Here we have used a ds2= 4T for the total solid angle subtending the sphere Ss. Finally, assuming that the volume sources are continuous, the volume integral over Vs vanishes E(r,)= JmxV'G+=V'G-joij'Gdv'+ G-jo(×mG]ds (dI·mVG (6.7) A similar formula for H can be derived by placing a magnetic dipole of moment pm at r=rp and proceeding as above. This leads to H(r,)=(J'xVG+mV'G-joe Ji) dv'+ 点x+(B下O+“+ (dI·E)VG (6.8) =i Jop We can also obtain this expression by substituting(6.7)into Faraday's law 6.1.2 The Sommerfeld radiation condition In 8 5.2.2 we found that if the potentials are not to be influenced by effects that are infinitely removed, then they must obey a radiation condition. We can make the same argument about the fields from(6.7) and(6.8). Let us allow one of the excluding surfaces, say Sw, to recede to infinity(enclosing all of the sources as it expands). AS Sw-00 any contributions from the fields on this surface to the fields at r should vanish Letting Sw be a sphere centered at the origin, we note that ft= -f and that as 4r VG(rr; w)=R jki Substituting these expressions into(6.7)we find that xVG+(·EvG [(×E)×P+(f·E)门] jop(fx [rakE E ②2001 by CRC Press LLCHere we have used   d = 4π for the total solid angle subtending the sphere Sδ . Finally, assuming that the volume sources are continuous, the volume integral over Vδ vanishes and we have E˜(r,ω) =  V −J˜i m × ∇ G + ρ˜i ˜ c ∇ G − jωµ˜ J˜i G  dV + +  N n=1  Sn (nˆ × E˜) × ∇ G + (nˆ · E˜)∇ G − jωµ(˜ nˆ × H˜ )G  d S − −  N n=1 1 jω˜ c na+nb (dl · H˜ )∇ G. (6.7) A similar formula for H˜ can be derived by placing a magnetic dipole of moment p˜ m at r = rp and proceeding as above. This leads to H˜ (r,ω) =  V J˜i × ∇ G + ρ˜i m µ˜ ∇ G − jω˜ c J˜i m G  dV + +  N n=1  Sn (nˆ × H˜ ) × ∇ G + (nˆ · H˜ )∇ G + jω˜ c (nˆ × E˜)G  d S + +  N n=1 1 jωµ˜ na+nb (dl · E˜)∇ G. (6.8) We can also obtain this expression by substituting (6.7) into Faraday’s law. 6.1.2 The Sommerfeld radiation condition In § 5.2.2 we found that if the potentials are not to be influenced by effects that are infinitely removed, then they must obey a radiation condition. We can make the same argument about the fields from (6.7) and (6.8). Let us allow one of the excluding surfaces, say SN , to recede to infinity (enclosing all of the sources as it expands). As SN → ∞ any contributions from the fields on this surface to the fields at r should vanish. Letting SN be a sphere centered at the origin, we note that nˆ = −rˆ and that as r → ∞ G(r|r ; ω) = e− jk|r−r | 4π|r − r | ≈ e− jkr 4πr , ∇ G(r|r ; ω) = Rˆ 1 + jkR 4π R2  e− jkR ≈ −rˆ 1 + jkr r  e− jkr 4πr . Substituting these expressions into (6.7) we find that lim SN→S∞ SN (nˆ × E˜) × ∇ G + (nˆ · E˜)∇ G − jωµ(˜ nˆ × H˜ )G  d S ≈ lim r →∞  2π 0  π 0 (rˆ × E˜) × rˆ + (rˆ · E˜)rˆ  1 + jkr r  + jωµ(˜ rˆ × H˜ ) e− jkr 4πr r2 sin θ dθ dφ ≈ lim r →∞  2π 0  π 0  r jkE˜ + jωµ(˜ rˆ × H˜ )  + E˜ e− jkr 4π sin θ dθ dφ .