Here, we describe a first-principles derivation of the macroscopic Poynting vector, heating rate, and stored
energy in arbitrary composite media formed by dielectric and metallic inclusions, taking into account the
effects of artificial magnetism, bianisotropy, as well as spatial dispersion. Starting from the microscopic
Maxwell’s equations in an arbitrary periodic structured material, we demonstrate that in some situations it is
possible to obtain a mathematically exact relation between quadratic expressions of the microscopic fields, such as the cell-averaged microscopic Poynting vector, and the macroscopic electromagnetic fields and the
effective dielectric function.