the electric displacement field
Put a dielectric inside a capacitor and its molecules polarize, sprouting bound charges that partly cancel the applied field. Rather than track those bound charges explicitly, we bundle their effect into a new field. The electric displacement field D is that bookkeeping device — the auxiliary field whose sources are only the free charges you actually control.
The displacement field is defined as D = epsilon_0 E + P, where E is the total electric field and P is the polarization density (dipole moment per unit volume). Its point is a clean version of Gauss's law: div D = rho_free, meaning the flux of D through a closed surface counts only the FREE charge enclosed, with the messy bound charges automatically absorbed into P. In a linear dielectric D is proportional to E, D = epsilon E, where epsilon = epsilon_0 epsilon_r is the permittivity and epsilon_r the relative permittivity (dielectric constant).
D is enormously convenient because in a problem you usually know the free charge — the charge you deposited on the plates — but not the bound charge, which the material decides. Writing Gauss's law in terms of D lets you find the field from the free charge alone, then recover E. A crucial caveat: D is a defined auxiliary, not the fundamental field. It is E that pushes charges and appears in the Lorentz force; and unlike E, the field D can have a nonzero curl, so it is not conservative and there is no potential for it.
A parallel-plate capacitor holds free surface charge density sigma_free on its plates. Then D between the plates is simply D = sigma_free, whatever dielectric fills the gap — the displacement field is fixed by the free charge alone. Inserting a dielectric of constant epsilon_r leaves D unchanged but reduces the actual field to E = D/(epsilon_0 epsilon_r).
D = epsilon_0 E + P, with div D = rho_free: Gauss's law counting only the free charge.
D is an auxiliary field, not the fundamental one — it is E, not D, that exerts force on charges. And div D = rho_free does not mean D is fully determined by free charge alone: unlike E, D can have a nonzero curl, so you also need boundary conditions or a constitutive relation to fix it.