Reciprocity between Generalized Mutual Impedances for Closed or Open Circuits

Let two wires L1 and L2 be considered and electromotive forces E
P1
and E
P2
be applied at two small gaps Pl and P2 of the wires, which can either be part of a loop or an open circuit (antennas). The two circuits are assumed to be placed in an isotropic medium. A general expression is obtained between E
P1
and E
P2
and the corresponding currents I
P1
and I
P2
at the gaps. In case the electromotive forces are sinusoidal and of the same frequency, the above expression leads to generalized self- and mutual impedances. The mutual impedances are shown to be reciprocal, the demonstration being derived from the reciprocity theorem in the form given to it by Ballantine. The formulas are applied to different cases, such as quasi-stationary current distribution, closely or loosely coupled antennas, and wave projectors. Attention is drawn to the conditions assumed in the demonstration for its validity.