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# Electric displacement field

## The meaning of «electric displacement field»

In physics, the electric displacement field (denoted by D) or electric induction is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in the related concept of displacement current in dielectrics. In free space, the electric displacement field is equivalent to flux density, a concept that lends understanding to Gauss's law. In the International System of Units (SI), it is expressed in units of coulomb per meter square (C⋅m−2).

In a dielectric material, the presence of an electric field E causes the bound charges in the material (atomic nuclei and their electrons) to slightly separate, inducing a local electric dipole moment. The electric displacement field "D" is defined as

where ε 0 {\displaystyle \varepsilon _{0}} is the vacuum permittivity (also called permittivity of free space), 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:

In this equation, ρ f {\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 ρ b {\displaystyle \rho _{\text{b}}} is the density of all those charges that are part of a dipole, each of which is neutral. 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. 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 ρ f {\displaystyle \rho _{\text{f}}} at the edges.

Separate the total volume charge density into free and bound charges:

The density can be rewritten as a function of the polarization P:

The polarization P is defined to be a vector field whose divergence yields the density of bound charges ρb in the material. The electric field satisfies the equation:

Electrostatic forces on ions or electrons in the material are governed by the electric field E in the material via the Lorentz Force. Also, D is not determined exclusively by the free charge. As E has a curl of zero in electrostatic situations, it follows that