A current density j 1 (r) exp(it ) produces fields E 1 (r) exp(it) and B 1 (r) exp(it ). A second current density

A current density j₁(r) exp(−iωt ) produces fields E₁(r) exp(−iωt) and B₁(r) × exp(−iωt ). A second current density j₂ exp(−iωt ) produces fields E₂(r) exp(−iωt) and B₂(r) × exp(−iωt ).

(a) If V is a volume bounded by a surface S, prove the Lorentz reciprocity theorem:

(b) Specialize to a spherical volume which both encloses the current sources and is large enough that all the fields on S can be taken to be radiation fields. Prove that

(c) Specialize further to time-harmonic point electric dipole sources located at r₁ and r₂. If pk is the dipole moment of the kth dipole, prove that

Lorentz reciprocity is used to prove that the angular distribution of power radiated by an antenna in broadcast mode is identical to the angular distribution of power absorbed by the same antenna in receiving mode.

[dr@2] Ho V dr (EjE -j) = fas S ds (E x B E x B). -