# What is permittivity

1. Jul 24, 2014

### Greg Bernhardt

Definition/Summary

Permittivity, $\varepsilon$, of a material is its ability to separate charge when a voltage difference is applied. It equals the ratio of the electric displacement field to the total electric field in the material: $\boldsymbol{D}\,=\,\varepsilon\boldsymbol{E}$

It is capacitance times length per cross-section area, and is measured in units of farads per metre ($F/m$), with dimensions $Q^2T^2/ML^3$

$\varepsilon_0$, the permittivity of the vacuum, is defined as $10^7/4\pi c^2\ F/m$

Relative permittivity (or dielectric constant) of a material is $\kappa\,=\,\varepsilon_r\ =\ \varepsilon/\varepsilon_0$ (a dimensionless number). Its excess over 1 is the electric susceptibility, $\chi_e$, and measures its ability to polarise: $\boldsymbol{P}\,=\,(\varepsilon_r-1)\varepsilon_0\boldsymbol{E}\,=\,\chi_e\varepsilon_0\boldsymbol{E}$

Technically, permittivity is a tensor, whose input vector ($\boldsymbol{E}$) and output vector ($\boldsymbol{D}$) are generally in different directions. But in most materials it is a multiple of the unit tensor, and may be treated as a scalar.

Equations

Free-charge field (electric displacement field):

$$\boldsymbol{D}\,=\,\varepsilon\boldsymbol{E}\,=\,\varepsilon_0\boldsymbol{E}\,+\,\boldsymbol{P}$$

Bound-charge field (minus the polarisation field):

$$-\boldsymbol{P}\,=\,-(\varepsilon -\varepsilon_0)\boldsymbol{E}\,=\,-(\varepsilon_r-1)\varepsilon_0\boldsymbol{E}\,=\,-\chi_e\varepsilon_0\boldsymbol{E}$$

Capacitance of parallel-plate capacitor, area $A$, small separation $d$:

$$C = \varepsilon A/d$$

Energy density:

$$(\varepsilon\boldsymbol{E})\cdot\boldsymbol{E}/2\ =\ \boldsymbol{D}\cdot\boldsymbol{E}/2$$

In a material with speed of light $v\text{ and permeability }\mu$:

$$\varepsilon\,\mu\ =\ 1/v^2$$

Extended explanation

"Capacitivity":

Permittivity is capacitance times length per cross-section area (just as conductivity is conductance times length per cross-section area, and resistivity is resistance times cross-section area per length): double the cross-section area of the material, and we double the capacitance: double the length, and we halve the capacitance.

For this reason, permittivity could be (but isn't) called "capacitivity" (and similarly, permeability could be, but isn't, called "inductivity").

A common definition:
"Permittivity is an expression of how much electrical charge material can store when subjected to an electrical field."

This is highly misleading. No charge is stored in the material (this is obvious from the fact that the vacuum does not store charge, yet it has a permittivity). A material (other than the vacuum) already has stored positive and negative charges, and an electric field will cause those charges to separate slightly: this is polarisation. The ability to do so is the relative susceptibility of the material, $\varepsilon_r-1$

The charge stored is not in the material, but on the surface of conductors separated by the material.

An increase in permittivity:

Increases capacitance.

Decreases E (for a fixed charge distribution, ie constant D), and decreases stored energy.

Increases D (for a fixed applied voltage difference, ie constant E), and increases stored energy.

Stored energy:

Even in a vacuum, separating two equal and opposite charges uses energy, which is recoverable by bringing the charges nearer again. This energy is regarded as stored energy, in the field between the charges. Its density is $\varepsilon_0E^2/2$

When a material "replaces" the vacuum, to the (above) energy of separation must be added the energy used to polarise the material: internal equal and opposite charges in the material will be separated locally, creating a field (the polarisation field) opposing the primary field. This energy is literally stored in the material. Its density is $-\boldsymbol{P}\cdot\boldsymbol{E}/2\,=\,\chi_eE^2/2\,=\,(\varepsilon\,-\,\varepsilon_0)E^2/2$

The total energy stored, in the field and in the material, has density $\varepsilon E^2/2\,=\,\boldsymbol{D}\cdot\boldsymbol{E}/2$

(Dimensionally, D·E is coulombs per area times volts per length = coulomb-volts per volume = joules per volume.)

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