Definition for polarization P and magnetization M fully consistent with Maxwell's equations

Dealing with the project of metamaterials scientists often have to design circuit elements at a subwavelength (or ``microscopic'') scale. At that scale, they use the set of Maxwell's equations in free-space, and neither permittivity ε nor permeability μ are formally defined. However, the objective is to use the unit cells in order to build a bulk material with some desired ``macroscopic'' properties. At that scale the set of Maxwell's equations in matter is adopted. To pass from one approach to the other is not obvious. In this paper we analyse the classic definitions of polarization P and magnetization M, highlighting their limits. Then we propose a definition for P and M fully consistent with Maxwell's equations at any scale.