,000,,m ed0 med0 )nd01m)n0d1J,J

E0H0（无介质情况

H

J

jE0EdHd（有介质情况）令HpHdH0EpEdE00

E0JmjH0HdJjEd0EdJmjHJ,Jn,V0,HpHdJ,Jn,V0,0jEdj(EdEp0j()EdjEp e ed0 EpEdE0jHdjH0jHdj(HdHp)j()HdjHp me med0 n(BdBd)n[Hd(HdM)]n(HdHd)nM 0 0m

n

n(H

Hd

n(

B1

(11)n

()nH0 0

n(DdDd)n[Ed(

0n(E2E1)nPnJ,JnJ,J,,V00Sd nPn(EdEd)n( 2 1

(11)n

()nE00 0

Ap(r)Ap(r) J(r)G(r,r)dV

0 Jm(r)G( Jm(r)G(r,r)dV

(r,

pm(r)1 m(r)G(r,S 0

S 0Ep

jAp1nJnJ,J,,V00S1Hp

jApm 0Ed

E

jAp

10

HdH0pmjApm1 EieEejkxcosysin斜投射到位于yoz平

Jm

eEejk(xcosysin eeEejky xOzyjkE0dzdycosejkysinejkrxOzy jkE0dzdycosejkysinejkrysindEez

E0

cosejkrysinysinEejkE0cose

ejkysinsinejkE0cosejkr 2r

a/za/

a/ya/

ejkysinsinsinkasinsinEa2

2 e

ejkr

a SnSn2(r)k2(r)SS(rG(r,rS0

G0(r,

SSG(r,rSG(r,rS(r0

SSE(r)SE(r)G(r,rE(rS[E(rS0G(r,r0SH(r)S[H(rSG(r,r0G(r,rS(r0nnEk2EjJJHk2HjJmE(r)

[jJ(r)Jm(r)]G(r,r)dV S[jnHS(r)nES(r)nES(r)]G0(r,H(r)[

m(r)

(r J(r

]G0(r,

S[jnES(r)nHS(r)nHS(r)]G0(r,E(E(r)S[jnHS(r)nES(r)nES(r)]G0(r,H(r)S[jnES(r)nHS(r)nHS(r)]G0(r,E(r)jVJ(r)G0(r,r)dVSG0(r,r)[nE(r)]dSjSG0(r,r)[nH(r)VJ(r)G0(r,r)dVE(r)SG0(r,r)[nE(r)]dSjSG0(r,r)[nHE(r)SG0(r,r)[nE(r)]dSjSG0(r,r)[nHHH(r)jSG0(r,r)[nE(r)]dSSG0(r,r)[nH哪一种公式均要求表面S必须是封闭。如果仅对非封闭表 空间辐射电磁波。因此，在S内填充理想导电体和导磁Hj(rEj(r)H(H)(E)E(r)E(r)ejk0L(r设 H(r) (r)ejk0L(r E(r)[E(r)jkL(r)E(r)]ejk0L(r E(r)[E(r)jkL(r)E(r)]ejk0L(r LE0H0/k0(1jk0)L(E0)(1jk0)(L(H)(1jk)(H

LH0E0/k0LE0H0/k0L LH0cE00L0L

cH0 L L(LE)n2 L(LE)0

L2 n2 0(L2n2) 0L n 1Re[EH]1 H 1Re