An often overlooked term in the application of the Poynting theorem around a receiving antenna

When applying the Poynting theorem to the total field over a surface enclosing a receiving antenna, several terms emerge. These include self terms, relating either to the incident or scattered fields as well as a cross term. The terms are sometimes quoted in the literature without proper reference to the cross term, resulting erroneously in violation of energy conservation. In order to reinstate conservation of energy, some ad-hoc remedies have been recently proposed. Below, we attempt to show an orderly development of the Poynting theorem as a whole that needs no further assumptions and naturally satisfies energy conservation.