Inelectromagnetism, theabsolute permittivity, often simply calledpermittivity and denoted by the Greek letterε (epsilon), is a measure of the electricpolarizability of adielectric material. A material with high permittivity polarizes more in response to an applied electric field than a material with low permittivity, thereby storing more energy in the material. Inelectrostatics, the permittivity plays an important role in determining thecapacitance of acapacitor.
In the simplest case, theelectric displacement fieldD resulting from an appliedelectric fieldE is
More generally, the permittivity is a thermodynamicfunction of state.[1] It can depend on thefrequency,magnitude, anddirection of the applied field. TheSI unit for permittivity isfarad permeter (F/m).
The permittivity is often represented by therelative permittivityεr which is the ratio of the absolute permittivityε and thevacuum permittivityε0[2]
This dimensionless quantity is also often and ambiguously referred to as thepermittivity. Another common term encountered for both absolute and relative permittivity is thedielectric constant which has been deprecated in physics and engineering[3] as well as in chemistry.[4]
By definition, a perfect vacuum has a relative permittivity of exactly 1 whereas atstandard temperature and pressure, air has a relative permittivity ofεr air ≡κair ≈ 1.0006 .
Relative permittivity is directly related toelectric susceptibility (χ) by
otherwise written as
The term "permittivity" was introduced in the 1880s byOliver Heaviside to complementThomson's (1872) "permeability".[5][irrelevant citation] Formerly written asp, the designation withε has been in common use since the 1950s.
The SI unit of permittivity isfarad per meter (F/m or F·m−1).[6]
Inelectromagnetism, theelectric displacement fieldD represents the distribution of electric charges in a given medium resulting from the presence of an electric fieldE. This distribution includes charge migration and electricdipole reorientation. Its relation to permittivity in the very simple case oflinear, homogeneous,isotropic materials with"instantaneous" response to changes in electric field is:
where the permittivityε is ascalar. If the medium isanisotropic, the permittivity is asecond 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 anonlinear 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 infarads per meter (F/m or A2·s4·kg−1·m−3). The displacement fieldD is measured in units ofcoulombs persquare meter (C/m2), while the electric fieldE is measured involts per meter (V/m).D andE describe the interaction between charged objects.D is related to thecharge densities associated with this interaction, whileE is related to theforces andpotential differences.
The vacuum permittivityεo (also calledpermittivity of free space or theelectric constant) is the ratioD/E infree space. It also appears in theCoulomb force constant,
where
The constantsc andµ0 were both defined in SI units to have exact numerical values until the2019 revision of the SI. Therefore, until that date,ε0 could be also stated exactly as a fraction,even if the result was irrational (because the fraction containedπ).[9] In contrast, the ampere was a measured quantity before 2019, but since then the ampere is now exactly defined and it isμ0 that is an experimentally measured quantity (with consequent uncertainty) and therefore so is the new 2019 definition ofε0 (c remains exactly defined before and since 2019).
The linear permittivity of a homogeneous material is usually given relative to that of free space, as a relative permittivityεr (also calleddielectric constant, although this term is deprecated and sometimes only refers to the static, zero-frequency relative permittivity). In an anisotropic material, the relative permittivity may be a tensor, causingbirefringence. The actual permittivity is then calculated by multiplying the relative permittivity byεo:
whereχ (frequently writtenχe) is the electric susceptibility of the material.
The susceptibility is defined as the constant of proportionality (which may be atensor) relating anelectric fieldE to the induceddielectricpolarization densityP such that
whereεo is theelectric permittivity of free space.
The susceptibility of a medium is related to its relative permittivityεr by
So in the case of a vacuum,
The susceptibility is also related to thepolarizability of individual particles in the medium by theClausius-Mossotti relation.
Theelectric displacementD is related to the polarization densityP by
The permittivityε andpermeabilityµ of a medium together determine thephase velocityv =c/n ofelectromagnetic radiation through that medium:
The capacitance of a capacitor is based on its design and architecture, meaning it will not change with charging and discharging. The formula for capacitance in aparallel plate capacitor is written as
where is the area of one plate, is the distance between the plates, and is the permittivity of the medium between the two plates. For a capacitor with relative permittivity, it can be said that
Permittivity is connected to electric flux (and by extension electric field) throughGauss's law. Gauss's law states that for a closedGaussian surface,S,
where is the net electric flux passing through the surface, is the charge enclosed in the Gaussian surface, is the electric field vector at a given point on the surface, and is a differential area vector on the Gaussian surface.
If the Gaussian surface uniformly encloses an insulated, symmetrical charge arrangement, the formula can be simplified to
where represents the angle between the electric field lines and the normal (perpendicular) toS.
If all of the electric field lines cross the surface at 90°, the formula can be further simplified to
Because the surface area of a sphere is the electric field a distance away from a uniform, spherical charge arrangement is
This formula applies to the electric field due to a point charge, outside of a conducting sphere or shell, outside of a uniformly charged insulating sphere, or between the plates of a spherical capacitor.
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 aconvolution of the electric field at previous times with time-dependent susceptibility given byχ(Δt). The upper limit of this integral can be extended to infinity as well if one definesχ(Δt) = 0 forΔt < 0. An instantaneous response would correspond to aDirac delta function susceptibilityχ(Δt) =χδ(Δt).
It is convenient to take theFourier transform with respect to time and write this relationship as a function of frequency. Because of theconvolution 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 thedispersion properties of the material.
Moreover, the fact that the polarization can only depend on the electric field at previous times (i.e. effectivelyχ(Δt) = 0 forΔt < 0), a consequence ofcausality, imposesKramers–Kronig constraints on the susceptibilityχ(0).
As opposed to the response of a vacuum, the response of normal materials to external fields generally depends on thefrequency of the field. This frequency dependence reflects the fact that a material's polarization does not change instantaneously when an electric field is applied. The response must always becausal (arising after the applied field), which can be represented by a phase difference. For this reason, permittivity is often treated as a complex function of the(angular) frequencyω of the applied field:
(sincecomplex numbers allow specification of magnitude and phase). The definition of permittivity therefore becomes
where
The response of a medium to static electric fields is described by the low-frequency limit of permittivity, also called the static permittivityεs (alsoεDC):
At the high-frequency limit (meaning optical frequencies), the complex permittivity is commonly referred to asε∞ (or sometimesεopt[11]). At theplasma frequency and below, dielectrics behave as ideal metals, with electron gas behavior. The static permittivity is a good approximation for alternating fields of low frequencies, and as the frequency increases a measurable phase differenceδ emerges betweenD andE. The frequency at which the phase shift becomes noticeable depends on temperature and the details of the medium. For moderate field strength (Eo),D andE 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 choice of sign for time-dependence,e−iωt, dictates the sign convention for the imaginary part of permittivity.
The complex permittivity is usually a complicated function of frequencyω, since it is a superimposed description ofdispersion phenomena occurring at multiple frequencies. The dielectric functionε(ω) must have poles only for frequencies with positive imaginary parts, and therefore satisfies theKramers–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,ε″, 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 theeigenvalues 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 ofphoton absorption, which is directly related to the imaginary part of the optical dielectric functionε(ω). The optical dielectric function is given by the fundamental expression:[12]
In this expression,Wc,v(E) represents the product of theBrillouin zone-averaged transition probability at the energyE with the jointdensity of states,[13][14]Jc,v(E);φ is a broadening function, representing the role of scattering in smearing out the energy levels.[15]In general, the broadening is intermediate betweenLorentzian andGaussian;[16][17]for an alloy it is somewhat closer to Gaussian because of strong scattering from statistical fluctuations in the local composition on a nanometer scale.
According to theDrude model of magnetized plasma, a more general expression which takes into account the interaction of the carriers with an alternating electric field at millimeter and microwave frequencies in an axially magnetized semiconductor requires the expression of the permittivity as a non-diagonal tensor:[18]
Ifε2 vanishes, then the tensor is diagonal but not proportional to the identity and the medium is said to be a uniaxial medium, which has similar properties to auniaxial crystal.
| εr″/εr′ | Currentconduction | Fieldpropagation |
|---|---|---|
| 0 | perfect dielectric lossless medium | |
| low-conductivity material poor conductor | low-loss medium good dielectric | |
| lossy conducting material | lossy propagation medium | |
| high-conductivity material good conductor | high-loss medium poor dielectric | |
| perfect conductor |
Materials can be classified according to their complex-valued permittivityε, upon comparison of its realε′ and imaginaryε″ components (or, equivalently,conductivity,σ, when accounted for in the latter). Aperfect conductor has infinite conductivity,σ = ∞, while aperfect dielectric is a material that has no conductivity at all,σ = 0; this latter case, of real-valued permittivity (or complex-valued permittivity with zero imaginary component) is also associated with the namelossless media.[19] Generally, when we consider the material to be alow-loss dielectric (although not exactly lossless), whereas is associated with agood conductor; such materials with non-negligible conductivity yield a large amount ofloss that inhibit the propagation of electromagnetic waves, thus are also said to belossy media. Those materials that do not fall under either limit are considered to be general media.
In the case of a lossy medium, i.e. when the conduction current is not negligible, the total current density flowing is:
where
Note that this is using the electrical engineering convention of thecomplex conjugate ambiguity; the physics/chemistry convention involves the complex conjugate of these equations.
The size of thedisplacement current is dependent on thefrequencyω of the applied fieldE; there is no displacement current in a constant field.
In this formalism, the complex permittivity is defined as:[20][21]
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 non-linear 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 calledsoakage orbattery action. For some dielectrics, such as many polymer films, the resulting voltage may be less than 1–2% of the original voltage. However, it can be as much as 15–25% in the case ofelectrolytic capacitors orsupercapacitors.
In terms ofquantum mechanics, permittivity is explained byatomic andmolecular interactions.
At low frequencies, molecules in polar dielectrics are polarized by an applied electric field, which induces periodic rotations. For example, at themicrowave frequency, the microwave field causes the periodic rotation of water molecules, sufficient to breakhydrogen bonds. The field does work against the bonds and the energy is absorbed by the material asheat. 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 farultraviolet (UV) frequency. 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).
At high frequencies (such as UV and above), molecules cannot relax, and the energy is purely absorbed by atoms, excitingelectron energy levels. Thus, these frequencies are classified asionizing radiation.
While carrying out a completeab initio (that is, first-principles) 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. TheDebye model and theLorentz model use a first-order and second-order (respectively) lumped system parameter linear representation (such as an RC and an LRC resonant circuit).
The relative permittivity 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 ofdielectric spectroscopy, covering nearly 21 orders of magnitude from 10−6 to 1015hertz. Also, by usingcryostats and ovens, the dielectric properties of a medium can be characterized over an array of temperatures. In order to study systems for such diverse excitation fields, a number of measurement setups are used, each adequate for a special frequency range.
Various microwave measurement techniques are outlined in Chenet al.[22] Typical errors for theHakki–Coleman method employing a puck of material between conducting planes are about 0.3%.[23]
At infrared and optical frequencies, a common technique isellipsometry.Dual polarisation interferometry is also used to measure the complex refractive index for very thin films at optical frequencies.
For the 3D measurement of dielectric tensors at optical frequency, Dielectric tensor tomography can be used.[24]
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