Electric displacement field

In any material, if there is an inversion center then the charge at, for instance, ${\displaystyle +x}$ and ${\displaystyle -x}$ are the same. This means that there is no dipole. If an electric field is applied to an insulator, then (for instance) the negative charges can move slightly towards the positive side of the field, and the positive charges in the other direction. This leads to an induced dipole which is described as a polarization. There can be slightly different movements of the negative electrons and positive nuclei in molecules, or different displacements of the atoms in an ionic compound. Materials which do not have an inversion center display piezoelectricity and always have a polarization; in others spatially varying strains can break the inversion symmetry and lead to polarization, the flexoelectric effect. Other stimuli such as magnetic fields can lead to polarization in some materials, this being called the magnetoelectric effect.

Definition

The electric displacement field "D" is defined as$${\displaystyle \mathbf{D}\equiv {\epsilon }_0\mathbf{E}+\mathbf{P},}$$where ${\displaystyle {\epsilon }_0}$ is the vacuum permittivity (also called permittivity of free space), E is the electric field, and P is the (macroscopic) density of the permanent and induced electric dipole moments in the material, called the polarization density.

The displacement field satisfies Gauss's law in a dielectric: $${\displaystyle \mathrm{\nabla }\cdot \mathbf{D}=\rho -{\rho }_{\text{b}}={\rho }_{\text{f}}}$$

In this equation, ${\displaystyle {\rho }_{\text{f}}}$ is the number of free charges per unit volume. These charges are the ones that have made the volume non-neutral, and they are sometimes referred to as the space charge. This equation says, in effect, that the flux lines of D must begin and end on the free charges. In contrast ${\displaystyle {\rho }_{\text{b}}}$, which is called the bound charge, is an effective density of the charges that are part of a dipole. In the example of an insulating dielectric between metal capacitor plates, the only free charges are on the metal plates and dielectric contains only dipoles. The net, unbalanced bound charge at the metal/dielectric interface balances the charge on the metal plate. If the dielectric is replaced by a doped semiconductor or an ionised gas, etc, then electrons move relative to the ions, and if the system is finite they both contribute to ${\displaystyle {\rho }_{\text{f}}}$ at the edges.

D is not determined exclusively by the free charge. As E has a curl of zero in electrostatic situations, it follows that $${\displaystyle \mathrm{\nabla }\times \mathbf{D}=\mathrm{\nabla }\times \mathbf{P}}$$

The effect of this equation can be seen in the case of an object with a "frozen in" polarization like a bar electret, the electric analogue to a bar magnet. There is no free charge in such a material, but the inherent polarization gives rise to an electric field, demonstrating that the D field is not determined entirely by the free charge. The electric field is determined by using the above relation along with other boundary conditions on the polarization density to yield the bound charges, which will, in turn, yield the electric field.

In a linear, homogeneous, isotropic dielectric with instantaneous response to changes in the electric field, P depends linearly on the electric field, $${\displaystyle \mathbf{P}={\epsilon }_0\chi \mathbf{E},}$$ where the constant of proportionality ${\displaystyle \chi }$ is called the electric susceptibility of the material. Thus $${\displaystyle \mathbf{D}={\epsilon }_0(1+\chi )\mathbf{E}={\epsilon }_0{\epsilon }_r\mathbf{E}=\epsilon \mathbf{E}}$$ where ε_r = 1 + χ is the relative permittivity of the material, and ε is the permittivity.

In linear, homogeneous, isotropic media, ε is a constant. However, in linear anisotropic media it is a tensor, and in nonhomogeneous media it is a function of position inside the medium. It may also depend upon the electric field (nonlinear materials) and have a time dependent response. Explicit time dependence can arise if the materials are physically moving or changing in time (e.g. reflections off a moving interface give rise to Doppler shifts). A different form of time dependence can arise in a time-invariant medium, as there can be a time delay between the imposition of the electric field and the resulting polarization of the material. In this case, P is a convolution of the impulse response susceptibility χ and the electric field E. Such a convolution takes on a simpler form in the frequency domain: by Fourier transforming the relationship and applying the convolution theorem, one obtains the following relation for a linear time-invariant medium: $${\displaystyle \mathbf{D}(\omega )=\epsilon (\omega )\mathbf{E}(\omega ),}$$ where ${\displaystyle \omega }$ is the frequency of the applied field. The constraint of causality leads to the Kramers–Kronig relations, which place limitations upon the form of the frequency dependence. The phenomenon of a frequency-dependent permittivity is an example of material dispersion. In fact, all physical materials have some material dispersion because they cannot respond instantaneously to applied fields, but for many problems (those concerned with a narrow enough bandwidth) the frequency-dependence of ε can be neglected.

At a boundary, ${\displaystyle ({\mathbf{D}}_{\mathbf{1}}-{\mathbf{D}}_{\mathbf{2}})\cdot \hat{\mathbf{n}}=D_{1,\perp }-D_{2,\perp }={\sigma }_{\text{f}}}$, where σ_f is the free charge density and the unit normal ${\displaystyle \hat{\mathbf{n}}}$ points in the direction from medium 2 to medium 1.