Lorentz lemma and reciprocity relations

We assume that two sets of sources, ja, j and j are located within a domain Q and that both generate electromagnetic fields of the same frequency w. We denote by and E jH the fields produced by a-type and 6-typc sources, 

Calculating the dot products of equation (8.129) with E , equation (8.132) with H , equation (8.130) with H, and equation (8.131) with E , we obtain the expression 

Now let the radius R of the ball Or tend to infinity. The surface integral over a sphere Cr vanishes due to radiation conditions (8.85) and (8.86), and we arrive at a mathematical formulation of the Lorentz lemma  

We now can use the Lorentz lemma to derive the reciprocity relations for the Green s electromagnetic tensor. Let us assume that the electric dipoles with moments a and b are located at points with the radius-vectors r and r , 

Substituting expressions (8.135) through (8.137) into the Lorentz lemma (8.134), we find 

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