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waar '''P''' de [[polarisatie (elektriciteit)|polarisatie]] van het medium is en <math>\chi</math> de elektrische susceptibiliteit van het medium. De relatieve permittiviteit and susceptibiliteit van een stof blijken gerelateerd: <math>\varepsilon_{r} = \chi + 1</math>.
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=== Complex permittivity ===
[[Categorie:Elektromagnetisme]]
[[Image:Dielectric responses.jpg|thumb|right|400px|A dielectric permittivity spectrum over a wide range of frequencies. Various processes are labeled on the image: ionic and dipolar relaxation, and atomic and electronic resonances at higher energies.]]
[[Categorie:Stofeigenschap]]
Opposed to vacuum, the response of real materials to external fields generally depends on the [[frequency]] of the field. This frequency dependence reflects the fact that a material's polarization does not respond instantaneously to an applied field. The response must always be ''causal'' (arising after the applied field). For this reason permittivity is often treated as a complex function of the frequency of the applied field <math>\omega</math>, <math>\varepsilon \rightarrow \widehat{\varepsilon}(\omega)</math>. The definition of permittivity therefore becomes
:<math>D_{0}e^{i \omega t} = \widehat{\varepsilon}(\omega) E_{0} e^{i \omega t},</math>
where <math>D_{0}</math> and <math>E_{0}</math> are the amplitudes of the displacement and electrical fields, respectively, <math>i=\sqrt{-1}</math> is the imaginary unit. The response of a medium to static electric fields is described by the low-frequency limit of permittivity, also called the static permittivity or [[dielectric constant]] <math>\varepsilon_{s}</math> (also <math>\varepsilon_{DC}</math>):
:<math>\varepsilon_{s} = \lim_{\omega \rightarrow 0} \widehat{\varepsilon}(\omega).</math>
At the high-frequency limit, the complex permittivity is commonly referred to as ε<sub>∞</sub>. At the plasma frequency and above, dielectrics behave as ideal metals, with electron gas behavior. The static permittivity is a good approximation for altering fields of low frequencies, and as the frequency increases a measureable phase difference <math>\delta</math> emerges between '''D''' and '''E'''. The frequency at which the phase shift becomes noticeable depends on temperature and the details of the medium. For moderate fields strength (<math>E_{0}</math>), '''D''' and '''E''' remain proportional, and
:<math>\widehat{\varepsilon} = \frac{D_0}{E_0}e^{i\delta} = |\varepsilon|e^{i\delta}.</math>
Since the response of materials to alternating fields is characterized by a complex permittivity, it is natural to separate its real and imaginary parts, which is done by convention in the following way:
:<math>\widehat{\varepsilon}(\omega) = \varepsilon'(\omega) - i\varepsilon''(\omega) = \frac{D_0}{E_0} \left( \cos\delta - i\sin\delta \right). </math>
In the equation above, <math>\varepsilon''</math> is the imaginary part of the permittivity, which is related to the rate at which energy is absorbed by the medium (converted into thermal energy, et cetera). The real part of the permittivity, <math>\varepsilon'</math>.
The complex permittivity is usually a complicated function of frequency ω, since it is a sumperimposed description of [[dispersion (optics)|dispersion]] phenomena occurring at multiple frequencies. The dielectric function <math>\varepsilon(\omega)</math> must have poles only for frequencies with positive imaginary parts, and therefore satisfies the [[Kramers-Kronig relations]]. However, in the narrow frequency ranges that are often studied in practice, the permittivity can be approximated as frequency-independent or by model functions.
At a given frequency, the imaginary part of <math>\widehat{\varepsilon}</math> leads to absorption loss if it is positive (in the above sign convention) and gain if it is negative. More generally, the imaginary parts of the eigenvalues of the anisotropic dielectric tensor should be considered.
=== Classification of materials ===
Materials can be classified according to their permittivity. Those with a permittivity that has a negative real part <math>\varepsilon'</math> are considered to be [[metal]]s, in which no propagating electromagnetic waves exists. Those with a positive real part are [[dielectric]]s.
A ''perfect dielectric'' is a material that exhibits a displacement current only, therefore it stores and returns electrical energy as if it were an ideal [[battery (electricity)|battery]]. In the case of lossy medium, i.e. when the conduction current is not negligible, the total current density flowing is:
:<math> J_{tot} = J_c + J_d = \sigma E + i \omega \varepsilon_0 \varepsilon_r E = i \omega \varepsilon_0 \widehat{\varepsilon} E </math>
where
:''σ'' is the [[conductivity]] of the medium;
:ε<sub>''r''</sub> is the relative permittivity.
The size of the displacement current is dependent on the frequency ω of the applied field ''E''; there is no displacement current in a constant field.
In this formalism, the complex permittivity <math>\widehat{\varepsilon}</math> is defined as:
:<math> \widehat{\varepsilon} = \varepsilon_r - i \frac{\sigma}{\varepsilon_0 \omega} </math>
=== Dielectric absorption processes ===
In general, the absorption of electromagnetic energy by dielectrics is covered by a few different mechanisms that influence the shape of the permittivity as a function of frequency:
* [[Relaxation]] effects associated with permanent and induced molecular dipoles. At low frequencies the field changes slowly enough to allow dipoles to reach equilibrium before the field has measurably changed. For frequencies at which dipole orientations cannot follow the applied field due to the [[viscosity]] of the medium, absorption of the field's energy leads to energy dissipation. The mechanism of dipoles relaxing is called [[dielectric relaxation]] and for ideal dipoles is described by classic [[Debye relaxation]]
* [[Resonance]] effects, which arise from the rotations or vibrations of atoms, ions, or electrons. These processes are observed in the neighborhood of their characteristic absorption frequencies.
=== Quantum-mechanical interpretation ===
Quantum-mechanically speaking, there are distinct regions of atomic and molecular interactions, microscopically, that account for the macroscopic behavior we label as ''permittivity''. At low frequencies in polar dielectrics, molecules are polarized by an applied electric field, which induces periodic rotations.
For example, at the microwave frequency, the microwave field causes the periodic rotation of water molecules, 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 ultraviolet (UV) wavelengths.
At UV and above, and at high frequencies in general, the frequencies are too high for molecules to relax in, and thus the energy is purely absorbed by atoms, exciting electron energy levels. At the plasma frequency, the electrons are fully ionized, and will conduct electricity. At moderate frequencies, where the energy content is not high enough to affect electrons directly, yet too high for rotational aspects, the energy is absorbed in terms of resonant molecular vibrations. In water, this is where the absorptive index starts to drop sharply, and the minimum of the imaginary permittivity is at the frequency of blue light (optical regime). This is why water is blue, and also why sunlight does not damage water-containing organs such as the [[eye]].
While carrying out a complete ''ab initio'' or first-principles modelling 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 relaxation|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).
== Permittivity measurements ==
The dielectric constant of a material can be found by a variety of static electrical measurements. The complex permittivity is evaluated over a wide range of frequencies by using different variants of [[dielectric spectroscopy]], covering nearly 21 decades from 10<sup>-6</sup> to 10<sup>15</sup> Hz. Also, by using [[cryostat]]s and ovens, the dielectric properties of a medium can be characterized over an array of temperatures. In order to study systems for such diverse exciting fields, a number of measurement setups are used, each adequate for a special frequency range.
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* low-frequency time domain measurements (10<sup>-6</sup>-10<sup>3</sup> Hz)
* low frequency frequency domain measurements (10<sup>-5</sup>-10<sup>6</sup> Hz)
* reflective coaxial methods (10<sup>6</sup>-10<sup>10</sup> Hz)
* transmission coaxial method (10<sup>8</sup>-10<sup>11</sup> Hz)
* quasi-optical methods (10<sup>9</sup>-10<sup>10</sup> Hz)
* Fourier-transform methods (10<sup>11</sup>-10<sup>15</sup> Hz)
==See also==
* [[Dielectric constant]]
* [[Dielectric spectroscopy]]
* [[SI electromagnetism units]]
== Suggested readings ==
* ''Theory of Electric Polarization: Dielectric Polarization'', C.J.F. Bötthcer, ISBN: 0-44441-579-3
* ''Dielectrics and Waves'' edited by A. von Hippel, Arthur R., ISBN: 0-89006-803-8
== External links ==
*[http://www.newton.dep.anl.gov/newton/askasci/1993/physics/PHY48.HTM What is the significance of permittivity of free space?]
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[[categorie:elektromagnetisme]]
[[categorie:stofeigenschap]]
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[[bg:Диелектрична проницаемост]]
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