Electric displacement field

In physics, the electric displacement field (denoted by D), also called electric flux density, is a vector field that appears in Maxwell's equations.

It plays a major role in the physics of phenomena such as the capacitance of a material, the response of dielectrics to an electric field, how shapes can change due to electric fields in piezoelectricity or flexoelectricity as well as the creation of voltages and charge transfer due to elastic strains.

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.

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.

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:

This equation says, in effect, that the flux lines of D must begin and end on the free charges.

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

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.

[1] In a linear, homogeneous, isotropic dielectric with instantaneous response to changes in the electric field, P depends linearly on the electric field,

In linear, homogeneous, isotropic media, ε is a constant.

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:

The constraint of causality leads to the Kramers–Kronig relations, which place limitations upon the form of the frequency dependence.

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.

, where σf is the free charge density and the unit normal

[1] The earliest known use of the term is from the year 1864, in James Clerk Maxwell's paper A Dynamical Theory of the Electromagnetic Field.

Maxwell introduced the term D, specific capacity of electric induction, in a form different from the modern and familiar notations.

[3] It was Oliver Heaviside who reformulated the complicated Maxwell's equations to the modern form.

It wasn't until 1884 that Heaviside, concurrently with Willard Gibbs and Heinrich Hertz, grouped the equations together into a distinct set.

In both cases, the free charges are only on the metal capacitor plates.

Since the flux lines D end on free charges, and there are the same number of uniformly distributed charges of opposite sign on both plates, then the flux lines must all simply traverse the capacitor from one side to the other.

is the free surface charge density on the positive plate.

If the space between the capacitor plates is filled with a linear homogeneous isotropic dielectric with permittivity

and either the voltage difference between the plates will be smaller by this factor, or the charge must be higher.

The partial cancellation of fields in the dielectric allows a larger amount of free charge to dwell on the two plates of the capacitor per unit of potential drop than would be possible if the plates were separated by vacuum.

If the distance d between the plates of a finite parallel plate capacitor is much smaller than its lateral dimensions we can approximate it using the infinite case and obtain its capacitance as