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In
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (
epsilon Epsilon (, ; uppercase , lowercase or lunate ; el, έψιλον) is the fifth letter of the Greek alphabet, corresponding phonetically to a mid front unrounded vowel or . In the system of Greek numerals it also has the value five. It was de ...
), is a measure of the electric
polarizability Polarizability usually refers to the tendency of matter, when subjected to an electric field, to acquire an electric dipole moment in proportion to that applied field. It is a property of all matter, considering that matter is made up of elementar ...
of a
dielectric In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mater ...
. 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. In
electrostatics Electrostatics is a branch of physics that studies electric charges at rest ( static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amb ...
, the permittivity plays an important role in determining the
capacitance Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
of a
capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passivity (engineering), passive electronic component with two termi ...
. In the simplest case, the
electric displacement field In physics, the electric displacement field (denoted by D) or electric induction is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in ...
D resulting from an applied
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field f ...
E is :\mathbf = \varepsilon \mathbf. More generally, the permittivity is a thermodynamic
function of state In the thermodynamics of equilibrium, a state function, function of state, or point function for a thermodynamic system is a mathematical function relating several state variables or state quantities (that describe equilibrium states of a syste ...
. It can depend on the
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz The hertz ...
,
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
, and direction of the applied field. The SI unit for permittivity is
farad The farad (symbol: F) is the unit of electrical capacitance, the ability of a body to store an electrical charge, in the International System of Units (SI). It is named after the English physicist Michael Faraday (1791–1867). In SI base unit ...
per
meter The metre ( British spelling) or meter ( American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pre ...
(F/m). The permittivity is often represented by the
relative permittivity The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insulat ...
''ε''r which is the ratio of the absolute permittivity ''ε'' and the
vacuum permittivity Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
''ε''0 :\kappa = \varepsilon_\mathrm = \frac. 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 as well as in chemistry. By definition, a perfect vacuum has a relative permittivity of exactly 1 whereas at
standard temperature and pressure Standard temperature and pressure (STP) are standard sets of conditions for experimental measurements to be established to allow comparisons to be made between different sets of data. The most used standards are those of the International Union o ...
, air has a relative permittivity of ''κ''air ≈ 1.0006. Relative permittivity is directly related to
electric susceptibility In electricity (electromagnetism), the electric susceptibility (\chi_; Latin: ''susceptibilis'' "receptive") is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applied ...
(''χ'') by :\chi = \kappa - 1 otherwise written as :\varepsilon = \varepsilon_\mathrm \varepsilon_0 = (1+\chi)\varepsilon_0 The term "permittivity" was introduced in the 1880s by
Oliver Heaviside Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vec ...
to complement Thomson's (1872) " permeability". Formerly written as ''p'', the designation with ''ε'' has been in common use since the 1950s.


Units

The standard SI unit for permittivity is
farad The farad (symbol: F) is the unit of electrical capacitance, the ability of a body to store an electrical charge, in the International System of Units (SI). It is named after the English physicist Michael Faraday (1791–1867). In SI base unit ...
per meter (F/m or F·m−1). :\frac = \frac = \frac = \frac= \frac


Explanation

In
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
, the
electric displacement field In physics, the electric displacement field (denoted by D) or electric induction is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in ...
represents the distribution of electric charges in a given medium resulting from the presence of an electric field . This distribution includes charge migration and electric
dipole In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways: *An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system i ...
reorientation. Its relation to permittivity in the very simple case of ''linear, homogeneous,
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
'' materials with ''"instantaneous" response'' to changes in electric field is: :\mathbf=\varepsilon \mathbf where the permittivity is a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...
. If the medium is
anisotropic Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
, the permittivity is a second rank
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
. 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 The farad (symbol: F) is the unit of electrical capacitance, the ability of a body to store an electrical charge, in the International System of Units (SI). It is named after the English physicist Michael Faraday (1791–1867). In SI base uni ...
per meter (F/m or A2·s4·kg−1·m−3). The displacement field is measured in units of coulombs per
square meter The square metre ( international spelling as used by the International Bureau of Weights and Measures) or square meter ( American spelling) is the unit of area in the International System of Units (SI) with symbol m2. It is the area of a square ...
(C/m2), while the electric field is measured in
volt The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827). Defin ...
s per meter (V/m). and describe the interaction between charged objects. is related to the ''charge densities'' associated with this interaction, while is related to the ''forces'' and ''potential differences''.


Vacuum permittivity

The vacuum permittivity (also called permittivity of free space or the electric constant) is the ratio in
free space A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
. It also appears in the Coulomb force constant, :k_\text = \frac Its value is :\varepsilon_0 \ \stackrel\ \frac \approx 8.854\,187\,8128(13)\times 10^\text where * is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit for ...
in free space, * is the
vacuum permeability The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, ...
. The constants and were both defined in SI units to have exact numerical values until the
2019 redefinition of the SI base units In 2019, four of the seven SI base units specified in the International System of Quantities were redefined in terms of natural physical constants, rather than human artifacts such as the standard kilogram. Effective 20 May 2019, the 144th ...
. Therefore, until that date, could be also stated exactly as a fraction, \tfrac = \tfrac\text even if the result was irrational (because the fraction contained ). In contrast, the ampere was a measured quantity before 2019, but since then the ampere is now exactly defined and it is that is an experimentally measured quantity (with consequent uncertainty) and therefore so is the new 2019 definition of ( remains exactly defined before and since 2019).


Relative permittivity

The linear permittivity of a homogeneous material is usually given relative to that of free space, as a relative permittivity (also called
dielectric constant The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insulat ...
, 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 Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefringe ...
. The actual permittivity is then calculated by multiplying the relative permittivity by : :\varepsilon = \varepsilon_\mathrm \varepsilon_0 = (1+\chi)\varepsilon_0, where (frequently written ) is the electric susceptibility of the material. The susceptibility is defined as the constant of proportionality (which may be a
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
) relating an
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field f ...
to the induced
dielectric In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mater ...
polarization density In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is ...
such that :\mathbf = \varepsilon_0\chi\mathbf, where is the electric permittivity of free space. The susceptibility of a medium is related to its relative permittivity by :\chi = \varepsilon_\mathrm - 1. So in the case of a vacuum, :\chi = 0. The susceptibility is also related to the
polarizability Polarizability usually refers to the tendency of matter, when subjected to an electric field, to acquire an electric dipole moment in proportion to that applied field. It is a property of all matter, considering that matter is made up of elementar ...
of individual particles in the medium by the Clausius-Mossotti relation. The
electric displacement In physics, the electric displacement field (denoted by D) or electric induction is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in ...
is related to the polarization density by :\mathbf = \varepsilon_0\mathbf + \mathbf = \varepsilon_0 (1+\chi) \mathbf = \varepsilon_\mathrm \varepsilon_0 \mathbf. The permittivity and permeability of a medium together determine the
phase velocity The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
of
electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) ligh ...
through that medium: :\varepsilon \mu = \frac.


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 in a
parallel plate capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of a c ...
is written as :C = \varepsilon \ \frac where A is the area of one plate, d is the distance between the plates, and \varepsilon is the permittivity of the medium between the two plates. For a capacitor with relative permittivity \kappa, it can be said that :C = \kappa \ \varepsilon_0 \frac


Gauss's law

Permittivity is connected to electric flux (and by extension electric field) through
Gauss's law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it stat ...
. Gauss's law states that for a closed
Gaussian surface A Gaussian surface is a closed surface in three-dimensional space through which the flux of a vector field is calculated; usually the gravitational field, electric field, or magnetic field. It is an arbitrary closed surface (the boundary of a 3- ...
, :\Phi_E = \frac = \oint_S \mathbf \cdot \mathrm \mathbf where \Phi_E is the net electric flux passing through the surface, Q_\text is the charge enclosed in the Gaussian surface, \mathbf is the electric field vector at a given point on the surface, and \mathrm \mathbf 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 :EA \cos(\theta) = \frac where \theta represents the angle between the electric field lines and the normal (perpendicular) to . If all of the electric field lines cross the surface at 90°, the formula can be further simplified to :E = \frac Because the surface area of a sphere is 4 \pi r^2, the electric field a distance r away from a uniform, spherical charge arrangement is :E = \frac = \frac = \frac = \frac where k is the
Coulomb constant The Coulomb constant, the electric force constant, or the electrostatic constant (denoted , or ) is a proportionality constant in electrostatics equations. In SI base units it is equal to .Derived from ''k''e = 1/(4''πε''0) – It was named ...
(\sim 9.0 \times 10^9 \ \text/\text). 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 :\mathbf(t) = \varepsilon_0 \int_^t \chi\left(t - t'\right) \mathbf\left(t'\right) \, dt'. That is, the polarization is a
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''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 would correspond to a
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
susceptibility . It is convenient to take the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
with respect to time and write this relationship as a function of frequency. Because of the
convolution theorem In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g ...
, the integral becomes a simple product, :\mathbf(\omega) = \varepsilon_0 \chi(\omega) \mathbf(\omega). 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 for ), a consequence of
causality Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the ca ...
, 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
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz The hertz ...
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 of the applied field: :\varepsilon \rightarrow \hat(\omega) (since complex numbers allow specification of magnitude and phase). The definition of permittivity therefore becomes :D_0 e^ = \hat(\omega) E_0 e^, where * and are the amplitudes of the displacement and electric fields, respectively, * is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition and ...
, . The response of a medium to static electric fields is described by the low-frequency limit of permittivity, also called the static permittivity (also ): :\varepsilon_\mathrm = \lim_ \hat(\omega). At the high-frequency limit (meaning optical frequencies), the complex permittivity is commonly referred to as (or sometimes ). At the
plasma frequency Plasma oscillations, also known as Langmuir waves (after Irving Langmuir), are rapid oscillations of the electron density in conducting media such as plasmas or metals in the ultraviolet region. The oscillations can be described as an instability i ...
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 emerges between and . The frequency at which the phase shift becomes noticeable depends on temperature and the details of the medium. For moderate field strength (), and remain proportional, and :\hat = \frac = , \varepsilon, e^. 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: :\hat(\omega) = \varepsilon'(\omega) - i\varepsilon''(\omega) = \left, \frac \ \left( \cos \delta - i\sin \delta \right). where * is the real part of the permittivity; * is the imaginary part of the permittivity; * is the loss angle. 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 , since it is a superimposed description of dispersion phenomena occurring at multiple frequencies. The dielectric function 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, , 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 In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
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 A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always m ...
absorption, which is directly related to the imaginary part of the optical dielectric function . The optical dielectric function is given by the fundamental expression: : \varepsilon(\omega) = 1 + \frac\sum_\int W_(E) \bigl( \varphi (\hbar \omega - E) - \varphi( \hbar\omega + E) \bigr) \, dx. In this expression, represents the product of the
Brillouin zone In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice i ...
-averaged transition probability at the energy with the joint
density of states In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states i ...
, ; is a broadening function, representing the role of scattering in smearing out the energy levels. In general, the broadening is intermediate between Lorentzian and
Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
; 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 The Drude model of electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of electrons in materials (especially metals). Basically, Ohm's law was well established and stated that the current ''J'' and voltag ...
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. (see also Electro-gyration). :\mathbf(\omega) = \begin \varepsilon_1 & -i \varepsilon_2 & 0 \\ i \varepsilon_2 & \varepsilon_1 & 0 \\ 0 & 0 & \varepsilon_z \\ \end \operatorname(\omega) If 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

Materials can be classified according to their complex-valued permittivity , upon comparison of its real and imaginary components (or, equivalently,
conductivity Conductivity may refer to: *Electrical conductivity, a measure of a material's ability to conduct an electric current **Conductivity (electrolytic), the electrical conductivity of an electrolyte in solution **Ionic conductivity (solid state), elec ...
, , when accounted for in the latter). A ''
perfect conductor A perfect conductor or perfect electric conductor (PEC) is an idealized material exhibiting infinite electrical conductivity or, equivalently, zero resistivity (cf. perfect dielectric). While perfect electrical conductors do not exist in nature, th ...
'' has infinite conductivity, , while a ''
perfect dielectric In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mater ...
'' is a material that has no conductivity at all, ; this latter case, of real-valued permittivity (or complex-valued permittivity with zero imaginary component) is also associated with the name ''lossless media''. Generally, when we consider the material to be a ''low-loss dielectric'' (although not exactly lossless), whereas is associated with a ''good conductor''; such materials with non-negligible conductivity yield a large amount of
loss Loss may refer to: Arts, entertainment, and media Music * ''Loss'' (Bass Communion album) (2006) * ''Loss'' (Mull Historical Society album) (2001) *"Loss", a song by God Is an Astronaut from their self-titled album (2008) * Losses "(Lil Tjay son ...
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: :J_\text = J_\mathrm + J_\mathrm = \sigma E + i \omega \varepsilon' E = i \omega \hat E where * is the
conductivity Conductivity may refer to: *Electrical conductivity, a measure of a material's ability to conduct an electric current **Conductivity (electrolytic), the electrical conductivity of an electrolyte in solution **Ionic conductivity (solid state), elec ...
of the medium; *\varepsilon'=\varepsilon_0\varepsilon_r is the real part of the permittivity. *\hat=\varepsilon'-i\varepsilon''is the complex permittivity Note that this is using the electrical engineering convention of the Complex conjugate ambiguity; the physics/chemistry convention involves the complex conjugate of these equations. The size of the
displacement current In electromagnetism, displacement current density is the quantity appearing in Maxwell's equations that is defined in terms of the rate of change of , the electric displacement field. Displacement current density has the same units as electric ...
is dependent on the
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz The hertz ...
ω of the applied field ''E''; there is no displacement current in a constant field. In this formalism, the complex permittivity is defined as: :\hat = \varepsilon' \left(1 - i \frac\right) = \varepsilon' - i \frac 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 dipoles. At low frequencies the field changes slowly enough to allow dipoles to reach equilibrium before the field has measurably changed. For frequencies at which dipole orientations cannot follow the applied field because of the
viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inter ...
of the medium, absorption of the field's energy leads to energy dissipation. The mechanism of dipoles relaxing is called
dielectric relaxation In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mater ...
and for ideal dipoles is described by classic
Debye relaxation In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mater ...
. * Second are the resonance effects, which arise from the rotations or vibrations of atoms,
ion An ion () is an atom or molecule with a net electrical charge. The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conv ...
s, or
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kno ...
s. These processes are observed in the neighborhood of their characteristic absorption frequencies. 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 An electrolytic capacitor is a polarized capacitor whose anode or positive plate is made of a metal that forms an insulating oxide layer through anodization. This oxide layer acts as the dielectric of the capacitor. A solid, liquid, or gel e ...
s or
supercapacitor A supercapacitor (SC), also called an ultracapacitor, is a high-capacity capacitor, with a capacitance value much higher than other capacitors but with lower voltage limits. It bridges the gap between electrolytic capacitors and rechargeable ba ...
s.


Quantum-mechanical interpretation

In terms of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qu ...
, permittivity is explained by atomic and
molecular A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
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 is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency rang ...
frequency, the microwave field causes the periodic rotation of water molecules, sufficient to break
hydrogen bond In chemistry, a hydrogen bond (or H-bond) is a primarily electrostatic force of attraction between a hydrogen (H) atom which is covalently bound to a more electronegative "donor" atom or group (Dn), and another electronegative atom bearing a ...
s. The field does work against the bonds and the energy is absorbed by the material as
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
. 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 Ultraviolet (UV) is a form of electromagnetic radiation with wavelength from 10 nm (with a corresponding frequency around 30  PHz) to 400 nm (750  THz), shorter than that of visible light, but longer than X-rays. UV radiatio ...
(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 The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kno ...
energy levels. Thus, these frequencies are classified as ionizing radiation. While carrying out a complete ''
ab initio ''Ab initio'' ( ) is a Latin term meaning "from the beginning" and is derived from the Latin ''ab'' ("from") + ''initio'', ablative singular of ''initium'' ("beginning"). Etymology Circa 1600, from Latin, literally "from the beginning", from abl ...
'' (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 In thermodynamics and solid-state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat (Heat capacity) in a solid. It treats the vibrations of the atomic lattice (hea ...
and the
Lorentz model The Drude model of electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of electrons in materials (especially metals). Basically, Ohm's law was well established and stated that the current ''J'' and voltage ...
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 Dielectric spectroscopy (which falls in a subcategory of impedance spectroscopy) measures the dielectric properties of a medium as a function of frequency.Kremer F., Schonhals A., Luck W. Broadband Dielectric Spectroscopy. – Springer-Verlag, 200 ...
, covering nearly 21 orders of magnitude from 10−6 to 1015
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that one he ...
. Also, by using
cryostat A cryostat (from ''cryo'' meaning cold and ''stat'' meaning stable) is a device used to maintain low cryogenic temperatures of samples or devices mounted within the cryostat. Low temperatures may be maintained within a cryostat by using various r ...
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 domain Time domain refers to 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 cas ...
measurements (10−6 to 103 Hz) * Low-frequency
frequency domain In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a si ...
measurements (10−5 to 106 Hz) * Reflective coaxial methods (106 to 1010 Hz) * Transmission coaxial method (108 to 1011 Hz) * Quasi-optical methods (109 to 1010 Hz) *
Terahertz time-domain spectroscopy In physics, terahertz time-domain spectroscopy (THz-TDS) is a spectroscopic technique in which the properties of matter are probed with short pulses of terahertz radiation. The generation and detection scheme is sensitive to the sample's effect on ...
(1011 to 1013 Hz) * Fourier-transform methods (1011 to 1015 Hz) At infrared and optical frequencies, a common technique is
ellipsometry Ellipsometry is an optical technique for investigating the dielectric properties (complex refractive index or dielectric function) of thin films. Ellipsometry measures the change of polarization upon reflection or transmission and compares it to ...
.
Dual polarisation interferometry Dual-polarization interferometry (DPI) is an analytical technique that probes molecular layers adsorbed to the surface of a waveguide using the evanescent wave of a laser beam. It is used to measure the conformational change in proteins, or othe ...
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.


See also

*
Acoustic attenuation Acoustic attenuation is a measure of the energy loss of sound propagation in media. Most media have viscosity and are therefore not ideal media. When sound propagates in such media, there is always thermal consumption of energy caused by viscosity. ...
*
Density functional theory Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
*
Electric-field screening 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 (classical plasmas), electrolytes, and charge car ...
*
Green–Kubo relations The Green–Kubo relations ( Melville S. Green 1954, Ryogo Kubo 1957) give the exact mathematical expression for transport coefficients \gamma in terms of integrals of time correlation functions: :\gamma = \int_0^\infty \left\langle \dot(t) \dot ...
*
Green's function (many-body theory) In many-body theory, the term Green's function (or Green function) is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators. The name comes from t ...
*
Linear response function A 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. Because of its many applications in informatio ...
*
Rotational Brownian motion Rotational 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. The polarization of a dielectric material is a competit ...
* Electromagnetic permeability


Notes


References


Further reading

* C. J. F. Bottcher, O. C. von Belle & Paul Bordewijk (1973) ''Theory of Electric Polarization: Dielectric Polarization'', volume 1, (1978) volume 2, Elsevier . * Arthur R. von Hippel (1954) ''Dielectrics and Waves'' * Arthur von Hippel editor (1966) ''Dielectric Materials and Applications: papers by 22 contributors'' {{ISBN, 0-89006-805-4.


External links


Electromagnetism
a chapter from an online textbook Electric and magnetic fields in matter Physical quantities