Permittivity

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A dielectric medium showing orientation of charged particles creating polarization effects. Such a medium can have a lower ratio of electric flux to charge (more permittivity) than empty space Diel.png
A dielectric medium showing orientation of charged particles creating polarization effects. Such a medium can have a lower ratio of electric flux to charge (more permittivity) than empty space

In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ε (epsilon), is a measure of the electric polarizability of a dielectric. 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 electric field. In electrostatics, the permittivity plays an important role in determining the capacitance of a capacitor.

Contents

In the simplest case, the electric displacement field resulting from an applied electric field is

More generally, the permittivity is a thermodynamic function of state [1] . It can depend on the frequency, magnitude, and direction of the applied field. The SI unit for permittivity is farad per meter (F/m).

The permittivity is often represented by the relative permittivity which is the ratio of the absolute permittivity and the vacuum permittivity

.

This dimensionless quantity is also often and ambiguously referred to as the permittivity. Another common term encountered for both absolute and relative permittivity is the dielectric constant which has been deprecated in physics and engineering [2] as well as in chemistry. [3]

By definition, a perfect vacuum has a relative permittivity of exactly 1 whereas at STP, air has a relative permittivity of κair = 1.0006.

Relative permittivity is directly related to electric susceptibility (χ) by

otherwise written as

Units

The standard SI unit for permittivity is farad per meter (F/m or F·m−1). [4]

Explanation

In electromagnetism, the electric displacement field D represents how an electric field E influences the organization of electric 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 A2·s4·kg−1·m−3). The displacement field D is measured in units of coulombs per square meter (C/m2), 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.

Vacuum permittivity

The vacuum permittivity ε0 (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,

Its value is [5]

where

The constants c0 and μ0 were defined in SI units to have exact numerical values until redefinition of SI units in 2019. [7] (The approximation in the second value of ε0 above stems from π being an irrational number.)

Relative permittivity

The linear permittivity of a homogeneous material is usually given relative to that of free space, as a relative permittivity εr (also called dielectric 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, causing birefringence. The actual permittivity is then calculated by multiplying the relative permittivity by ε0:

where χ (frequently written χe) is the electric susceptibility of the material.

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 ε0 is the electric permittivity of free space.

The susceptibility of a medium is related to its relative permittivity εr by

So in the case of a vacuum,

The susceptibility is also related to the polarizability of individual particles in the medium by the Clausius-Mossotti 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 v = c/n of electromagnetic radiation through that medium:

Practical applications

Determining capacitance

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 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

Gauss's law

Permittivity is connected to electric flux (and by extension electric field) through Gauss's law. Gauss's law states that for a closed Gaussian 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 vector and the area vector.

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

where is Coulomb's constant (). 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.

Dispersion and causality

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 time-dependent susceptibility given by χt). The upper limit of this integral can be extended to infinity as well if one defines χt) = 0 for Δt < 0. An instantaneous response would correspond to a Dirac delta function susceptibility χt) = χδt).

It is convenient to take the Fourier transform with respect to time and write this relationship as a function of frequency. Because of 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. effectively χt) = 0 for Δt < 0), a consequence of causality, imposes Kramers–Kronig constraints on the susceptibility χ(0).

Complex permittivity

A dielectric permittivity spectrum over a wide range of frequencies. e' and e'' denote the real and the imaginary part of the permittivity, respectively. Various processes are labeled on the image: ionic and dipolar relaxation, and atomic and electronic resonances at higher energies. Dielectric responses.svg
A dielectric permittivity spectrum over a wide range of frequencies. ε and ε denote the real and the imaginary part of the permittivity, respectively. Various processes are labeled on the image: ionic and dipolar relaxation, and atomic and electronic resonances at higher energies.

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 change instantaneously when an electric field is applied. 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 of the (angular) frequency ω of the applied field:

(since complex 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 permittivity εs (also εDC):

At the high-frequency limit (meaning optical frequencies), the complex permittivity is commonly referred to as ε (or sometimes εopt [9] ). At the plasma 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 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 field strength (E0), 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 choice of sign for time-dependence, eiωt, 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 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 frequency-independent or by model functions.

At a given frequency, the imaginary part, 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 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: [10]

In this expression, Wc,v(E) represents the product of the Brillouin zone-averaged transition probability at the energy E with the joint density of states, [11] [12] Jc,v(E); φ is a broadening function, representing the role of scattering in smearing out the energy levels. [13] In general, the broadening is intermediate between Lorentzian and Gaussian; [14] [15] for an alloy it is somewhat closer to Gaussian because of strong scattering from statistical fluctuations in the local composition on a nanometer scale.

Tensorial permittivity

According to the Drude 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. [16] (see also Electro-gyration).

If ε2 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 a uniaxial crystal.

Classification of materials

Classification of materials based on permittivity
εr / εr Current conduction Field propagation
0 perfect dielectric
lossless medium
≪ 1low-conductivity material
poor conductor
low-loss medium
good dielectric
≈ 1lossy conducting materiallossy propagation medium
≫ 1high-conductivity material
good conductor
high-loss medium
poor dielectric
perfect conductor

Materials can be classified according to their complex-valued permittivity ε, upon comparison of its real ε and imaginary ε components (or, equivalently, conductivity, σ, when accounted for in the latter). A perfect conductor has infinite conductivity, σ = ∞, while a perfect 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 name lossless media. [17] Generally, when σ/ωε ≪ 1 we consider the material to be a low-loss dielectric (although not exactly lossless), whereas σ/ωε ≫ 1 is associated with a good conductor; such materials with non-negligible conductivity yield a large amount of loss that inhibit the propagation of electromagnetic waves, thus are also said to be lossy media. Those materials that do not fall under either limit are considered to be general media.

Lossy medium

In the case of a 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: [18] [19]

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 called soakage or battery 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 of electrolytic capacitors or supercapacitors.

Quantum-mechanical interpretation

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. 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, exciting electron energy levels. Thus, these frequencies are classified as ionizing radiation.

While carrying out a complete ab 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. The Debye model and the Lorentz model use a first-order and second-order (respectively) lumped system parameter linear representation (such as an RC and an LRC resonant circuit).

Measurement

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 of dielectric spectroscopy, covering nearly 21 orders of magnitude from 10−6 to 1015 hertz. 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 excitation 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.. [20] Typical errors for the Hakki-Coleman method employing a puck of material between conducting planes are about 0.3%. [21]

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.

See also

Notes

  1. Current practice of standards organizations such as NIST and BIPM is to use c0, rather than c, to denote the speed of light in a vacuum according to ISO 31. In the original Recommendation of 1983, the symbol c was used for this purpose. [6]

Related Research Articles

Nonlinear optics branch of physics

Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typically observed only at very high light intensities (values of atomic electric fields, typically 108 V/m) such as those provided by lasers. Above the Schwinger limit, the vacuum itself is expected to become nonlinear. In nonlinear optics, the superposition principle no longer holds.

Dielectric electrically poorly conducting or non-conducting, non-metallic substance of which charge carriers are generally not free to move

A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material as they do in an electrical conductor but only slightly shift from their average equilibrium positions causing dielectric polarization. Because of dielectric polarization, positive charges are displaced in the direction of the field and negative charges shift in the direction opposite to the field. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axes align to the field.

The wave impedance of an electromagnetic wave is the ratio of the transverse components of the electric and magnetic fields. For a transverse-electric-magnetic (TEM) plane wave traveling through a homogeneous medium, the wave impedance is everywhere equal to the intrinsic impedance of the medium. In particular, for a plane wave travelling through empty space, the wave impedance is equal to the impedance of free space. The symbol Z is used to represent it and it is expressed in units of ohms. The symbol η (eta) may be used instead of Z for wave impedance to avoid confusion with electrical impedance.

Poynting vector Measure of directional electromagnetic energy flux

In physics, the Poynting vector represents the directional energy flux of an electromagnetic field. The SI unit of the Poynting vector is the watt per square metre (W/m2). It is named after its discoverer John Henry Poynting who first derived it in 1884. Oliver Heaviside also discovered it independently.

Relative permittivity ratio of permittivity to the electric constant

The relative permittivity of a material is its (absolute) permittivity expressed as a ratio relative to the vacuum permittivity.

Gausss law Foundational law of electromagnetism

In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. The surface under consideration may be a closed one enclosing a volume such as a spherical surface.

Crystal optics is the branch of optics that describes the behaviour of light in anisotropic media, that is, media in which light behaves differently depending on which direction the light is propagating. The index of refraction depends on both composition and crystal structure and can be calculated using the Gladstone–Dale relation. Crystals are often naturally anisotropic, and in some media it is possible to induce anisotropy by applying an external electric field.

In physics, screening is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases, electrolytes, and charge carriers in electronic conductors . In a fluid, with a given permittivity ε, composed of electrically charged constituent particles, each pair of particles interact through the Coulomb force as

In physics, the dissipation factor (DF) is a measure of loss-rate of energy of a mode of oscillation in a dissipative system. It is the reciprocal of quality factor, which represents the "quality" or durability of oscillation.

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Polarization density physical quantity

In classical electromagnetism, polarization density is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is said to be polarized. The electric dipole moment induced per unit volume of the dielectric material is called the electric polarization of the dielectric.

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The optical conductivity is a material property, which links the current density to the electric field for general frequencies. In this sense, this linear response function is a generalization of the electrical conductivity, which is usually considered in the static limit, i.e., for a time-independent electric field. While the static electrical conductivity is vanishingly small in insulators, the optical conductivity always remains finite in some frequency intervals ; the total optical weight can be inferred from sum rules. The optical conductivity is closely related to the dielectric function, the generalization of the dielectric constant to arbitrary frequencies.

Surface plasmon polariton

Surface plasmon polaritons (SPPs) are electromagnetic waves that travel along a metal–dielectric or metal–air interface, practically in the infrared or visible-frequency. The term "surface plasmon polariton" explains that the wave involves both charge motion in the metal and electromagnetic waves in the air or dielectric ("polariton").

In optics, Miller's rule is an empirical rule which gives an estimate of the order of magnitude of the nonlinear coefficient.

In optics, the Ewald–Oseen extinction theorem, sometimes referred to as just "extinction theorem", is a theorem that underlies the common understanding of scattering. It is named after Paul Peter Ewald and Carl Wilhelm Oseen, who proved the theorem in crystalline and isotropic media, respectively, in 1916 and 1915. Originally, the theorem applied to scattering by an isotropic dielectric objects in free space. The scope of the theorem was greatly extended to encompass a wide variety of bianisotropic media.

References

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Further reading