Four Poynting Theorems

TL;DR

This paper systematically analyzes four different Poynting theorems by deriving energy continuity equations and propagation equations for each flux vector, providing insights into their physical interpretations and interactions with media.

Contribution

It introduces a comprehensive treatment of all four possible electromagnetic flux vectors, deriving their energy and propagation equations with minimal media restrictions.

Findings

Derivation of energy continuity equations for each flux vector

Analysis of how each flux vector relates to medium response

Development of propagation equations using directional fields

Abstract

The Poynting vector is an invaluable tool for analysing electromagnetic problems. However, even a rigorous stress-energy tensor approach can still leave us with the question: is it best defined as $\Vec{E} \cross \Vec{H}$ or as $\Vec{D} \cross \Vec{B}$? Typical electromagnetic treatments provide yet another perspective: they regard $\Vec{E} \cross \Vec{B}$ as the appropriate definition, because $\Vec{E}$ and $\Vec{B}$ are taken to be the fundamental electromagnetic fields. The astute reader will even notice the fourth possible combination of fields: i.e. $\Vec{D} \cross \Vec{H}$. Faced with this diverse selection, we have decided to treat each possible flux vector on its merits, deriving its associated energy continuity equation but applying minimal restrictions to the allowed host media. We then discuss each form, and how it represents the response of the medium. Finally, we derive a propagation equation for each flux vector using a directional fields approach; a useful result which enables further interpretation of each flux and its interaction with the medium.