The differential forms of Rumsey reaction and the corresponding reciprocal equations in the unified space and time form

Rumsey proposed the concept of reaction to describe the interaction between two sets of time-harmonic electromagnetic sources through the fields produced by sources, and expressed the Lorentz Reciprocity Theorem in a concise form of < 1,2 > = < 2,1 > , in 1954. Later, Lindell et al. extended the Rumsey’s reaction using the differential form, which contains two terms. The spatial term is the traditional Rumsey’s reaction term, which represents the electromagnetic power density reaction (scalar reaction), and the time-related term is the electromagnetic force density reaction (vector reaction). In this paper, the corresponding unified space and time form of the reciprocal equation is derived for Lindell’s reaction term by the differential form of the electromagnetic field equation, under the assumption that there is only electric source. However, the special form of the reciprocity equation < 1,2 > = ± < 2,1 > does not exist. Thus, we derived a new unified space and time equation of energy–momentum reciprocity and its special form. From this, a new generalized reaction is defined.