Chapter 6 Integral solutions of Maxwells equations 6. 1 Vector Kirchoff solution: method of stratton and chu One of the most powerful tools for the analysis of electromagnetics problems is the integral solution to Maxwells equations formulated by Stratton and Chu [187, 188 These authors used the vector Green's theorem to solve for e and h in much the same way as is done in static fields with the scalar Green's theorem. An alternative approach is to use the Lorentz reciprocity theorem of$ 4. 10. 2, as done by Fradin[ 74. The reciprocity approach allows the identification of terms arising from surface discontinuities, which must be added to the result obtained from the other approach [187 6.1.1 The Stratton-Chu formula Consider an isotropic, homogeneous medium occupying a bounded region V in space The medium is described by permeability i(o), permittivity E(@), and conductivity o() The region V is bounded by a surface S, which can be multiply-connected so that S is the union of several surfaces S1,..., SN as shown in Figure 6.1: these are used to exclude unknown sources and to formulate the vector Huygens principle. Impressed electric and magnetic sources may thus reside both inside and outside V We wish to solve for the electric and magnetic fields at a point r within V. To do this we employ the Lorentz reciprocity theorem(4. 173), written here using the frequency-domain fields as an integral over primed coordinates ea(r,o)x Hb(r,a)-Eb(r,o)x Ha(r, o) [Eb(r,o)·Ja(r,a)-Ea(r,a)·J6(r,o)- (6.1) Hr,o)·Jma(r,o)+H2(r,o)·Jmb(r,o)]dv (6.2) Note that the negative sign on the left arises from the definition of f as the inward normal to V as shown in Figure 6. 1. We place an electric Hertzian dipole at the point r= rp where we wish to compute the field, and set E,= En and H,= H, in the reciprocity theorem,where Ep and H, are the fields produced by the dipole(5. 88)-(5.89) IpG(rrp: o), Ep(r, o)==Vx(V x IpG(rrp; o))) 4 ②2001 by CRC Press LLCChapter 6 Integral solutions of Maxwell’s equations 6.1 Vector Kirchoff solution: method of Stratton and Chu One of the most powerful tools for the analysis of electromagnetics problems is the integral solution to Maxwell’s equations formulated by Stratton and Chu [187, 188]. These authors used the vector Green’s theorem to solve for E˜ and H˜ in much the same way as is done in static fields with the scalar Green’s theorem. An alternative approach is to use the Lorentz reciprocity theorem of § 4.10.2, as done by Fradin [74]. The reciprocity approach allows the identification of terms arising from surface discontinuities, which must be added to the result obtained from the other approach [187]. 6.1.1 The Stratton–Chu formula Consider an isotropic, homogeneous medium occupying a bounded region V in space. The medium is described by permeability µ(ω) ˜ , permittivity (ω) ˜ , and conductivity σ(ω) ˜ . The region V is bounded by a surface S, which can be multiply-connected so that S is the union of several surfaces S1,... , SN as shown in Figure 6.1; these are used to exclude unknown sources and to formulate the vector Huygens principle. Impressed electric and magnetic sources may thus reside both inside and outside V. We wish to solve for the electric and magnetic fields at a point r within V. To do this we employ the Lorentz reciprocity theorem (4.173), written here using the frequency-domain fields as an integral over primed coordinates: − S E˜ a(r ,ω) × H˜ b(r ,ω) − E˜ b(r ,ω) × H˜ a(r ,ω) · nˆ d S =  V E˜ b(r ,ω) · J˜a(r ,ω) − E˜ a(r ,ω) · J˜b(r ,ω)− (6.1) H˜ b(r ,ω) · J˜ma(r ,ω) + H˜ a(r ,ω) · J˜mb(r ,ω) dV . (6.2) Note that the negative sign on the left arises from the definition of nˆ as the inward normal to V as shown in Figure 6.1. We place an electric Hertzian dipole at the point r = rp where we wish to compute the field, and set E˜ b = E˜ p and H˜ b = H˜ p in the reciprocity theorem, where E˜ p and H˜ p are the fields produced by the dipole (5.88)–(5.89): H˜ p(r,ω) = jω∇ × [p˜G(r|rp; ω)], (6.3) E˜ p(r,ω) = 1 ˜ c ∇ ×  ∇ × [p˜G(r|rp; ω)]  . (6.4)