the macroscopic Maxwell equations
Inside matter, the true microscopic field wiggles violently from atom to atom, and tracking every bound charge and bound current is hopeless. The macroscopic Maxwell equations are the practical rewrite: average over the atomic jitter, hide the bound sources inside the auxiliary fields D and H, and keep only the free charges and currents you actually control.
The four macroscopic equations are div D = rho_free, div B = 0, curl E = -dB/dt, and curl H = J_free + dD/dt, where D = epsilon_0 E + P and H = B/mu_0 - M. Compare them to the vacuum equations: the two source equations now involve D and H and are driven only by FREE charge and current, because the bound charge (-div P) and bound current (curl M plus dP/dt) have been quietly folded into D and H. The two source-free equations, Faraday's law and div B = 0, are untouched — no material can create magnetic monopoles or change the structural equations.
This is the form of Maxwell's equations that engineers and optical physicists actually use, because in the lab you set the free charge and free current, not the bound response of the material. To solve anything you must adjoin the constitutive relations (D = epsilon E, B = mu H) that say how the medium responds, plus boundary conditions at interfaces. The whole scheme is exact — it is just the microscopic theory reorganized — but the price of hiding the bound sources is that you now carry two extra fields, D and H, and must know the material to close the system.
For a light wave inside glass, you solve the macroscopic equations with curl H = dD/dt (no free current) and the constitutive relation D = epsilon E. The permittivity epsilon exceeds epsilon_0, so the wave speed drops to c/n with n = sqrt(epsilon/epsilon_0) the refractive index — refraction emerges directly from the macroscopic form plus a constitutive relation.
div D = rho_free, div B = 0, curl E = -dB/dt, curl H = J_free + dD/dt, with D = epsilon_0 E + P, H = B/mu_0 - M.
The macroscopic equations are not a new physical law — they are the microscopic equations spatially averaged, exact but useless without constitutive relations to specify the material. Only the SOURCE equations change (free sources, D and H); Faraday's law and div B = 0 are identical to the vacuum form.