J/JA S Figure 6. 1: Geometry used to derive the Stratton-Chu formula We also let Ea=E and Ha=h, where E and h are the fields produced by the impressed sources Ja=J and Jma=Jm within V that we wish to find at r=rp. Since the dipole fields are singular at r=rp, we must exclude the point rp with a small spherical surface Ss surrounding the volume Va as shown in Figure 6. 1. Substituting these fields into(6.2) we obtain ,、匡B一Eד=⊥,国了-B,:(6 A useful identity involves the spatially-constant vector p and the Greens function G(rrp) V××(cp=vv·(Cp]-v2(Gp) =v·(Gp)}-pv2G =V师p·VG)+pk2G (6.6) where we have used V G=-k-G for r*rp We begin by computing the terms on the left side of(6.5). We suppress the rde- pendence of the fields and also the dependencies of G(rrp). Substituting from(6.3)we E x Hp]. nds xVx(p]·的 Using f.xVx(Gp=mx(VG×p)=('xE)·(VG×p) we can write E X Hp]- nds= ja ×E]×vGdS'. ②2001 by CRC Press LLCFigure 6.1: Geometry used to derive the Stratton–Chu formula. We also let E˜ a = E˜ and H˜ a = H˜ , where E˜ and H˜ are the fields produced by the impressed sources J˜a = J˜i and J˜ma = J˜i m within V that we wish to find at r = rp. Since the dipole fields are singular at r = rp, we must exclude the point rp with a small spherical surface Sδ surrounding the volume Vδ as shown in Figure 6.1. Substituting these fields into (6.2) we obtain − S+Sδ E˜ × H˜ p − E˜ p × H˜  · nˆ d S =  V−Vδ E˜ p · J˜i − H˜ p · J˜i m  dV . (6.5) A useful identity involves the spatially-constant vector p˜ and the Green’s function G(r |rp): ∇ × ∇ × (Gp˜)  = ∇ [∇ · (Gp˜)] − ∇2 (Gp˜) = ∇ [∇ · (Gp˜)] − p˜∇2G = ∇ (p˜ · ∇ G) + p˜ k2G, (6.6) where we have used ∇2G = −k2G for r = rp. We begin by computing the terms on the left side of (6.5). We suppress the r de￾pendence of the fields and also the dependencies of G(r |rp). Substituting from (6.3) we have S+Sδ [E˜ × H˜ p] · nˆ d S = jω S+Sδ E˜ × ∇ × (Gp˜)  · nˆ d S . Using nˆ · [E˜ × ∇ × (Gp˜)] = nˆ · [E˜ × (∇ G × p˜)] = (nˆ × E˜) · (∇ G × p˜) we can write S+Sδ [E˜ × H˜ p] · nˆ d S = jωp˜ · S+Sδ [nˆ × E˜ ] × ∇ GdS .