The modified Poynting theorem and the concept of mutual energy

Abstract

The goal of this article is to derive the reciprocity theorem, mutual energy theorem from Poynting theorem instead of from Maxwell equation. In this way the reciprocity theorem will become the energy theorem. In order to realize this purpose the followings have been done. The Poynting theorem is generalized to the modified Poynting theorem. In the modified Poynting theorem the electromagnetic field is superimposition of different electromagnetic fields including the field of retarded potential and advanced potential, electric/magnetic mirrored field, time-reversed field, time-offset field, space-offset field. The media epsilon (permittivity) and mu (permeability) can also be different in the different fields. The concept of mutual energy is introduced which is the difference between the total energy and self-energy. First we try to derive the mutual energy theorem from complex Poynting theorem, it is failed. As a side effect we obtained the mixed mutual energy theorem. We applied the average process to derive the mutual energy theorem from Poynting theorem. This is derivation is not strictly. Then we derive the mutual energy from Fourier domain, instead of obtained the mutual energy theorem from time-domain. We obtain the time-reversed mutual energy theorem. A time-reverse transform needed to further derive the mutual energy theorem. The time-reverse transform contains some information from Maxwell equation, hence the derivation is not a purely derivation from Poynting theorem. Then we derive the mutual energy theorem in time-domain. Using the modified Poynting theorem with the concept of the mutual energy. The instantaneous modified mutual energy theorem is derived.Applying time-offset transform and time integral to the instantaneous modified mutual energy theorem, the time-correlation modified mutual energy theorem is obtained. Assume there are two electromagnetic fields one is retarded potential and one is advanced potential, the convolution reciprocity theorem can be derived. Corresponding to the modified time-correlation mutual energy theorem and the time-convolution reciprocity theorem in Fourier domain, there is the modified mutual energy theorem and the Lorentz reciprocity theorem. Hence all mutual energy theorem and the reciprocity theorems are put in one frame of the concept of the mutual energy. The inner product is introduced for two different electromagnetic fields in both time domain and Fourier domain. The concept of inner product of electromagnetic fields simplifies the theory of the wave expansion. The concept of reaction is re-explained as the mutual energy of two fields with retarded potential and advanced potential.