# Electric flux density

Physical size
Surname Electric flux density
Formula symbol ${\ displaystyle {\ vec {D}}}$
Size and
unit system
unit dimension
SI A · s · m -2 L −2 · T · I
Gauss ( cgs ) Fr · cm −2 M 1/2 · L −1/2 · T −1
esE ( cgs ) Fr · cm −2 M 1/2 · L −1/2 · T −1
emE ( cgs ) abC · cm -2 L -3/2 x M 1/2

The electrical flux density - also known as electrical excitation , dielectric displacement , displacement density or displacement flux density - describes the density of the electrical field lines in relation to a surface. It is a physical quantity of electrostatics and electrodynamics and according to the international system of units is given in the unit coulomb per square meter  (C / m²). ${\ displaystyle {\ vec {D}}}$

The electrical flux density is a vectorial , i.e. directed quantity - in contrast to the scalar area charge density  σ, which is specified in the same unit.

If there is an electrical voltage between two points and in space , one speaks of different potentials in and . In between there are equipotential surfaces , i.e. H. closed areas with constant potential. The electrical flux lines are at right angles to these equipotential surfaces. According to the definition of the electric field strength , positive charges are the source of the electric flow, negative charges are the sink. ${\ displaystyle P_ {1}}$${\ displaystyle P_ {2}}$${\ displaystyle P_ {1}}$${\ displaystyle P_ {2}}$

## Connection with the electric flow

The electric flux formed by any surface  A passes through, is equal to the surface integral of the electric flux density D . Only that portion of the electrical flow that is normal to area  A contributes . Mathematically this is expressed by means of vectors and by the operation of the scalar product (inner product): ${\ displaystyle {\ mathit {\ Psi}}}$

${\ displaystyle {\ mathit {\ Psi}} = \ int \ limits _ {A} {\ vec {D}} \ cdot \ mathrm {d} {\ vec {A}}}$

The electric flow through a closed surface is therefore equal to the electric charge enclosed by this surface :

${\ displaystyle \ oint _ {A} {\ vec {D}} \; \ cdot \ mathrm {d} {\ vec {A}} = \ int _ {V} \ rho \; \ mathrm {d} V = Q}$

The change in the electrical flux density over time is shown as Maxwell's displacement current in the expanded Amperian law .

## Relationship with the electric field strength

In general, the electrical flux density can be written as the sum of the polarization and the product of the electrical field strength with the electrical field constant (dielectric constant or permittivity of the vacuum) : ${\ displaystyle {\ vec {P}}}$ ${\ displaystyle {\ vec {E}}}$${\ displaystyle \ varepsilon _ {0}: = 8 {,} 854187817 \ ldots \ cdot 10 ^ {- 12} \ mathrm {Fm} ^ {- 1}}$

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

In the case of a vacuum , the polarization disappears, then the electric flux density is simply related to the electric field strength:

${\ displaystyle {\ vec {D}} = \ varepsilon _ {0} {\ vec {E}}}$

In the case of a linear, isotropic medium as a dielectric , in which the electrical flux density is measured, the relationship between the electrical flux density and the electrical field strength changes by the relative permittivity : ${\ displaystyle \ varepsilon _ {\ mathrm {r}}}$

${\ displaystyle {\ vec {D}} = \ varepsilon _ {\ mathrm {r}} \ varepsilon _ {0} {\ vec {E}}}$

In the case of an anisotropic medium , as is typical for single crystals , the electrical flux density no longer necessarily points in the direction of the electrical field strength. If the two quantities are linearly related, then a second level tensor can be specified as a clear proportionality factor:

${\ displaystyle {\ vec {D}} = {\ begin {pmatrix} \ varepsilon _ {11} & \ varepsilon _ {12} & \ varepsilon _ {13} \\\ varepsilon _ {21} & \ varepsilon _ { 22} & \ varepsilon _ {23} \\\ varepsilon _ {31} & \ varepsilon _ {32} & \ varepsilon _ {33} \ end {pmatrix}} \ varepsilon _ {0} {\ vec {E}} }$

One consequence of such a relationship is birefringence .

Ferroelectrics , which retain part of their polarization after a strong field is applied, are an example of non-linear behavior between the electric field and flux ; further examples can be found in non-linear optics .

## Electric flux density in the plate capacitor

In the plate capacitor with parallel plates, the electrical flux density points in the direction of the surface normal of the capacitor plates . Your amount is included:

${\ displaystyle | {\ vec {D}} | = {\ frac {Q} {A}}}$.

It is

• ${\ displaystyle Q}$the amount of charge on a plate capacitor
• ${\ displaystyle A}$ the area of ​​its panels.

Alternatively, because of the proportionality of the electric field strength and surface charge density, this can also be written as ${\ displaystyle E}$ ${\ displaystyle \ sigma = {\ frac {Q} {A}}}$

${\ displaystyle | {\ vec {D}} | = \ varepsilon _ {\ mathrm {r}} \, \ varepsilon _ {0} \, | {\ vec {E}} |}$