Reciprocity (electromagnetism) - Lorentz Reciprocity

Lorentz Reciprocity

Specifically, suppose that one has a current density that produces an electric field and a magnetic field, where all three are periodic functions of time with angular frequency ω, and in particular they have time-dependence . Suppose that we similarly have a second current at the same frequency ω which (by itself) produces fields and . The Lorentz reciprocity theorem then states, under certain simple conditions on the materials of the medium described below, that for an arbitrary surface S enclosing a volume V:

Equivalently, in differential form (by the divergence theorem):

This general form is commonly simplified for a number of special cases. In particular, one usually assumes that and are localized (i.e. have compact support), and that there are no incoming waves from infinitely far away. In this case, if one integrates over all space then the surface-integral terms cancel (see below) and one obtains:

This result (along with the following simplifications) is sometimes called the Rayleigh-Carson reciprocity theorem, after Lord Rayleigh's work on sound waves and an extension by John R. Carson (1924; 1930) to applications for radio frequency antennas. Often, one further simplifies this relation by considering point-like dipole sources, in which case the integrals disappear and one simply has the product of the electric field with the corresponding dipole moments of the currents. Or, for wires of negligible thickness, one obtains the applied current in one wire multiplied by the resulting voltage across another and vice versa; see also below.

Another special case of the Lorentz reciprocity theorem applies when the volume V entirely contains both of the localized sources (or alternatively if V intersects neither of the sources). In this case: