高等電磁理論-重要定理和原理

第4章重要定理和原理惟一性定邊界面上nEnH結(jié)論：區(qū)域V JE,V

nE 證明：假設(shè)有兩組解（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(附加條件：所有源位于有限區(qū)域內(nèi)。

(E0H0)ndSn

Sn

Sn

Sn

Sn

Sn

S2k21）邊界S上的給定或2）邊界S上的n則區(qū)域中的是由源唯一確定2k2

u2vdV

uvdVSuSuv*Sk22

2dVV得 Im(k2)V得

S

影響邊值問題轉(zhuǎn)換為 鏡像電流元Il

RIl

EE0 dRrIldRr E

sine0sine0

R Il在邊界面z0上:0

EE0EIldRdIlrIldRdIlrREE，1R1R1kRkrkdIlrEdIlrEdxdIl

j0Ilsinejkrejkd

sinejkrejkd 故E j0Ilsincos(kdcos)edRrdRr鏡像電流元IlIlIlE0

j0040

sinejkRIl

E

4R

sin

ejkR合成波電場：EE0 在邊界面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理導(dǎo)J。E理導(dǎo)J電 磁流元（電流元）

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源在體積V[(JaEbJE)

JmH)]dVSS(EaHbEbHa)dS

mm

JaE,HnSaaVE,JbbbJbS為無限大球面時，面SE0HS(EaHbEbHa)dS0SEaEb)dS 源在體積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，則Jm dV

Jm SenEaenEb0S(EaHbEbHa)dS

EJE)(Jm JmH)]dV JS導(dǎo)Eb表面的電流源JS不產(chǎn)生輻射場。證明：設(shè)JS產(chǎn)生的場為EHIbl

,

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

J,J JE,JE,H2J2E,nSJ1J11J2J2nSJmS(EE21VJSn(Η22將區(qū)域VJJm用分界面SJ

E JSnH來替代，且將區(qū)域V1內(nèi)的場設(shè)為零，則區(qū)域V2討論：若區(qū)域V2內(nèi)無源，則其中的場由分界面S JmE 和 nH nnE,HJmEnEHSSJnSH(r)1

nH(r)ejkRdS

E(r)

1

nE(r)ejkRd

E(r) H

H(r) E

SchelknoffS區(qū)域內(nèi)V2的場由2Jm產(chǎn)生SE(r)1nE(r)ejkRdJmJmESJSnnE0HV2 H(r)

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

JmJmESJSnnE0HV2? 的均勻平面波E=exE0e-jkz在z=0面S的等效源為JS－exE0/Z0JS

－eyE0，證明此等效源在0

EZ0H

H1kZ考慮

2Ek2E

zS考慮JS

(?H?H)J

zs 2Hk2Hs

?E)J

z(?H?H)

z

z

?

ejkz zE

H

ejkz z

ejkz z

SJ S

ejkz z

?

ejkz zE

H x?E0ejkz z

0ejkz z

jkz

0?E0ejkz0

zE z

H z

zSS

=（2）磁壁，緊貼磁壁的電流源為JS－exE0Z0Z0為波阻

JJ

考慮

Sn(?H)

z

z EeEe

H

E0ey0E (ab) ln(bJmEnee

zln(b

ln(b bImSezπ db SzS

πU(b2a2

終端開路同軸

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電 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為入射場（已知設(shè)定，可應(yīng)用空間的

能應(yīng)用空間的公式計算例：設(shè)理想導(dǎo)電小球的半徑為a，且ai ezE0ejkxiEkEka由于a JmnEiee 0對于球面上寬度為dlad0d(I

mS)

madπa2

sin2

πa3Esin3S總等效磁流環(huán)：ImSS

d(ImS)

πsin3d00

πa3E

IljImSj4πa3

IlIlj4πa3 ESe2

Il

sine

2k2a3Esine HSeES散射總功率：PS

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

E0H0（無介質(zhì)情況

H

J

jE0EdHd（有介質(zhì)情況）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哪一種公式均要求表面S必須是封閉。如果僅對非封閉表 空間輻射電磁波。因此，在S內(nèi)填充理想導(dǎo)電體和導(dǎo)磁Hj(rEj(r)H(H)(E)E(r)E(r)ejk0L(r設(shè) 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