Permittivity

Overview

Electromagnetism

Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

,

**absolute permittivity**is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field

Electric field

In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

affects, and is affected by, a dielectric

Dielectric

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 in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...

medium. The permittivity of a medium describes how much electric field (more correctly, flux) is 'generated' per unit charge. Less electric flux exists in a medium with a high permittivity (per unit charge) due to polarization effects.

Unanswered Questions

Encyclopedia

In electromagnetism

,

affects, and is affected by, a dielectric

medium. The permittivity of a medium describes how much electric field (more correctly, flux) is 'generated' per unit charge. Less electric flux exists in a medium with a high permittivity (per unit charge) due to polarization effects. Permittivity is directly related to electric susceptibility

, which is a measure of how easily a dielectric polarizes

in response to an electric field

. Thus, permittivity relates to a material's ability to transmit (or "permit") an electric field.

In SI

units, permittivity ε is measured in farad

s per meter (F/m); electric susceptibility χ is dimensionless. They are related to each other through

where ε

, the electric displacement field

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

s per meter (V/m).

. It also appears in the Coulomb force constant 1/4πε

Its value is

where

Constants c

. (The approximation in the second value of ε

.)

, although this 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 ε

where

The susceptibility is defined as the constant of proportionality (which may be a tensor

) relating an electric fieldIn physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

polarization density

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 Clausius-Mossotti relation

.

The electric displacement

The permittivity ε and permeability

µ of a medium together determine the phase velocity

v = c/n of electromagnetic radiation

through that medium:

That is, the polarization is a convolution

of the electric field at previous times with time-dependent susceptibility given by . The upper limit of this integral can be extended to infinity as well if one defines for . An instantaneous response corresponds to Dirac delta function

susceptibility .

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. for ), a consequence of causality

, imposes Kramers–Kronig constraints on the susceptibility .

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

The response of a medium to static electric fields is described by the low-frequency limit of permittivity, also called the static permittivity ε

):

At the high-frequency limit, the complex permittivity is commonly referred to as ε

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

It is important to realize that the choice of sign for time-dependence, , 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 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:

In this expression, W

-averaged transition probability at the energy E with the joint density of states

, J

; for an alloy it is somewhat closer to Gaussian because of strong scattering from statistical fluctuations in the local composition on a nanometer scale.

.

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:

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 capacitor

s or supercapacitor

s.

, permittivity is explained by atom

ic 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 bond

s. 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 1st-order and 2nd-order (respectively) lumped system parameter linear representation (such as an RC and an LRC resonant circuit).

, covering nearly 21 orders of magnitude from 10

. 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 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.. Typical errors for the Hakki-Coleman method employing a puck of material between conducting planes are about 0.3%.

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.

Electromagnetism

Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

,

**absolute permittivity**is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric fieldElectric field

In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

affects, and is affected by, a dielectric

Dielectric

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 in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...

medium. The permittivity of a medium describes how much electric field (more correctly, flux) is 'generated' per unit charge. Less electric flux exists in a medium with a high permittivity (per unit charge) due to polarization effects. Permittivity is directly related to electric susceptibility

Electric susceptibility

In electromagnetism, the electric susceptibility \chi_e is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applied electric field...

, which is a measure of how easily a dielectric polarizes

Polarization density

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...

in response to an electric field

Electric field

In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

. Thus, permittivity relates to a material's ability to transmit (or "permit") an electric field.

In SI

Si

Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

units, permittivity ε is measured in farad

Farad

The farad is the SI unit of capacitance. The unit is named after the English physicist Michael Faraday.- Definition :A farad is the charge in coulombs which a capacitor will accept for the potential across it to change 1 volt. A coulomb is 1 ampere second...

s 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 = 8.85… × 10^{−12}F/m is the vacuum permittivity.## Explanation

In electromagnetismElectromagnetism

Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

, 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 dipoleDipole

In physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...

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

Scalar (physics)

In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...

. If the medium is anisotropic, the permittivity is a second rank tensor

Tensor

Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

.

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

Nonlinear optics

Nonlinear optics is the branch of optics that describes the behavior of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light...

, 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 voltVolt

The volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...

s 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 ε_{0}(also called**permittivity of free space**or the**electric constant**) is the ratio**D**/**E**in free spaceVacuum

In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...

. It also appears in the Coulomb force constant 1/4πε

_{0}.Its value is

where

- c
_{0}is the speed of lightSpeed of lightThe speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...

in free space, - µ
_{0}is the vacuum permeabilityVacuum permeabilityThe physical constant μ0, commonly called the vacuum permeability, permeability of free space, or magnetic constant is an ideal, physical constant, which is the value of magnetic permeability in a classical vacuum...

.

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 ampereAmpere

The ampere , often shortened to amp, is the SI unit of electric current and is one of the seven SI base units. It is named after André-Marie Ampère , French mathematician and physicist, considered the father of electrodynamics...

. (The approximation in the second value of ε

_{0}above stems from π being an irrational numberIrrational number

In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

.)

## Relative permittivity

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 constantDielectric constant

The relative permittivity of a material under given conditions reflects the extent to which it concentrates electrostatic lines of flux. In technical terms, it is the ratio of the amount of electrical energy stored in a material by an applied voltage, relative to that stored in a vacuum...

, although this sometimes only refers to the static, zero-frequency relative permittivity). In an anisotropic material, the relative permittivity may be a tensor, causing birefringence

Birefringence

Birefringence, or double refraction, is the decomposition of a ray of light into two rays when it passes through certain anisotropic materials, such as crystals of calcite or boron nitride. The effect was first described by the Danish scientist Rasmus Bartholin in 1669, who saw it in calcite...

. The actual permittivity is then calculated by multiplying the relative permittivity by ε

_{0}:where

- χ (frequently written χ
_{e}) is the electric susceptibility of the material.

The susceptibility is defined as the constant of proportionality (which may be a tensor

Tensor

Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

) relating an electric field

Electric field

**E**to the induced dielectricDielectric

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 in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...

polarization density

**P**such thatwhere 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

Polarizability

Polarizability is the measure of the change in a molecule's electron distribution in response to an applied electric field, which can also be induced by electric interactions with solvents or ionic reagents. It is a property of matter...

of individual particles in the medium by the Clausius-Mossotti relation

Clausius-Mossotti relation

The Clausius–Mossotti relation is named after the Italian physicist Ottaviano-Fabrizio Mossotti, whose 1850 book analyzed the relationship between the dielectric constants of two different media, and the German physicist Rudolf Clausius, who gave the formula explicitly in his 1879 book in the...

.

The electric displacement

**D**is related to the polarization density**P**byThe permittivity ε and permeability

Permeability (electromagnetism)

In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...

µ of a medium together determine the phase velocity

Phase velocity

The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...

v = c/n of electromagnetic radiation

Electromagnetic radiation

Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

through that medium:

## 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 isThat is, the polarization is a convolution

Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...

of the electric field at previous times with time-dependent susceptibility given by . The upper limit of this integral can be extended to infinity as well if one defines for . An instantaneous response corresponds to Dirac delta function

Dirac delta function

The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

susceptibility .

It is more convenient in a linear system to take the Fourier transform

Continuous Fourier transform

The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, known as a frequency spectrum. For instance, the transform of a musical chord made up of pure notes is a mathematical representation of the amplitudes of the individual notes that make...

and write this relationship as a function of frequency. Due to the convolution theorem

Convolution theorem

In mathematics, the convolution theorem states that under suitableconditions the Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain equals point-wise multiplication in the other domain...

, 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

Dispersion (optics)

In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency, or alternatively when the group velocity depends on the frequency.Media having such a property are termed dispersive media...

properties of the material.

Moreover, the fact that the polarization can only depend on the electric field at previous times (i.e. for ), a consequence of causality

Causality

Causality is the relationship between an event and a second event , where the second event is understood as a consequence of the first....

, imposes Kramers–Kronig constraints on the susceptibility .

### Complex permittivity

As opposed to the response of a vacuum, the response of normal materials to external fields generally depends on the frequencyFrequency

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of 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

Angular frequency

In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...

of the applied field ω, . The definition of permittivity therefore becomes

where

- D
_{0}and E_{0}are the amplitudes of the displacement and electrical fields, respectively, - i is the imaginary unitImaginary unitIn mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...

, i^{ 2}= −1.

The response of a medium to static electric fields is described by the low-frequency limit of permittivity, also called the static permittivity ε

_{s}(also ε_{DC}):

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 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 fields strength (E_{0}),**D**and**E**remain proportional, andSince 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

- ε" is the imaginary part of the permittivity, which is related to the dissipation (or loss) of energy within the medium.
- ε' is the real part of the permittivity, which is related to the stored energy within the medium.

It is important to realize that the choice of sign for time-dependence, , 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

Dispersion (optics)

In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency, or alternatively when the group velocity depends on the frequency.Media having such a property are termed dispersive media...

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 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

Photon

In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

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:

In this expression, W

_{cv}(E) represents the product of the Brillouin zoneBrillouin zone

In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. The boundaries of this cell are given by planes related to points on the reciprocal lattice. It is found by the same method as for the Wigner–Seitz cell in the Bravais lattice...

-averaged transition probability at the energy E with the joint density of states

Density of states

In solid-state and condensed matter physics, the density of states of a system describes the number of states per interval of energy at each energy level that are available to be occupied by electrons. Unlike isolated systems, like atoms or molecules in gas phase, the density distributions are not...

, J

_{cv}(E); φ is a broadening function, representing the role of scattering in smearing out the energy levels. In general, the broadening is intermediate between Lorentzian and GaussianGAUSSIAN

Gaussian is a computational chemistry software program initially released in 1970 by John Pople and his research group at Carnegie-Mellon University as Gaussian 70. It has been continuously updated since then...

; for an alloy it is somewhat closer to Gaussian because of strong scattering from statistical fluctuations in the local composition on a nanometer scale.

### Classification of materials

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 low-loss 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 capacitorCapacitor

A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...

.

### Lossy medium

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

- σ is the conductivity of the medium;
- ε' is the real part of the permittivity. is the complex permittivity

The size of the displacement current

Displacement current

In electromagnetism, displacement current is a quantity that is defined in terms of the rate of change of electric displacement field. Displacement current has the units of electric current density, and it has an associated magnetic field just as actual currents do. However it is not an electric...

is dependent on the frequency

Frequency

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of 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:

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:

- First, are the relaxation effects associated with permanent and induced molecular dipolesDipoleIn physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...

. 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 viscosityViscosityViscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher 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.

- Second are the resonance effectsResonanceIn physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...

, which arise from the rotations or vibrations of atoms, ionIonAn ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge. The name was given by physicist Michael Faraday for the substances that allow a current to pass between electrodes in a...

s, or electronElectronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s. These processes are observed in the neighborhood of their characteristic absorption frequenciesAbsorption (electromagnetic radiation)In physics, absorption of electromagnetic radiation is the way by which the energy of a photon is taken up by matter, typically the electrons of an atom. Thus, the electromagnetic energy is transformed to other forms of energy for example, to heat. The absorption of light during wave propagation is...

.

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 capacitor

Electrolytic capacitor

An electrolytic capacitor is a type of capacitor that uses an electrolyte, an ionic conducting liquid, as one of its plates, to achieve a larger capacitance per unit volume than other types. They are often referred to in electronics usage simply as "electrolytics"...

s or supercapacitor

Supercapacitor

An electric double-layer capacitor , also known as supercapacitor, supercondenser, electrochemical double layer capacitor, or ultracapacitor, is an electrochemical capacitor with relatively high energy density. Their energy density is typically hundreds of times greater than conventional...

s.

### Quantum-mechanical interpretation

In terms of quantum mechanicsQuantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, permittivity is explained by atom

Atom

The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

ic and molecular

Molecule

A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...

interactions.

At low frequencies, molecules in polar dielectrics are polarized by an applied electric field, which induces periodic rotations. For example, at the microwave

Microwave

Microwaves, a subset of radio waves, have wavelengths ranging from as long as one meter to as short as one millimeter, or equivalently, with frequencies between 300 MHz and 300 GHz. This broad definition includes both UHF and EHF , and various sources use different boundaries...

frequency, the microwave field causes the periodic rotation of water molecules, sufficient to break hydrogen bond

Hydrogen bond

A hydrogen bond is the attractive interaction of a hydrogen atom with an electronegative atom, such as nitrogen, oxygen or fluorine, that comes from another molecule or chemical group. The hydrogen must be covalently bonded to another electronegative atom to create the bond...

s. The field does work against the bonds and the energy is absorbed by the material as heat

Heat

In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

. 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

Electron

The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

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

Ionizing radiation

Ionizing radiation is radiation composed of particles that individually have sufficient energy to remove an electron from an atom or molecule. This ionization produces free radicals, which are atoms or molecules containing unpaired electrons...

.

While carrying out a complete ab initio

Ab initio

ab initio is a Latin term used in English, meaning from the beginning.ab initio may also refer to:* Ab Initio , a leading ETL Tool Software Company in the field of Data Warehousing.* ab initio quantum chemistry methods...

(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 1st-order and 2nd-order (respectively) lumped system parameter linear representation (such as an RC and an LRC resonant circuit).

## Measurement

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 spectroscopyDielectric spectroscopy

Dielectric spectroscopy , and also known as electrochemical impedance spectroscopy, measures the dielectric properties of a medium as a function of frequency...

, covering nearly 21 orders of magnitude from 10

^{−6}to 10^{15}HzHertz

The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

. Also, by using cryostat

Cryostat

A cryostat is a device used to maintain cold cryogenic temperatures. Low temperatures may be maintained within a cryostat by using various refrigeration methods, most commonly using cryogenic fluid bath such as liquid helium. Hence it is usually assembled into a vessel, similar in construction...

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 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.. Typical errors for the Hakki-Coleman method employing a puck of material between conducting planes are about 0.3%.

- Low-frequency time domainTime domainTime domain is a term used to describe the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the case of continuous time, or at various...

measurements (10^{−6}-10^{3}Hz) - Low-frequency frequency domainFrequency domainIn electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....

measurements (10^{−5}-10^{6}Hz) - Reflective coaxial methods (10
^{6}-10^{10}Hz) - Transmission coaxial method (10
^{8}-10^{11}Hz) - Quasi-optical methods (10
^{9}-10^{10}Hz) - Fourier-transform methods (10
^{11}-10^{15}Hz)

At infrared and optical frequencies, a common technique is ellipsometry

Ellipsometry

Ellipsometry is an optical technique for the investigation of the dielectric properties of thin films....

. Dual polarisation interferometry

Dual Polarisation Interferometry

Dual polarization interferometry is an analytical technique that can probe molecular scale layers adsorbed to the surface of a waveguide by using the evanescent wave of a laser beam confined to the waveguide...

is also used to measure the complex refractive index for very thin films at optical frequencies.

## See also

- Density functional theoryDensity functional theoryDensity functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...
- Electric-field screening
- Green-Kubo relationsGreen-Kubo relationsThe Green–Kubo relations give the exact mathematical expression for transport coefficients in terms of integrals of time correlation functions.-Thermal and mechanical transport processes:...
- Green's function (many-body theory)Green's function (many-body theory)In many-body theory, the term Green's function is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators....
- Linear response functionLinear response functionA linear response function describes the input-output relationship of a signal transducer such as a radio turning electromagnetic waves into music or a neuron turning synaptic input into a response...
- Rotational Brownian motionRotational Brownian motionRotational Brownian motion is the random change in the orientation of a polar molecule due to collisions with other molecules. It is an important element of theories of dielectric materials....
- Electromagnetic permeabilityPermeability (electromagnetism)In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...

## Further reading

- Theory of Electric Polarization: Dielectric Polarization, C.J.F. Böttcher, ISBN 0-444-41579-3
- Dielectrics and Waves edited by von Hippel, Arthur R., ISBN 0-89006-803-8
- Dielectric Materials and Applications edited by Arthur von Hippel, ISBN 0-89006-805-4.

## External links

- Electromagnetism, a chapter from an online textbook
- What's all this trapped charge stuff . . ., A different approach to some capacitor problems