View source for Relative permittivity

{{Short description|Measure of the electric polarizability of a dielectric, compared with that of a vacuum}}
{{Relative permittivity table}}
[[File:Water relative static permittivity.svg|thumb|right|Temperature dependence of the relative static permittivity of water]]

The '''relative permittivity''' (in older texts, '''dielectric constant''') is the [[permittivity]] of a material expressed as a ratio with the [[vacuum permittivity|electric permittivity of a vacuum]]. A [[dielectric]] is an insulating material, and the dielectric constant of an insulator measures the ability of the insulator to store electric energy in an electrical field.

Permittivity is a material's property that affects the [[Coulomb force]] between two point charges in the material. Relative permittivity is the factor by which the electric field between the charges is decreased relative to vacuum.

Likewise, relative permittivity is the ratio of the [[capacitance]] of a [[capacitor]] using that material as a [[dielectric]], compared with a similar capacitor that has vacuum as its dielectric. Relative permittivity is also commonly known as the dielectric constant, a term still used but deprecated by standards organizations in engineering<ref name=IEEE1997/> as well as in chemistry.<ref name="IUPAC"/>

== Definition ==
Relative permittivity is typically denoted as {{math|''ε''<sub>r</sub>(''ω'')}} (sometimes {{math|''κ''}}, lowercase [[kappa]]) and is defined as
: <math>\varepsilon_\text{r}(\omega) = \frac{\varepsilon(\omega)}{\varepsilon_{0}},</math>
where ''ε''(''ω'') is the [[complex number|complex]] frequency-dependent [[permittivity]] of the material, and ''ε''<sub>0</sub> is the [[vacuum permittivity]].

Relative permittivity is a [[dimensionless quantity|dimensionless]] number that is in general [[complex number|complex-valued]]; its real and imaginary parts are denoted as:<ref name=ChenVaradan2004>{{cite book|author1=Linfeng Chen |author2=Vijay K. Varadan |name-list-style=amp |year=2004|title=Microwave electronics: measurement and materials characterization |url=https://books.google.com/books?id=2oA3po4coUoC&pg=PA8 |publisher=John Wiley and Sons |isbn=978-0-470-84492-2 |doi=10.1002/0470020466 |page=8, eq.(1.15)}}</ref>
:<math> \varepsilon_\text{r}(\omega) = \varepsilon_\text{r}'(\omega) - i \varepsilon_\text{r}''(\omega). </math>

The relative permittivity of a medium is related to its [[electric susceptibility]], {{math|''χ''<sub>e</sub>}}, as {{math|1=''ε''<sub>r</sub>(''ω'') = 1 + ''χ''<sub>e</sub>}}.

In anisotropic media (such as non cubic crystals) the relative permittivity is a second rank [[tensor]].

The relative permittivity of a material for a [[frequency]] of zero is known as its '''static relative permittivity'''.

=== Terminology ===
The historical term for the relative permittivity is ''dielectric constant''. It is still commonly used, but has been deprecated by standards organizations,<ref name=IEEE1997>{{cite journal |author=[[IEEE]] Standards Board|url=http://ieeexplore.ieee.org/servlet/opac?punumber=5697|title=IEEE Standard Definitions of Terms for Radio Wave Propagation | journal=IEEE STD 211-1997 |year=1997 |page=6}}</ref><ref name="IUPAC">{{cite journal |last=Braslavsky |first=S.E.|url=http://iupac.org/publications/pac/2007/pdf/7903x0293.pdf |title=Glossary of terms used in photochemistry (IUPAC recommendations 2006)|journal=Pure and Applied Chemistry|volume=79 |issue=3 |year=2007 |pages=293–465|doi=10.1351/pac200779030293|s2cid=96601716}}</ref> because of its ambiguity, as some older reports used it for the absolute permittivity ''ε''.<ref name=IEEE1997/><ref>{{cite book
|last = King
|first = Ronold W. P.
|author-link = Ronold W. P. King
|title = Fundamental Electromagnetic Theory
|publisher = Dover
|year = 1963
|location = New York
|page = 139}}</ref><ref name=Jackson/> The permittivity may be quoted either as a static property or as a frequency-dependent variant, in which case it is also known as the ''dielectric function''. It has also been used to refer to only the real component ''ε''′<sub>r</sub> of the complex-valued relative permittivity.{{citation needed|date=September 2013}}

=== Physics ===
In the causal theory of waves, permittivity is a complex quantity. The imaginary part corresponds to a phase shift of the polarization {{math|'''P'''}} relative to {{math|'''E'''}} and leads to the attenuation of electromagnetic waves passing through the medium.  By definition, the linear relative [[vacuum permittivity|permittivity of vacuum]] is equal to 1,<ref name=Jackson>
{{cite book 
|author=John David Jackson
|title=Classical Electrodynamics
|url=https://archive.org/details/classicalelectro00jack_697
|url-access=limited
|edition=Third
|publisher= Wiley
|location=New York
|year=1998
|isbn=978-0-471-30932-1
|page=[https://archive.org/details/classicalelectro00jack_697/page/n177 154]
}}</ref> that is {{nowrap|1=''ε'' = ''ε''<sub>0</sub>}}, although there are theoretical [[Quantum_vacuum_state#Non-linear_permittivity|nonlinear quantum effects in vacuum]] that become non-negligible at high field strengths.<ref name=Mourou>{{cite journal|doi=10.1103/RevModPhys.78.309|title=Optics in the relativistic regime|year=2006|last1=Mourou|first1=Gerard A.|journal=Reviews of Modern Physics|volume=78|issue=2|page=309|bibcode=2006RvMP...78..309M}}</ref>

The following table gives some typical values.
{|class="wikitable"
|+ Low-frequency relative permittivity of some common solvents
|-
! colspan="2" |Solvent
! Relative permittivity
! Temperature
|-
|C<sub>6</sub>H<sub>6</sub>
| [[benzene]] || 2.3 || 298 K  (25 °C)
|-
|Et<sub>2</sub>O
| [[diethyl ether]] || 4.3 || 293 K  (20 °C)
|-
|(CH<sub>2</sub>)<sub>4</sub>O
| [[tetrahydrofuran]] (THF) || 7.6 || 298 K  (25 °C)
|-
|CH<sub>2</sub>Cl<sub>2</sub>
| [[dichloromethane]] || 9.1 || 293 K  (20 °C)
|-
|NH<sub>3</sub>(''liq'')
| [[ammonia|liquid ammonia]] || 17 || 273 K  (0 °C)
|-
|C<sub>2</sub>H<sub>5</sub>OH
| [[ethanol]] || 24.3 || 298 K  (25 °C)
|-
|CH<sub>3</sub>OH
| [[methanol]] || 32.7 || 298 K  (25 °C)
|-
|CH<sub>3</sub>NO<sub>2</sub>
| [[nitromethane]] || 35.9 || 303 K  (30 °C)
|-
|HCONMe<sub>2</sub>
| [[dimethyl formamide]] (DMF) || 36.7 || 298 K  (25 °C)
|-
|CH<sub>3</sub>CN
| [[acetonitrile]] || 37.5 || 293 K  (20 °C)
|-
|H<sub>2</sub>O
| [[water]] || 78.4|| 298 K  (25 °C)
|-
|HCONH<sub>2</sub>
| [[formamide]] || 109 || 293 K  (20 °C)
|}

The relative low frequency permittivity of [[ice]] is ~96 at −10.8&nbsp;°C, falling to 3.15 at high frequency, which is independent of temperature.<ref>{{cite journal |doi=10.3189/S0022143000018840|title=Dielectric Properties of Ice and Snow–a Review |year=1965 |last1=Evans |first1=S. |journal=Journal of Glaciology |volume=5 |issue=42 |pages=773–792 |s2cid=227325642 |doi-access=free }}</ref>  It remains in the range 3.12–3.19 for frequencies between about 1&nbsp;MHz and the far infrared region.<ref>{{citation |title=A summary of the complex dielectric permittivity of ice in the megahertz range and its applications for radar sounding of polar ice sheets |author1=Fujita, Shuji |author2=Matsuoka, Takeshi |author3=Ishida, Toshihiro |author4=Matsuoka, Kenichi |author5=Mae, Shinji |url=https://eprints.lib.hokudai.ac.jp/dspace/bitstream/2115/32469/1/P185-212.pdf }}</ref>

== Measurement ==
The relative static permittivity, ''ε''<sub>r</sub>, can be measured for static [[electric field]]s as follows: first the [[capacitance]] of a test [[capacitor]], ''C''<sub>0</sub>, is measured with vacuum between its plates. Then, using the same capacitor and distance between its plates, the capacitance ''C'' with a [[dielectric]] between the plates is measured. The relative permittivity can be then calculated as
: <math>\varepsilon_\text{r} = \frac{C}{C_0}.</math>

For time-variant [[electromagnetic field]]s, this quantity becomes [[frequency]]-dependent. An indirect technique to calculate ''ε''<sub>r</sub> is conversion of radio frequency [[S-parameter]] measurement results. A description of frequently used S-parameter conversions for determination of the frequency-dependent ''ε''<sub>r</sub> of dielectrics can be found in this bibliographic source.<ref>{{cite web|url=https://cdn.rohde-schwarz.com/pws/dl_downloads/dl_application/00aps_undefined/RAC-0607-0019_1_5E.pdf|title= Measurement of Dielectric Material Properties|first1=CheeYaw|last1=Kuek|publisher=R&S}}</ref> Alternatively, resonance based effects may be employed at fixed frequencies.<ref>{{Cite journal | last1 = Costa | first1 = F. | last2 = Amabile | first2 = C. | last3 = Monorchio | first3 = A. | last4 = Prati | first4 = E. | title = Waveguide Dielectric Permittivity Measurement Technique Based on Resonant FSS Filters | doi = 10.1109/LMWC.2011.2122303 | journal = IEEE Microwave and Wireless Components Letters | volume = 21 | issue = 5 | page = 273 | year = 2011 | s2cid = 34515302 | url = https://zenodo.org/record/894374 }}</ref>

==Applications==

=== Energy ===
The relative permittivity is an essential piece of information when designing [[capacitor]]s, and in other circumstances where a material might be expected to introduce [[capacitance]] into a circuit. If a material with a high relative permittivity is placed in an [[electric field]],  the magnitude of that field will be measurably reduced within the volume of the dielectric. This fact is commonly used to increase the capacitance of a particular capacitor design. The layers beneath etched conductors in printed circuit boards ([[printed circuit board|PCBs]]) also act as dielectrics.

=== Communication ===
Dielectrics are used in [[radio frequency]] (RF) transmission lines. In a [[coaxial]] cable, [[polyethylene]] can be used between the center conductor and outside shield. It can also be placed inside waveguides to form [[dielectric resonator filter|filters]]. [[Optical fibers]] are examples of ''dielectric [[waveguide]]s''. They consist of dielectric materials that are purposely doped with impurities so as to control the precise value of ''ε''<sub>r</sub> within the cross-section. This controls the [[refractive index]] of the material and therefore also the optical modes of transmission. However, in these cases it is technically the relative permittivity that matters, as they are not operated in the electrostatic limit.

=== Environment ===
The relative permittivity of air changes with temperature, humidity, and barometric pressure.<ref>5×10<sup>−6</sup>/°C, 1.4×10<sup>−6</sup>/%RH and 100×10<sup>−6</sup>/atm respectively. See [http://repository.tudelft.nl/islandora/object/uuid:e2234250-950d-4eb5-9f2e-b5b8e67af2e5/datastream/OBJ/download A Low Cost Integrated Interface for Capacitive Sensors], Ali Heidary, 2010, Thesis, p. 12. {{ISBN|9789461130136}}.</ref> Sensors can be constructed to detect changes in capacitance caused by changes in the relative permittivity. Most of this change is due to effects of temperature and humidity as the barometric pressure is fairly stable. Using the capacitance change, along with the measured temperature, the relative humidity can be obtained using engineering formulas.

=== Chemistry ===
The relative static permittivity of a solvent is a relative measure of its [[chemical polarity]]. For example, [[water (molecule)|water]] is very polar, and has a relative static permittivity of 80.10 at 20&nbsp;°C while ''n''-[[hexane]] is non-polar, and has a relative static permittivity of 1.89 at 20&nbsp;°C.<ref>{{RubberBible86th}}</ref> This information is important when designing separation, [[sample preparation]] and [[chromatography]] techniques in [[analytical chemistry]].

The correlation should, however, be treated with caution. For instance, [[dichloromethane]] has a value of ''ε''<sub>r</sub> of [[dichloromethane (data page)|9.08]] (20&nbsp;°C) and is rather poorly soluble in water (13{{nbsp}}g/L or 9.8{{nbsp}}mL/L at 20&nbsp;°C); at the same time, [[tetrahydrofuran]] has its ''ε''<sub>r</sub> = [[tetrahydrofuran (data page)|7.52]] at 22&nbsp;°C, but it is completely miscible with water.  In the case of tetrahydrofuran, the oxygen atom can act as a [[hydrogen bond]] acceptor; whereas dichloromethane cannot form hydrogen bonds with water.<!-- The commonly known "like-dissolves-like" principle could be useful here, as the probable reason for the discrepancy is the specific interaction between the oxygen atoms, as the THF could be treated as a homologue of water. -->

This is even more remarkable when comparing the ''ε''<sub>r</sub> values of [[acetic acid]] (6.2528)<ref name="gaussian">AE. Frisch, M. J. Frish, F. R. Clemente, G. W. Trucks. Gaussian 09 User's Reference. Gaussian, Inc.: Walligford, CT, 2009.- p. 257.</ref> and that of [[iodoethane]] (7.6177).<ref name="gaussian" /> The large numerical value of ''ε''<sub>r</sub> is not surprising in the second case, as the [[iodine]] atom is easily polarizable; nevertheless, this does not imply that it is polar, too (electronic [[polarizability]] prevails over the orientational one in this case).

== Lossy medium ==
Again, similar as for [[lossy medium|absolute permittivity]], relative permittivity for lossy materials can be formulated as:
: <math> \varepsilon_\text{r} = \varepsilon_\text{r}' - \frac{i\sigma}{\omega\varepsilon_0}, </math>
in terms of a "dielectric conductivity" ''σ'' (units S/m, [[siemens (unit)|siemens]] per meter), which "sums over all the dissipative effects of the material; it may represent an actual [[Electrical conductivity|[electrical] conductivity]] caused by migrating charge carriers and it may also refer to an energy loss associated with the dispersion of ''ε''′ [the real-valued permittivity]" (<ref name=ChenVaradan2004/> p.&nbsp;8). Expanding the [[angular frequency]] {{nowrap|1=''ω'' = 2π''c''{{nnbsp}}/{{nnbsp}}''λ''}} and the [[electric constant]] {{nowrap|1=''ε''<sub>0</sub> = 1{{nnbsp}}/{{nnbsp}}''µ''<sub>0</sub>''c''<sup>2</sup>}}, which reduces to:
: <math> \varepsilon_\text{r} = \varepsilon_\text{r}' - i\sigma\lambda\kappa, </math>
where ''λ'' is the wavelength, ''c'' is the speed of light in vacuum and {{nowrap|1=''κ'' = ''µ''<sub>0</sub>''c''{{nnbsp}}/{{nnbsp}}2π}} = 59.95849&nbsp;Ω ≈ 60.0&nbsp;Ω is a newly introduced constant (units [[ohm]]s, or reciprocal [[siemens (unit)|siemens]], such that ''σλκ'' = ''ε''<sub>r</sub> remains unitless).

== Metals ==
Permittivity is typically associated with [[dielectric materials]], however metals are described as having an effective permittivity, with real relative permittivity equal to one.<ref name=Lourtioz>{{cite book |last=Lourtioz |first=J.-M. |url=https://books.google.com/books?id=vSszZ2WuG_IC&pg=PA121 |title=Photonic Crystals: Towards Nanoscale Photonic Devices |year=2005 |publisher=Springer |isbn=978-3-540-24431-8 |display-authors=etal |pages=121–122}} equation&nbsp;(4.6), page&nbsp;121</ref> In the high-frequency region, which extends from radio frequencies to the far [[infrared]] and [[Terahertz radiation|terahertz]] region, the plasma frequency of the electron gas is much greater than the electromagnetic propagation frequency, so the refractive index ''n'' of a metal is very nearly a purely imaginary number. In the low frequency regime, the effective relative permittivity is also almost purely imaginary: It has a very large imaginary value related to the conductivity and a comparatively insignificant real-value.<ref>Lourtioz (2005), equations&nbsp;(4.8)–(4.9), page&nbsp;122</ref>

== See also ==
{{div col|colwidth=22em}}
* [[Curie temperature]]
* [[Dielectric spectroscopy]]
* [[Dielectric strength]]
* [[Electret]]
* [[Ferroelectricity]]
* [[Green–Kubo relations]]
* [[High-κ dielectric]]
* [[Kramers–Kronig relation]]
* [[Linear response function]]
* [[Low-κ dielectric]]
* [[Loss tangent]]
* [[Permittivity]]
* [[Refractive index]]
* [[Permeability (electromagnetism)]]
{{div col end}}

== References ==
{{Reflist|30em}}

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{{DEFAULTSORT:Relative Permittivity}}
[[Category:Electricity]]
[[Category:Electric and magnetic fields in matter]]
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