# Permittivity

The permittivity of a medium is an intensive physical quantity that describes how an electric field affects and is affected by the medium. In electromagnetism one can define an electric displacement field D, which represents how an applied electric field E will influence the organization of electrical charges in the medium, including charge migration and electric dipole reorientation. Its relation to permittivity is given by

[itex]\mathbf{D}=\varepsilon \cdot \mathbf{E}[itex],

where ε is a scalar if the medium is isotropic or a 3 by 3 matrix otherwise.

A high permittivity tends to reduce any electric field present. Therefore the capacitance of a capacitor can be increased by increasing the permittivity of the dielectric material inside it. Permittivity can be looked at as the quality of a material that allows it to store electrical charge. A given amount of material with high permittivity can store more charge than a material with lower permittivity.

Permittivity can take a real or complex value. In general, it is not a constant, as it can vary with the position in the medium, the frequency of the field applied, humidity, temperature, and other parameters.

In SI units, permittivity is measured in farads per metre (F/m). The displacement field 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 that simplify Maxwell's Equations, since current theory believes that in vacuum D and E represent the same phenomena. Schemes can be devised whereby both quantities have the same units, and [itex]\varepsilon[itex] is a dimensionless quantity or 1.

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## Vacuum permittivity

The permittivity of a material is usually given relative to that of vacuum, as a relative permittivity, [itex]\varepsilon_{r}[itex]. The actual permittivity is then calculated by multiplying the relative permittivty by [itex]\varepsilon_{0}[itex]:

[itex]\varepsilon = \varepsilon_r \varepsilon_0[itex]

Vacuum permittivity [itex]\varepsilon_{0}[itex] ("the permittivity of free space") is the ratio D/E in vacuum. It also appears in Coulomb's law as a part of the Coulomb force constant, [itex]\frac{1}{ 4 \pi \epsilon_0} [itex], which expresses the attraction between two unit charges in vacuum.

[itex]\varepsilon_0 = \frac{1}{c^2\mu_0} = 8.8541878176\ldots \times 10^{-12} F/m[itex],

where [itex]c[itex] is the speed of light and [itex]\mu_0[itex] is the permeability of vacuum. All three of these constants are exactly defined in SI units.

## Permittivity in media

Dielectric constant of some materials at room temperature
Material Dielectric constant
Vacuum 1 (by definition)
Air 1.0005
Teflon 2
Paper 3
Rubber 7
Methyl Alcohol 30
Water 80
Barium Titanate 1200

In the common case of isotropic media, D and E are parallel vectors and [itex]\varepsilon[itex] is a scalar, but in general anisotropic media this is not the case and [itex]\varepsilon[itex] is a rank-2 tensor (causing birefringence). The permittivity [itex]\varepsilon[itex] and magnetic permeability [itex]\mu[itex] of a medium together determine the phase velocity v of electromagnetic radiation through that medium:

[itex]\varepsilon \mu = \frac{1}{v^2}[itex]

When an electric field is applied to a medium, a current flows. The total current flowing in a real medium is in general made of two parts: a conduction and a displacement current. The displacement current can be thought of as the elastic response of the material to the applied electric field. As the magnitude of 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. The electric displacement can be separated into a vacuum contribution and one arising from the material by

[itex]\mathbf{D} = \varepsilon_{0} \mathbf{E} + \mathbf{P} = \varepsilon_{0} \mathbf{E} + \varepsilon_{0}\chi\mathbf{E} = \varepsilon_{0} \mathbf{E} \left( 1 + \chi \right)[itex],

where P is the polarization of the medium and [itex]\chi[itex] its electric susceptibility. It follows that the relative permittivity and susceptibility of a sample are related, [itex]\varepsilon_{r} = \chi + 1[itex].

### Complex permittivity

Missing image
Dielectric_responses.jpg
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.

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 [itex]\omega[itex], [itex]\varepsilon \rightarrow \hat{\varepsilon}(\omega)[itex]. The definition of permittivity therefore becomes

[itex]D_{0}e^{i \omega t} = \hat{\varepsilon}(\omega) E_{0} e^{i \omega t},[itex]

where [itex]D_{0}[itex] and [itex]E_{0}[itex] are the amplitudes of the electrical and displacement fields, respectively, [itex]i=\sqrt{-1}[itex] 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 [itex]\varepsilon_{s}[itex] (also [itex]\varepsilon_{DC}[itex]):

[itex]\varepsilon_{s} = \lim_{\omega \rightarrow 0} \hat{\varepsilon}(\omega)[itex]

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. The static permittivity is a good approximation for altering fields of low frequencies, and as the frequency increases a measureable phase difference [itex]\delta[itex] 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 ([itex]E_{0}[itex]), D and E remain proportional, and

[itex]\hat{\varepsilon} = \frac{D_0}{E_0}e^{i\delta} = |\varepsilon|e^{i\delta}[itex].

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:

[itex]\hat{\varepsilon}(\omega) = \varepsilon'(\omega) - i\varepsilon''(\omega) = \frac{D_0}{E_0} \left( cos\delta - i\sin\delta \right) [itex].

In the equation above, [itex]\varepsilon''[itex] is the imaginary part of the permittivity and is sometimes called dielectric dispersion, since it is directly related to the fraction of applied field's energy that is dissipated by the medium. The real part of the permittivity, [itex]\varepsilon'[itex], is related to the fraction of the energy absorbed by the medium.

The complex permittivity is usually a complicated function of frequency ω, since it is a sumperimposed description of dispersion phenomena occuring at multiple frequencies. The dielectric function [itex]\varepsilon(\omega)[itex] 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 permittivty can be approximated as frequency-independent ot by model functions.

At a given frequency, the imaginary part of [itex]\hat{\varepsilon}[itex] leads to absorption loss if it is positive (in the above sign convention) and gain if it is negativee. 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 [itex]\varepsilon'[itex] are considered to be metals, in which no propagating electromagnetic waves exists. Those with a positive real part are dielectrics.

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. In the case of lossy medium, i.e. when the conduction current is not negligible, the total current density flowing is:

[itex] J_{tot} = J_c + J_d = \sigma E + i \omega \varepsilon_0 \varepsilon_r E = i \omega \varepsilon_0 \hat{\varepsilon} E [itex]
σ is the conductivity of the medium
εr 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 [itex]\hat{\varepsilon}[itex] is defined as:

[itex] \hat{\varepsilon} = \varepsilon_r - i \frac{\sigma}{\varepsilon_0 \omega} [itex]

### 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, as 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 fast 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 effect electrons directly, yet too high for rotational aspects, the energy is absorbed in terms of resonant molecular vibrations. In water, this is where the absorbtive index starts to sharply frop, 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 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 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 measuremnts. The complex permittivity is evaluated over a wide range of frequencies by using different variants of dielectric spectroscopy, covering nearly 21 decades from 10-6 to 1015 Hz. Also, by using cryostats 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 measruement setups are used, eahc adequate for a special frequency range.

• low-frequency time domain measurements (10-6-103 Hz)
• low frequency frequency domain measurements (10-5-106 Hz)
• reflective coaxial methods (106-1010 Hz)
• transmission coaxial method (108-1011 Hz)
• quasi-optical methods methods (109-101 Hz)
• Fourier-transform methods (1011-1015 Hz)

## SI electricity units

• Theory of Electric Polarization: Dielectric Polarization, C.J.F. Btthcer, ISBN: 0-44441-579-3
• Dielectrcs and Wavesedited by A. von Hippel, Arthur R., ISBN: 0-89006-803-8

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