Permittivity

Permittivity, usually denoted by $\varepsilon$, is a constitutive quantity in electromagnetism that relates the electric field to electric displacement. In a linear, isotropic medium, this relationship is $\mathbf D=\varepsilon\mathbf E$. In matter, permittivity incorporates the response of bound electric charges to the field; in vacuum, the corresponding quantity is the vacuum permittivity $\varepsilon_0$. Depending on the medium and conditions, permittivity may be a real scalar, a complex frequency-dependent quantity, or a tensor rather than a single constant. (web.mit.edu)

Definition and units

The electric displacement field $\mathbf D$ is defined by

where $\mathbf P$ is the electric polarization density: electric dipole moment per unit volume. For a linear, isotropic material without spontaneous polarization,

where $\chi_{\mathrm e}$ is the dimensionless electric susceptibility. Consequently,

These are macroscopic relationships: they describe fields averaged over distances large compared with atomic dimensions, rather than the rapidly varying fields around individual particles. (web.mit.edu)

In the International System of Units (SI), absolute permittivity has units of farads per metre:

Relative permittivity is the dimensionless ratio

The expression dielectric constant commonly denotes relative permittivity, especially its real, low-frequency value. The word “constant” does not imply independence from frequency or other measurement conditions. Absolute and relative permittivity must therefore be distinguished when reporting numerical values. (physics.nist.gov)

Vacuum permittivity

Vacuum permittivity, also called the electric constant, appears in Coulomb’s law and Maxwell’s equations. It is related to vacuum magnetic permeability $\mu_0$ and the speed of light in vacuum $c$ by

The 2022 CODATA recommended value is

where the parenthesized digits express the standard uncertainty in the last quoted digits. (physics.nist.gov)

Following the SI revision that took effect on 20 May 2019, the numerical values of vacuum permittivity and vacuum permeability are no longer exact. The elementary charge $e$, Planck constant $h$, and $c$ have exact defining values, whereas the fine-structure constant $\alpha$ is experimentally determined. The relation

therefore gives $\varepsilon_0$ a measurement uncertainty. (bipm.org)

Microscopic origin in matter

A material’s dielectric response arises from the redistribution or reorientation of electric charge. Several mechanisms can contribute:

• Electronic polarization: displacement of an atom’s or molecule’s electron distribution relative to its nuclei.
• Ionic polarization: relative displacement of positively and negatively charged constituents, particularly in ionic solids.
• Orientational polarization: partial alignment of permanent electric dipoles against thermal disorder.
• Interfacial or space-charge polarization: accumulation of charge at interfaces or other electrically nonuniform regions. (ocw.mit.edu)

Each mechanism has a characteristic response time. A contribution that responds to a slowly varying field may become ineffective when the field oscillates too rapidly. This explains why a material’s low-frequency permittivity need not equal its optical-frequency permittivity. Polarization measurements thus provide information about molecular motion, charge redistribution, and dielectric loss. (ocw.mit.edu)

Frequency dependence and dielectric loss

For a time-varying field, polarization generally depends on the field’s earlier values rather than only its instantaneous value. In a linear, spatially local medium, this dependence can be represented by a response integral in time. A Fourier transform converts that relationship into

where $\omega$ is angular frequency. Thus, frequency-dependent permittivity is a compact description of a material’s delayed electrical response. (ocw.mit.edu)

Using the harmonic convention $e^{i\omega t}$, complex permittivity is commonly written

Here the asterisk labels the complex quantity; it does not signify complex conjugation. The real part describes the in-phase displacement response, while the loss component $\varepsilon''$ describes dissipation. With the opposite time convention, $e^{-i\omega t}$, the imaginary-part sign reverses. The convention must therefore accompany any interpretation of that sign. (tsapps.nist.gov)

For an isotropic medium and a peak-amplitude electric-field phasor $\mathbf E_0$, the average power dissipated per unit volume by the loss included in the complex permittivity is

Where $\varepsilon'>0$, the loss tangent is conventionally

It expresses dielectric loss relative to the in-phase response. (web.mit.edu)

Electrical conduction may be represented separately or absorbed into an effective complex permittivity. For a real, frequency-independent electrical conductivity $\sigma$, under the $e^{i\omega t}$ convention,

Consequently, a measured loss component can contain both polarization loss and conduction loss. If conductivity has already been included in effective permittivity, adding it again as a separate loss term would double-count its contribution. (academy.cba.mit.edu)

Anisotropic and nonlinear media

In an anisotropic material, electric displacement need not be parallel to the electric field. Permittivity is then a second-rank tensor, represented in a chosen coordinate system by a matrix:

Different field directions can therefore produce different responses. The tensor’s components may themselves be complex and frequency-dependent. (ocw.mit.edu)

The scalar relation $D=\varepsilon E$ also requires qualification in nonlinear materials. When displacement does not vary proportionally with field, the ratio $D/E$ is not generally the same as the incremental response $dD/dE$. Materials exhibiting ferroelectricity may possess spontaneous polarization and a response dependent on their electrical history. Their reported permittivity can consequently refer to a small-signal measurement about a specified operating state, rather than a universal material constant. (web.mit.edu)

Electrostatic fields and capacitance

In macroscopic electrostatics, Gauss’s law takes the form

where $\rho_{\mathrm{free}}$ is free-charge density. In a linear dielectric this becomes

Permittivity therefore influences both field strength and field distribution. At interfaces, the normal component of $\mathbf D$ changes according to the free surface charge, while the tangential electric field is continuous in electrostatic conditions. (web.mit.edu)

For an ideal parallel-plate capacitor completely filled by a uniform dielectric,

where $C$ is capacitance, $A$ is plate area, and $d$ is plate separation; edge effects are neglected. Relative permittivity is therefore the factor by which filling the same ideal capacitor increases its capacitance over the vacuum value. Capacitance depends on both material and geometry, whereas permittivity describes the material response under specified conditions. (ocw.mit.edu)

A larger permittivity does not always mean a smaller electric field. At fixed free charge in this capacitor geometry, $E=D/\varepsilon$ decreases as permittivity increases. At fixed voltage, however, $E=V/d$ remains unchanged and the stored charge increases instead. The electrical constraint is essential to interpreting the effect. (web.mit.edu)

Electromagnetic waves and applications

For a homogeneous, isotropic, lossless medium, the electromagnetic phase velocity is

and the refractive index satisfies

For a nonmagnetic optical medium, $\mu_{\mathrm r}\approx1$, giving $n^2\approx\varepsilon_{\mathrm r}$ at the same frequency. Complex permittivity requires a complex wave description and accounts for attenuation as well as phase propagation. Its frequency dependence contributes to dispersion. A static dielectric constant cannot generally be substituted into an optical calculation. (ocw.mit.edu)

Permittivity is consequently important in capacitor design, electronic substrates, microwave components, and optical materials. High-permittivity materials increase capacitance within a given geometry, while low dielectric loss is important where dissipation must be limited. Materials with spontaneous polarization provide particularly strong dielectric responses, but their response must be characterized together with its field and frequency dependence. (electroceramics.scripts.mit.edu)

Measurement and interpretation

At low frequencies, permittivity can be inferred from capacitance and electrical impedance measurements using a specimen of known geometry. At microwave frequencies, methods include transmission/reflection measurements, open-ended coaxial probes, resonant cavities, and free-space measurements. Different methods suit different specimen shapes, frequency ranges, and loss levels. (tsapps.nist.gov)

Reported values should specify frequency, temperature, measurement direction where relevant, and whether conduction is included. Sample dimensions, air gaps, surface properties, and instrument calibration contribute to measurement uncertainty. Air gaps can produce a substantial bias because their electrical response combines with that of the specimen. An experimentally inferred permittivity must therefore be interpreted together with the measurement model and specimen conditions. (nist.gov)

References

1. 2022 CODATA adjustmentphysics.nist.gov
2. CODATA recommended values of the fundamental physical constants: 2022tsapps.nist.gov
3. The SI - BIPMbipm.org
4. SI Brochure - 9th ed./version 3.02bipm.org
5. 013 Electromagnetics and Applications, Course Notesocw.mit.edu
6. 097(UG) Fundamentals of Photonicsocw.mit.edu
7. Electromagnetic Fields and Energy: 6.4 Constitutive Laws of Polarizationweb.mit.edu
8. Electromagnetic Fields and Energy: 6.5 Fields in the Presence of Electrically Linear Dielectricsweb.mit.edu
9. Electromagnetic Fields and Energy: 11.5 Electromagnetic Dissipationweb.mit.edu
10. lecture_10.pdf — Electronic and Mechanical Properties of Materialsocw.mit.edu
11. lecture_11.pdf — Electronic and Mechanical Properties of Materialsocw.mit.edu