Electric displacement field

In physics, the electric displacement field or electric flux density is a vector-valued field $\mathbf {D}$ that appears in Maxwell's equations. It accounts for the effects of bound charges within materials. "D" stands for "displacement," as in the related concept of displacement current in dielectrics.

Definition

In general, D is defined by the relation

$\mathbf {D} =\varepsilon _{0}\ \mathbf {E} \ +\ \mathbf {P}$

where E is the electric field, $\varepsilon _{0}$ is the vacuum permittivity, and P is the polarization density of the material.

In most ordinary materials, however, D may be calculated with the simpler formula

$\mathbf {D} =\varepsilon \mathbf {E}$

where $\varepsilon$ is the permittivity of the material; in linear isotropic media this will be a constant, and in linear anisotropic media it will be a rank 2 tensor (a matrix)

Displacement field in a capacitor

Consider an infinite parallel plate capacitor placed in space (or in a medium) with no free charges present except on the capacitor. In SI units, the charge density on the plates is equal to the value of the D field between the plates. This follows directly from Gauss's law, by integrating over a small rectangular box straddling the plate of the capacitor:

$\oint _{A}\mathbf {D} \cdot d\mathbf {A} =Q$

The part of the box inside the capacitor plate has no field, so that part of the integral is zero. On the sides of the box, $d\mathbf {A}$ is perpendicular to the field, so that part of the integral is also zero, leaving:

$|\mathbf {D} |={\frac {Q}{A}}$

which is the charge density on the plate.

Units

In the standard SI system of units D is measured in coulombs per square meter (C/m²).

This choice of units results in one of the simplest forms of the Ampère-Maxwell equation:

$\nabla \times \mathbf {H} =\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}$

If one chooses both B and H to be measured in teslas, and E and D to be measured in newtons per coulomb, then the formula is modified to be:

$\nabla \times \mathbf {H} =\mu _{0}\mathbf {J} +{\frac {1}{c^{2}}}{\frac {\partial \mathbf {D} }{\partial t}}$

Therefore it is seen as being preferential to express B & H, and D & E in different sets of units.

Choice of units has differed in history, for instance in the electromagnetic system of scientific units, in which the unit of charge is defined such that $1/4\pi \varepsilon _{0}=1$ (dimensionless), D and E are expressed in the same units.