# Electric displacement

In physics, the **electric displacement**, also known as **dielectric displacement** and usually denoted by its first letter **D**, is a vector field in a non-conducting medium, a dielectric. The displacement **D** is proportional to an external electric field **E** in which the dielectric is placed.

In SI units the proportionality is,

where ε_{0} is the electric constant and ε_{r} is the relative permittivity of the dielectric. In Gaussian units ε_{0} does not occur and can be put equal to unity in this equation. In vacuum the dimensionless quantity ε_{r} = 1 (both for SI and Gaussian units) and **D** is simply related (SI), or equal (Gaussian), to **E**. Often **D** is termed an auxiliary field with **E** the principal field.

An other auxiliary field is the electric polarization **P** of the dielectric,

The vector field **P** describes the polarization (small separation of the charges on each molecule of the dielectric) occurring in a slab of dielectric when it is brought into an electric field. The relative permittivity ε_{r} is greater than one for any insulator; this is due to the polarization of the dielectric which gives an opposing electric field.

The electric displacement appears in the following macroscopic Maxwell equation (in SI),

where the symbol **∇**⋅ gives the divergence of **D**(**r**) and ρ(**r**) is the charge density (charge per volume) at the point **r**.

In SI units **D** has the dimension charge per area: C/m^{2}, while in
Gaussian units **D** has the same dimension as **E**, i.e., statV/cm or dyne/statC.

## Relation of D to surface charge density σ

In the special case of a parallel-plate capacitor, often used to study and exemplify problems in electrostatics, the electric displacement *D* has an interesting interpretation. In that case *D* (the magnitude of vector **D**) is equal to the *true surface charge density* *σ*_{true} (the surface density on the plates of the right-hand capacitor in the figure).^{[1]}
In this figure two parallel-plate capacitors are shown that are identical, except for the matter between the plates: on the left no matter (vacuum), on the right a dielectric. Note in particular that the plates are held at the same voltage difference *V* and have the same area *A*. When the capacitor on the right discharges, it will deliver the total charge *Q*_{true} = *A*⋅*σ*_{true}. The one on the left will produce *Q*_{free} = *A*⋅*σ*_{free}.

To explain that *D* = *σ*_{true}, we recall that the relative permittivity may be defined as the ratio of the capacitances of two parallel-plate capacitors, (capacitance is total charge on the plates divided by voltage difference). Namely, the ratio of the capacitance *C* of a capacitor filled with dielectric to the capacitance *C*_{vac} of an identical capacitor in vacuum,

where we used again that the charge *Q* is *A*⋅σ. Clearly, the charge density on the plates increases by a factor ε_{r} when the dielectric is inserted in-between the plates. This means that the external source (of voltage *V*) must deliver a current during this insertion (it must move negative charge—electrons—from the positive plate to the negative plate).

The extra charge on the plates is compensated by the *polarization* of the dielectric, that is, the build-up of a positive polarization surface charge density σ_{p} on the side of the negative plate and a negative surface charge density on the positive side. The total charge is conserved, for instance on the side of the positively charged plate:

(Here the minus sign appears because the polarization charge density σ_{p} is negative on the positive side of the capacitor).

Assuming that the plates are very much larger than the distance between the plates, we may apply the following formula for *E*_{vac} (the magnitude of the vector **E**_{vac}),

(This electric field strength does not depend on the distance of a field point to the plates: the electric field between the plates is *homogeneous*.)
Now

hence *D* depends only on the charge delivered upon discharge, *D* = *Q*_{true} / *A*.

It is of some interest to note that the polarization vector **P** (pointing from minus to plus polarization charges, i.e., parallel to **E**_{vac}) has magnitude *P* equal to the polarization surface charge density σ_{p}. Indeed, the magnitudes of the three parallel vectors are related by (in SI units),

## Tensor character of relative permittivity

As defined above, **D** and **E** are parallel, i.e., ε_{r} is a number (a scalar). For a *non-isotropic* dielectric ε_{r} may be a second rank tensor,

so that **D** and **E** are not necessarily parallel in non-isotropic dielectrics.

## Note

- ↑ The nomenclature of the several surface charge distributions is not standardized. Here we will follow by and large R. Kronig,
*Textbook of physics*, Pergamon Press London, New York (1959). (English translation from the Dutch*Leerboek der Natuurkunde*)