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# Permittivity

In electromagnetism, one can define a field D which is the electric displacement field (also electric flux density), which represents how an applied electric field E in a medium will influence the organization of the electrical charge in this medium (charge migration, reorientation of electric dipoles, ...). The permittivity ( ε ) links those two fields; the relationship between them is given by

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

where ε can be a scalar or a 3 by 3 matrix.

Permittivity can take a real or complex value, not necessarily constant; it can be a function, for example, of the position in the medium, the frequency of the field applied, the humidity, or the temperature.

In SI units, permittivity is measured in farads per metre (F/m). The displacement D is measured in units of coulombs per square metre (C/m2), while the electric field E is measured in volts per metre (V/m). It should be noted that these are conventions which simplify Maxwell's Equations, current theory believes that in a vacuum D and E represent the same phenomena. Schemes can be devised whereby both quantities have the same units, and epsilon is a dimensionless quantity or 1.

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## Vacuum permittivity

Vacuum permittivity, ε0, ("the permittivity of free space") is the ratio of the D / E in vacuum. It also appears in Coulomb's law as a part of the Coulomb force constant, $\frac{1}{ 4 \pi \epsilon_0}$, which expresses the attraction between two unit charges.

One can define the permittivity of a medium as a dimensionless relative permittivity, εr, or dielectric constant, normalized to the absolute vacuum permittivity, ε0, such that:

$\varepsilon = \varepsilon_r \varepsilon_0$

where:

$\varepsilon_0 = 8.85419 \times 10^{-12} F/m$

## Permittivity in media

In the common case of an isotropic medium, D and E are parallel vectors and ε is a scalar, but in more general anisotropic media this is not the case and ε is a rank-2 tensor (causing birefringence).

The permittivity ε and magnetic permeability μ of a medium together determine the phase velocity v of electromagnetic radiation through that medium:

$\varepsilon \mu = \frac{1}{v^2}$

In a vacuum, these are given by

$\varepsilon_0\mu_0 = \frac{1}{c^2}$

where

ε0 is the permittivity of free space, equal to 8.85419 10−12 F/m
μ0 is the magnetic constant, or permeability of a vacuum, equal to 4π×10−7 N·A−2
c is the speed of light in vacuum, 299,792,458 m/s.

When an electric field is applied, a current flows. The total current flowing in a real medium is in general made of two parts: a conduction current and a displacement one. The displacement current can be thought of as an elastic response which a material has to the applied electric field. As the electric field is increased, the displacement current is stored in the material, and when the electric field is decreased the material releases the displacement current.

### Classification of materials

Material behaviors can be classified according to their permittivity. Materials whose permittivity has a negative real part are considered metals (in which no propagating electromagnetic waves exist), and those with a positive real part are dielectrics.

A perfect dielectric is a material that shows displacement current only, so it stores and returns electrical energy as if it were an ideal 'battery'.

In case of lossy medium (i.e. when the conduction currents are not negligible) the total current density flowing is:

$J_{tot}=J_c+J_d=\sigma E + j \; \omega \varepsilon_0 \varepsilon_d E = j \; \omega \varepsilon_0 \varepsilon^* E$

where

$j = \sqrt{-1}\,$
σ is the conductivity (responsible for conduction current) of the medium
εd is the relative permittivity (responsible for displacement current).

The size of the displacement current is seen to be dependant on the frequency ω of the applied field E; there is no displacement current in a constant field.

In this formalism the complex permittivity ε* is defined as:

$\varepsilon^* = \varepsilon_d - j \frac{\sigma}{\varepsilon_0 \omega}$

### Real materials

For real materials, both the real and imaginary parts of the permittivity are more complicated functions of frequency ω; since this leads to dispersion of signals containing multiple frequencies, such materials are called dispersive. This frequency dependence reflects the fact that a material's polarization does not respond instantaneously to an applied field—because the response must always be causal (come after the applied field), the dielectric function ε(ω) must have poles only for ω with positive imaginary parts, and ε(ω) therefore satisfies the Kramers-Kronig relations. However, in the narrow frequency ranges that are often studied in practice, the dielectric constants can often be approximated as frequency-independent.

At a given frequency, the imaginary part of ε leads to absorption loss if it is negative (in the above sign convention for frequency) and gain if it is positive. (More generally, one looks at the imaginary parts of the eigenvalues of the anisotropic dielectric tensor.)

Frequency-dependent dielectrics, for example, water, have very challenging responses. John David Jackson's Classical Electrodynamics has a good plot of the frequency-dependent nature of complex electric permittivity, which can alternatively be expressed as the complex index of refraction, or the refractive index n(ω) and the absorption coefficient α(ω).

At the low-frequency limit, the complex permittivity is equivalent to the constant permittivity εDC. At the high-frequency limit, the complex permittivity is commonly referred to as ε. At the plasma frequency and above, dielectrics behave as ideal metals, with electron gas behavior.

### Quantum-mechanical interpretation

Quantum-mechanically speaking, there are distinct regions of atomic and molecular interactions, microscopically, that account for the macroscopic behavior we label permittivity. At low frequencies in polar dielectrics, the molecules are polarized by an applied electric field, and induce periodic rotations. For example, as the microwave frequency, the microwave field causes a periodic rotation in water molecules that are sufficient to break hydrogen bonds. The field does work against the bonds and the energy is absorbed by the material in terms of heat, which is why microwave ovens work very well for materials containing water. There are two maximums of the imaginary component (the absorptive index) of water, one at the microwave frequency, and the other at far UV. At UV and above, and at high frequencies in general, the frequencies are too fast for the big fat molecules to spin around, and thus the energy is purely absorbed by the atoms, knocking electrons up in energy levels. At the plasma frequency, the electrons are totally ionized, and will conduct electricity. At middle frequencies, where the energy content is not high enough to effect electrons directly, yet too high for the rotational aspects, the energy is absorbed in terms of molecular vibrations. In water, this is where we begin to see a sharp drop off in the absorbtive index, and the minimum of the imaginary permittivity is at blue (optical regime) for water. This is why water is blue, and also why sunlight does not boil our eyeballs when we look at it. This is called a transparency window, where light can pass through the material with minimal loss.

While carrying out a complete ab initio or first principles modeling is now computationally possible it has not been widely applied yet. Thus, a phenomological model is accepted as being an adequate method of capturing experimental behaviors. The Debye model and the Lorentz model use a 1st order and 2nd order (respectively) lumped system parameter linear representation (such as an RC and an LRC resonant circuit). The von Hippel book Dielectrcs and Waves covers this material well.

For example a representation of a frequency dependent model for the complex permittivity function using a Lorentz model with n resonances would be:

$\varepsilon(\omega) = \varepsilon_{\infty} + \sum _{i=1} ^{n} \frac{\Delta \varepsilon_i}{1 + 2i \delta_i \frac{\omega}{\omega_{0i}} - \left({\frac{\omega}{\omega_{0i}}}\right)^2}$

where:

$\Delta \varepsilon_i\,$ is the difference in the magnitude of the real component for this resonance,
$\delta_i\,$ is the damping coefficient for this resonance,
$\omega_{0i}\,$ is the i-th resonant frequency of the material.