Since this gives the contribution to the field in V from the fields on the surface receding to infinity, we expect that this term should be zero. If the medium has loss, then the exponential term decays and drives the contribution to zero. For a lossless medium the contributions are zero if lim rE(r,o)<∞, (69) limr[nf x H(r,o)+E(r,o]=0 To accompany(6. 8)we al lim rh(r,o)<∞, (6.11) lim r[nH(r,o)-fxE(r,o)=0 We refer to (6.9) and(611)as the finiteness conditions, and to(6.10) and(6. 12)as the Sommerfeld radiation condition, for the electromagnetic field. They show that far from the sources the fields must behave as a wave tem to the r-direction We shall se 86.2 that the waves are in fact spherical TEM waves 6.1.3 Fields in the excluded region: the extinction theorem The Stratton-Chu formula provides a solution for the field within the region V, external to the excluded regions. An interesting consequence of this formula, and one that helps us identify the equivalence principle, is that it gives the null resultH=E=0when evaluated at points within the excluded regions We can show this by considering two cases. In the first case we do not exclude the particular region Vm, but do exclude the remaining regions Vn, n# Then the electric field everywhere outside the remaining excluded regions(including at points within Vm) is,by(6.7), E(r,o)= JmxVG+2V'G-joij'Gdv+ G+(·E)VG-jd h)G]ds' drHVG In the second case we apply the Stratton-Chu formula only to Vn, and exclude all other regions. We incur a sign change on the surface and line integrals compared to the first case because the normal is now directed oppositely. By(6.7) we have E(r, o)= Jm xVG+=)dV E)V'G-joa(n'xhGdS'+ ②2001 by CRC Press LLCSince this gives the contribution to the field in V from the fields on the surface receding to infinity, we expect that this term should be zero. If the medium has loss, then the exponential term decays and drives the contribution to zero. For a lossless medium the contributions are zero if lim r→∞ rE˜(r,ω) < ∞, (6.9) lim r→∞ r ηrˆ × H˜ (r,ω) + E˜(r,ω) = 0. (6.10) To accompany (6.8) we also have lim r→∞ rH˜ (r,ω) < ∞, (6.11) lim r→∞ r ηH˜ (r,ω) − rˆ × E˜(r,ω) = 0. (6.12) We refer to (6.9) and (6.11) as the finiteness conditions, and to (6.10) and (6.12) as the Sommerfeld radiation condition, for the electromagnetic field. They show that far from the sources the fields must behave as a wave TEM to the r-direction. We shall see in § 6.2 that the waves are in fact spherical TEM waves. 6.1.3 Fields in the excluded region: the extinction theorem The Stratton–Chu formula provides a solution for the field within the region V, external to the excluded regions. An interesting consequence of this formula, and one that helps us identify the equivalence principle, is that it gives the null result H˜ = E˜ = 0 when evaluated at points within the excluded regions. We can show this by considering two cases. In the first case we do not exclude the particular region Vm, but do exclude the remaining regions Vn, n = m. Then the electric field everywhere outside the remaining excluded regions (including at points within Vm) is, by (6.7), E˜(r,ω) =  V+Vm −J˜i m × ∇ G + ρ˜i ˜ c ∇ G − jωµ˜ J˜i G  dV + +  n=m  Sn (nˆ × E˜) × ∇ G + (nˆ · E˜)∇ G − jωµ(˜ nˆ × H˜ )G  d S − −  n=m 1 jω˜ c na+nb (dl · H˜ )∇ G, r ∈ V + Vm. In the second case we apply the Stratton–Chu formula only to Vm, and exclude all other regions. We incur a sign change on the surface and line integrals compared to the first case because the normal is now directed oppositely. By (6.7) we have E˜(r,ω) =  Vm −J˜i m × ∇ G + ρ˜i ˜ c ∇ G − jωµ˜ J˜i G  dV − −  Sm (nˆ × E˜) × ∇ G + (nˆ · E˜)∇ G − jωµ(˜ nˆ × H˜ )G  d S + + 1 jω˜ c na+nb (dl · H˜ )∇ G, r ∈ Vm.