constitutive relations
Maxwell's macroscopic equations relate the four fields E, B, D and H, but they are not enough on their own — there are more fields than equations. The missing information is how a particular material responds to being pushed by a field. The constitutive relations supply exactly that, closing the system by connecting D to E, H to B, and current to field.
For a linear, isotropic medium the constitutive relations are D = epsilon E (the displacement responds to the electric field through the permittivity epsilon), B = mu H (the magnetic field responds through the permeability mu), and, for conductors, Ohm's law in local form J = sigma E (the current density responds through the conductivity sigma). These three material constants — epsilon, mu, sigma — encode everything the medium contributes; feed them to Maxwell's equations and the fields are fully determined. In vacuum they reduce to epsilon_0, mu_0 and zero conductivity.
The simple proportional form is only the leading approximation. Real materials can be anisotropic (epsilon becomes a tensor, so D need not be parallel to E, as in birefringent crystals), dispersive (epsilon and mu depend on frequency, which is why a prism splits colours), nonlinear (P grows with E^2, the basis of nonlinear optics), or nonlocal in space and time. Because they carry all the material physics, the constitutive relations — not Maxwell's equations, which are universal — are where the hard modelling of real media actually lives.
In a linear dielectric of relative permittivity epsilon_r, the constitutive relation D = epsilon_0 epsilon_r E fixes how much the material weakens the field: deposit the same free charge and the internal E is smaller by a factor epsilon_r than it would be in vacuum. Water, with epsilon_r about 80, screens electric fields dramatically — which is why it dissolves salts so well.
D = epsilon E, B = mu H, J = sigma E: material responses that close Maxwell's equations.
Constitutive relations are properties of MATTER, not laws of nature — they can be anisotropic, frequency-dependent, nonlinear or history-dependent, unlike the universal Maxwell equations. The simple D = epsilon E, B = mu H form assumes a linear, isotropic, non-dispersive medium.