In electromagnetism, permittivity is the measure of how much resistance is encountered when forming an electric field in a vacuum. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. Permittivity is determined by the ability of a material to polarize in response to the field, and thereby reduce the total electric field inside the material. Thus, permittivity relates to a material's ability to transmit (or "permit") an electric field.
It is directly related to electric susceptibility, which is a measure of how easily a dielectric polarizes in response to an electric field.
In SI units, permittivity ε is measured in farads per meter (F/m); electric susceptibility χ is dimensionless. They are related to each other through
where ε_{r} is the relative permittivity of the material, and ε_{0} = 8.85… × 10^{−12} F/m is the vacuum permittivity.
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In electromagnetism, the electric displacement field D represents how an electric field E influences the organization of electrical charges in a given medium, including charge migration and electric dipole reorientation. Its relation to permittivity in the very simple case of linear, homogeneous, isotropic materials with "instantaneous" response to changes in electric field is
where the permittivity ε is a scalar. If the medium is anisotropic, the permittivity is a second rank tensor.
In general, permittivity 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 a nonlinear medium, the permittivity can depend on the strength of the electric field. Permittivity as a function of frequency can take on real or complex values.
In SI units, permittivity is measured in farads per meter (F/m or A^{2}·s^{4}·kg^{−1}·m^{−3}). The displacement field D is measured in units of coulombs per square meter (C/m^{2}), while the electric field E is measured in volts per meter (V/m). D and E describe the interaction between charged objects. D is related to the charge densities associated with this interaction, while E is related to the forces and potential differences.
The vacuum permittivity ε_{0} (also called permittivity of free space or the electric constant) is the ratio D/E in free space. It also appears in the Coulomb force constant 1/4πε_{0}.
Its value is^{[1]}
where
Constants c_{0} and μ_{0} are defined in SI units to have exact numerical values, shifting responsibility of experiment to the determination of the meter and the ampere.^{[3]} (The approximation in the second value of ε_{0} above stems from π being an irrational number.)
The linear permittivity of a homogeneous material is usually given relative to that of free space, as a relative permittivity ε_{r} (also called dielectric constant, although this sometimes only refers to the static, zerofrequency relative permittivity). In an anisotropic material, the relative permittivity may be a tensor, causing birefringence. The actual permittivity is then calculated by multiplying the relative permittivity by ε_{0}:
where
The susceptibility is defined as the constant of proportionality (which may be a tensor) relating an electric field E to the induced dielectric polarization density P such that
where is the electric permittivity of free space.
The susceptibility of a medium is related to its relative permittivity by
So in the case of a vacuum,
The susceptibility is also related to the polarizability of individual particles in the medium by the ClausiusMossotti relation.
The electric displacement D is related to the polarization density P by
The permittivity ε and permeability µ of a medium together determine the phase velocity c of electromagnetic radiation through that medium:
In general, a material cannot polarize instantaneously in response to an applied field, and so the more general formulation as a function of time is
That is, the polarization is a convolution of the electric field at previous times with timedependent susceptibility given by χ(Δt). The upper limit of this integral can be extended to infinity as well if one defines χ(Δt) = 0 for Δt < 0. An instantaneous response corresponds to Dirac delta function susceptibility χ(Δt) = χδ(Δt).
It is more convenient in a linear system to take the Fourier transform and write this relationship as a function of frequency. Due to the convolution theorem, the integral becomes a simple product,
This frequency dependence of the susceptibility leads to frequency dependence of the permittivity. The shape of the susceptibility with respect to frequency characterizes the dispersion properties of the material.
Moreover, the fact that the polarization can only depend on the electric field at previous times (i.e. χ(Δt) = 0 for Δt < 0), a consequence of causality, imposes Kramers–Kronig constraints on the susceptibility χ(0).
As opposed to the response of a vacuum, the response of normal 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) which can be represented by a phase difference. For this reason permittivity is often treated as a complex function (since complex numbers allow specification of magnitude and phase) of the (angular) frequency of the applied field ω, . The definition of permittivity therefore becomes
where
It is important to realise that the choice of sign for timedependence dictates the sign convention for the imaginary part of permittivity. The signs used here correspond to those commonly used in physics, whereas for the engineering convention one should reverse all imaginary quantities.
The response of a medium to static electric fields is described by the lowfrequency limit of permittivity, also called the static permittivity ε_{s} (also ε_{DC} ):
At the highfrequency 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 measurable phase difference δ 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 (E_{0}), D and E remain proportional, and
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:
where
The complex permittivity is usually a complicated function of frequency ω, since it is a superimposed description of dispersion phenomena occurring at multiple frequencies. The dielectric function ε(ω) 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 frequencyindependent or by model functions.
At a given frequency, the imaginary part of 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.
In the case of solids, the complex dielectric function is intimately connected to band structure. The primary quantity that characterizes the electronic structure of any crystalline material is the probability of photon absorption, which is directly related to the imaginary part of the optical dielectric function ε(ω). The optical dielectric function is given by the fundamental expression:^{[5]}
In this expression, W_{cv}(E) represents the product of the Brillouin zoneaveraged transition probability at the energy E with the joint density of states,^{[6]}^{[7]} J_{cv}(E); φ is a broadening function, representing the role of scattering in smearing out the energy levels.^{[8]} In general, the broadening is intermediate between Lorentzian and Gaussian;^{[9]}^{[10]} for an alloy it is somewhat closer to Gaussian because of strong scattering from statistical fluctuations in the local composition on a nanometer scale.
Materials can be classified according to their permittivity and conductivity, σ. Materials with a large amount of loss inhibit the propagation of electromagnetic waves. In this case, generally when σ/(ωε) >> 1, we consider the material to be a good conductor. Dielectrics are associated with lossless or lowloss materials, where σ/(ωε) << 1. Those that do not fall under either limit are considered to be general media. A perfect dielectric is a material that has no conductivity, thus exhibiting only a displacement current. Therefore it stores and returns electrical energy as if it were an ideal capacitor.
In the case of lossy medium, i.e. when the conduction current is not negligible, the total current density flowing is:
where
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 is defined as^{[11]}:
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:
The above effects often combine to cause nonlinear effects within capacitors. For example, dielectric absorption refers to the inability of a capacitor that has been charged for a long time to completely discharge when briefly discharged. Although an ideal capacitor would remain at zero volts after being discharged, real capacitors will develop a small voltage, a phenomenon that is also called soakage or battery action. For some dielectrics, such as many polymer films, the resulting voltage may be less than 12% of the original voltage. However, it can be as much as 15  25% in the case of electrolytic capacitors or supercapacitors.
In terms of quantum mechanics, permittivity is explained by atomic and molecular interactions.
At low frequencies, molecules in polar dielectrics 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 as heat. This is why microwave ovens work very well for materials containing water. There are two maxima of the imaginary component (the absorptive index) of water, one at the microwave frequency, and the other at far ultraviolet (UV) frequency. It should be noted that both of these resonances are at higher frequencies than the operating frequency of microwave ovens.
At moderate frequencies, the energy is too high to cause rotation, yet too low to affect electrons directly, and is absorbed in the form 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 sunlight does not damage watercontaining organs such as the eye.^{[12]}
At high frequencies (such as UV and above), molecules cannot relax, and the energy is purely absorbed by atoms, exciting electron energy levels. Thus, these frequencies are classified as ionizing radiation.
While carrying out a complete ab initio (that is, firstprinciples) modelling is now computationally possible, it has not been widely applied yet. Thus, a phenomenological model is accepted as being an adequate method of capturing experimental behaviors. The Debye model and the Lorentz model use a 1storder and 2ndorder (respectively) lumped system parameter linear representation (such as an RC and an LRC resonant circuit).
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 orders of magnitude from 10^{−6} to 10^{15} 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 measurement setups are used, each adequate for a special frequency range.
Various microwave measurement techniques are outlined in Chen et al..^{[13]} Typical errors for the HakkiColeman method employing a puck of material between conducting planes are about 0.3%.^{[14]}
At infrared and optical frequencies, a common technique is ellipsometry. Dual polarisation interferometry is also used to measure the complex refractive index for very thin films at optical frequencies.
The electric susceptibility χ_{e} of a dielectric material is a measure of how easily it polarizes in response to an electric field. This, in turn, determines the electric permittivity of the material and thus influences many other phenomena in that medium, from the capacitance of capacitors to the speed of light.
It is defined as the constant of proportionality (which may be a tensor) relating an electric field E to the induced dielectric polarization density P such that
{\mathbf P}=\varepsilon_0\chi_e{\mathbf E},
where $\backslash ,\; \backslash varepsilon\_0$ is the electric permittivity of free space.
The susceptibility of a medium is related to its relative permittivity $\backslash ,\; \backslash varepsilon\_r$ by
So in the case of a vacuum,
The electric displacement D is related to the polarization density P by
\mathbf{D} \ = \ \varepsilon_0\mathbf{E} + \mathbf{P} \ = \ \varepsilon_0 (1+\chi_e) \mathbf{E} \ = \ \varepsilon_r \varepsilon_0 \mathbf{E}.
In general, a material cannot polarize instantaneously in response to an applied field, and so the more general formulation as a function of time is
That is, the polarization is a convolution of the electric field at previous times with timedependent susceptibility given by $\backslash chi\_e(\backslash Delta\; t)$. The upper limit of this integral can be extended to infinity as well if one defines $\backslash chi\_e(\backslash Delta\; t)\; =\; 0$ for $\backslash Delta\; t\; <\; 0$. An instantaneous response corresponds to Dirac delta function susceptibility $\backslash chi\_e(\backslash Delta\; t)\; =\; \backslash chi\_e\; \backslash delta(\backslash Delta\; t)$.
It is more convenient in a linear system to take the Fourier transform and write this relationship as a function of frequency. Due to the convolution theorem, the integral becomes a simple product,
This frequency dependence of the susceptibility leads to frequency dependence of the permittivity. The shape of the susceptibility with respect to frequency characterizes the dispersion properties of the material.
Moreover, the fact that the polarization can only depend on the electric field at previous times (i.e. $\backslash chi\_e(\backslash Delta\; t)\; =\; 0$ for $\backslash Delta\; t\; <\; 0$), a consequence of causality, imposes Kramers–Kronig constraints on the susceptibility $\backslash chi\_e(0)$.
