On the definition of the Poynting vector: A non-local derivation

We review a formalism developed previously to deal with the non-local (spatially dispersive) effective electromagnetic response of colloidal systems. We use this formalism to find, in the long wavelength limit, a general expression for the local energy conservation theorem in terms of parameters associated to the non-local response of the system. We find an explicit expression for the Poynting vector in terms of these non-local parameters and show that there are several ways to write it in terms of the electric permittivity and the magnetic permeability. The significance of this problem in the treatment of refraction is also discussed.