As I wrote earlier, this complicated set of equations cannot be solved for macroscopic matter, and fortunately that's not necessary, but you can use approximations, of which the simplest and often applicable in everyday life is linear response theory. Then you get macroscopic classical electromagnetics as you find it in almost all textbooks on the subject, although these very often omit to stress the basic fact that on a fundamental level, there is only one electromagnetic field with components [itex]\vec{E}[/itex] and [itex]\vec{B}[/itex], while [itex]\vec{D}[/itex] and [itex]\vec{H}[/itex] are auxiliary quantities denoting the average fields over macroscopic small, microscopic large regions, containing the effective average charge and current distributions of matter. In these macroscopic Maxwell equations only the external charges and currents brought into the system from outside and not taken care of by the average charge and current distributions of the matter., Even if we used natural units where μ 0 = ε 0 = 1, this distinction between B and H remains. In magnetic materials, B is not linearly related to H due to the magnetization term M. If B = H in space, then B ≠ H in magnetic materials. If voltage = d/dt ∫B·n dA, then how can curl H= σ·E = σ·voltage/length if B = H?, Which one depends on the medium & which one is medium-independent can vary. Between E/D, & H/B, it can go either way. Also, B is magnetic flux density, whereas H is magnetic field intensity. Elecric flux density, aka electric displacement, is called D, whereas electric field intensity is E. I hope I've helped. Claude.