View source for Vacuum permittivity

{{Short description|Absolute dielectric permittivity of free space}}
{{about|the electric constant|the analogous magnetic constant|Vacuum permeability|the ordinal number &epsilon;<sub>0</sub>|Epsilon numbers (mathematics)}}

{| class="wikitable floatright"
|-
! Value of ''ε''<sub>0</sub>!! Unit
|-
| {{physconst|eps0|unit=no|ref=no}} || [[Farad|F]]⋅[[Meter|m]]<sup>−1</sup>
|-
| {{val|55.26349406}} || [[Elementary charge|''e'']]<sup>2</sup>⋅[[Electronvolt|eV]]<sup>−1</sup>⋅[[μm]]<sup>−1</sup>
|}

'''Vacuum permittivity''', commonly denoted {{math|'''''ε''<sub>0</sub>'''}} (pronounced "epsilon nought" or "epsilon zero"), is the value of the [[Permittivity|absolute dielectric permittivity]] of [[classical vacuum]]. It may also be referred to as the '''permittivity of free space''', the '''electric constant''', or the distributed capacitance of the vacuum. It is an ideal (baseline) [[physical constant]]. Its [[CODATA]] value is:

:{{physconst|eps0|symbol=yes|ref=no}} ([[farad]]s per [[meter]]), with a relative uncertainty of {{physconst|eps0|runc=yes|after=.}}

It is a measure of how dense of an [[electric field]] is "permitted" to form in response to electric charges and relates the units for [[electric charge]] to mechanical quantities such as length and force.<ref>
{{cite encyclopedia | url = http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=121-11-03 | title = Electropedia: International Electrotechnical Vocabulary (IEC&nbsp;60050) | contribution = electric constant | publisher = International Electrotechnical Commission | location = Geneva | access-date = 26 March 2015 }}.</ref> For example, the force between two separated electric charges with spherical symmetry (in the [[Vacuum#Electromagnetism|vacuum of classical electromagnetism]]) is given by [[Coulomb's law]]:
:<math>F_\text{C} = \frac{1} {4 \pi \varepsilon_0} \frac{q_1 q_2} {r^2}</math>
Here, ''q''<sub>1</sub> and ''q''<sub>2</sub> are the charges, ''r'' is the distance between their centres, and the value of the constant fraction <math>1/(4\pi\varepsilon_0)</math> (known as the [[Coulomb constant]], ''k''<sub>e</sub>) is approximately 9 × 10<sup>9</sup> N⋅m<sup>2</sup>⋅C<sup>−2</sup>. Likewise, ''ε''<sub>0</sub> appears in [[Maxwell's equations]], which describe the properties of [[Electric field|electric]] and [[magnetic field]]s and [[electromagnetic radiation]], and relate them to their sources. In electrical engineering, ''ε''<sub>0</sub> itself is used as a unit to quantify the permittivity of various dielectric materials.

==Value==
The value of ''ε''<sub>0</sub> is ''defined'' by the formula<ref name=ep0>
The approximate numerical value is found at:
{{cite web |title= -: Electric constant, &epsilon;<sub>0</sub> |work=NIST reference on constants, units, and uncertainty: Fundamental physical constants |url=http://physics.nist.gov/cgi-bin/cuu/Value?ep0 |publisher=NIST |access-date=2012-01-22}} This formula determining the exact value of ''ε''<sub>0</sub> is found in Table 1, p. 637 of {{cite journal |title=<u>Table 1: Some exact quantities relevant to the 2006 adjustment</u>'' in ''CODATA recommended values of the fundamental physical constants: 2006 |url=http://physics.nist.gov/cuu/Constants/RevModPhys_80_000633acc.pdf |journal =Rev Mod Phys |pages=633–729 |volume=80 |date=April–June 2008 |issue=2 |author1=PJ Mohr |author2=BN Taylor |author3=DB Newell |bibcode = 2008RvMP...80..633M |doi = 10.1103/RevModPhys.80.633 |arxiv = 0801.0028 }}

</ref>

:<math> \varepsilon_0 =\frac {1}{\mu_0 c^2}</math>

where ''c'' is the defined value for the [[speed of light]] in [[classical vacuum]] in [[International System of Units|SI units]],<ref name="SIBrochure9thEd">{{SIbrochure9th}}</ref>{{Rp|127}} and ''μ''<sub>0</sub> is the parameter that international Standards Organizations call the "[[magnetic constant]]" (commonly called vacuum permeability or the permeability of free space). Since ''μ''<sub>0</sub> has an approximate value 4π&nbsp;×&nbsp;10<sup>−7</sup>&nbsp;[[Henry (unit)|H]]/[[meter|m]],<ref>See the last sentence of the [http://physics.nist.gov/cuu/Units/ampere.html NIST definition of ampere].</ref> and ''c'' has the ''defined'' value {{gaps|299|792|458}}&nbsp;m⋅s<sup>−1</sup>, it follows that ''ε''<sub>0</sub> can be expressed numerically as

:<math>\begin{align}
  \varepsilon_0 & \approx \frac{1}{\left(4 \pi \times 10^{-7} \, \textrm{N/A}^{2}\right)\left(299792458 \, \textrm{m/s}\right)^2} \\[2pt]
                & = \frac{625000}{22468879468420441 \pi} \, \textrm{ F/m} \\[2pt]
                & \approx 8.85418781762039 \times 10^{-12} \, \textrm{F}{\cdot}\textrm{m}^{-1}
\end{align}</math>
:(or [[Ampere|A]]<sup>2</sup>⋅[[second|s]]<sup>4</sup>⋅[[kilogram|kg]]<sup>−1</sup>⋅[[meter|m]]<sup>−3</sup> in [[SI base unit]]s, or [[Coulomb|C]]<sup>2</sup>⋅[[Newton (unit)|N]]<sup>−1</sup>⋅[[meter|m]]<sup>−2</sup> or [[Coulomb|C]]⋅[[Volt|V]]<sup>−1</sup>⋅[[meter|m]]<sup>−1</sup> using other SI coherent units).<ref name="CODATA">{{CODATA2006|url=http://physics.nist.gov/cgi-bin/cuu/Value?ep0}}.</ref><ref>A summary of the definitions of ''c'', ''μ''<sub>0</sub> and ''ε''<sub>0</sub> is provided in the 2006 CODATA Report: [http://physics.nist.gov/cuu/Constants/codata.pdf CODATA report, pp. 6–7]</ref>

The historical origins of the electric constant ''ε''<sub>0</sub>, and its value, are explained in more detail below.

===Redefinition of the SI units===
{{main article|2019 redefinition of the SI base units}}
The [[ampere]] was redefined by defining the [[elementary charge]] as an exact number of coulombs as from 20 May 2019,<ref name="SIBrochure9thEd" /> with the effect that the vacuum electric permittivity no longer has an exactly determined value in SI units. The value of the electron charge became a numerically defined quantity, not measured, making ''μ''<sub>0</sub> a measured quantity. Consequently, ''ε''<sub>0</sub>  is not exact. As before, it is defined by the equation {{nowrap|1=''ε''<sub>0</sub> = 1/(''μ''<sub>0</sub>''c''<sup>2</sup>)}}, and is thus determined by the value of ''μ''<sub>0</sub>, the [[Vacuum permeability|magnetic vacuum permeability]] which in turn is determined by the experimentally determined dimensionless [[fine-structure constant]] ''α'':
:<math> \varepsilon_0 = \frac {1}{\mu_0 c^2} = \frac {e^2}{2\alpha h c}\ ,</math>
with ''e'' being the [[elementary charge]], ''h'' being the [[Planck constant]], and ''c'' being the [[speed of light]] in [[Vacuum#In electromagnetism|vacuum]], each with exactly defined values. The relative uncertainty in the value of ''ε''<sub>0</sub> is therefore the same as that for the dimensionless [[fine-structure constant]], namely {{physconst|alpha|runc=yes|after=.}}

==Terminology==
Historically, the parameter ''ε''<sub>0</sub> has been known by many different names. The terms "vacuum permittivity" or its variants, such as  "permittivity in/of vacuum",<ref name=Sze>
{{Cite book
|author1=SM Sze  |author2=Ng KK
 |name-list-style=amp |title=Physics of semiconductor devices
|chapter=Appendix E
|page=788
|edition=Third
|year= 2007
|publisher=Wiley-Interscience
|location=New York
|isbn=978-0-471-14323-9
|chapter-url=http://worldcat.org/isbn/0-471-14323-5}}
</ref><ref name=Muller>
{{Cite book
|author1=RS Muller, Kamins TI  |author2=Chan M
 |name-list-style=amp |title=Device electronics for integrated circuits
|page=Inside front cover
|edition=Third
|year= 2003
|publisher=Wiley
|location=New York
|isbn=978-0-471-59398-0
|url=http://worldcat.org/isbn/0-471-59398-2
|no-pp=true}}
</ref> "permittivity of empty space",<ref name=Zemansky>{{Cite book
|author1=FW Sears, Zemansky MW  |author2=Young HD
 |name-list-style=amp | title=College physics
|url=https://books.google.com/books?id=AvVQAAAAMAAJ&q=zemansky+%22permittivity+of+empty+space%22  |publisher=Addison-Wesley |location=Reading, Mass.
|year=1985
|page=40
|isbn=978-0-201-07836-7}}</ref> or "permittivity of [[free space]]"<ref name="Saleh">B. E. A. Saleh and M. C. Teich, ''Fundamentals of Photonics'' (Wiley, 1991)</ref> are widespread. Standards Organizations worldwide now use "electric constant" as a uniform term for this quantity,<ref name="CODATA"/> and official standards documents have adopted the term (although they continue to list the older terms as synonyms).<ref>{{SIbrochure8th|page=12}}</ref><ref name="IUPAC">{{Cite journal|last=Braslavsky |first=S.E. | url=http://www.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; see p. 348 |doi=10.1351/pac200779030293 |s2cid=96601716 }}</ref>

Another historical synonym was "dielectric constant of vacuum", as "dielectric constant" was sometimes used in the past for the absolute permittivity.<ref name="Freie Universität Berlin">{{Cite web|title=Naturkonstanten |url=http://www.chemie.fu-berlin.de/chemistry/general/constants.html |publisher=[[Freie Universität Berlin]] }}</ref><ref>{{Cite book
  | last = King
  | first = Ronold W. P.
  | title = Fundamental Electromagnetic Theory
  | publisher = Dover
  | year = 1963
  | location = New York
  | page = 139}}</ref>  However, in modern usage "dielectric constant" typically refers exclusively to a [[relative permittivity]] ''ε''/''ε''<sub>0</sub> and even this usage is considered "obsolete" by some standards bodies in favor of [[relative static permittivity]].<ref name="IUPAC"/><ref>{{Cite book|author=[[IEEE]] Standards Board|title=IEEE Standard Definitions of Terms for Radio Wave Propagation|year=1997 |page=6|doi=10.1109/IEEESTD.1998.87897|isbn=978-0-7381-0580-2}}</ref> Hence, the term "dielectric constant of vacuum" for the electric constant ''ε''<sub>0</sub> is considered obsolete by most modern authors, although occasional examples of continuing usage can be found.

As for notation, the constant can be denoted by either ''&epsilon;''<sub>0</sub> or ''{{not a typo|&varepsilon;}}''<sub>0</sub>, using either of the common [[glyph]]s for the letter [[epsilon]].

==Historical origin of the parameter ''ε''<sub>0</sub>==
As indicated above, the parameter ''ε''<sub>0</sub> is a measurement-system constant. Its presence in the equations now used to define electromagnetic quantities is the result of the so-called "rationalization" process described below. But the method of allocating a value to it is a consequence of the result that Maxwell's equations predict that, in free space, electromagnetic waves move with the speed of light.  Understanding why ''ε''<sub>0</sub> has the value it does requires a brief understanding of the history.

===Rationalization of units===
The experiments of [[Charles-Augustin de Coulomb|Coulomb]] and others showed that the force ''F'' between two, equal, point-like "amounts" of electricity that are situated a distance ''r'' apart in free space, should be given by a formula that has the form

:<math> F = k_{\text{e}} \frac{Q^2}{r^2}, </math>

where ''Q'' is a quantity that represents the amount of electricity present at each of the two points, and ''k''<sub>e</sub> is the [[Coulomb constant]]. If one is starting with no constraints, then the value of ''k''<sub>e</sub> may be chosen arbitrarily.<ref name=Jackson>For an introduction to the subject of choices for independent units, see
{{Cite book
|author=John David Jackson
|title=Classical electrodynamics
|edition=Third
|chapter=Appendix on units and dimensions
|pages=775 ''et seq.''
|publisher= Wiley
|location=New York
|year=1999
|isbn=978-0-471-30932-1
|chapter-url=http://worldcat.org/isbn/047130932X}}
</ref> For each different choice of ''k''<sub>e</sub> there is a different "interpretation" of ''Q'': to avoid confusion, each different "interpretation" has to be allocated a distinctive name and symbol.

In one of the systems of equations and units agreed in the late 19th century, called the "centimeter–gram–second electrostatic system of units" (the cgs esu system), the constant ''k''<sub>e</sub> was taken equal to 1, and a quantity now called "[[Gaussian units#Unit of charge|Gaussian electric charge]]" ''q''<sub>s</sub> was defined by the resulting equation

:<math> F = \frac{{q_{\text{s}}}^2}{r^2}. </math>

The unit of Gaussian charge, the [[statcoulomb]], is such that two units, at a distance of 1 centimeter apart, repel each other with a force equal to the cgs unit of force, the [[dyne]]. Thus, the unit of Gaussian charge can also be written 1 dyne<sup>1/2</sup> cm. "Gaussian electric charge" is not the same mathematical quantity as modern ([[MKS system of units|MKS]] and subsequently the [[International System of Units|SI]]) electric charge and is not measured in coulombs.

The idea subsequently developed that it would be better, in situations of spherical geometry, to include a factor 4π in equations like Coulomb's law, and write it in the form:

:<math> F = k'_{\text{e}} \frac{{q'_{\text{s}}}^2}{4 \pi r^2}. </math>

This idea is called "rationalization". The quantities ''q''<sub>s</sub>&prime; and ''k''<sub>e</sub>&prime; are not the same as those in the older convention. Putting {{nowrap|1=''k''<sub>e</sub>&prime; = 1}} generates a unit of electricity of different size, but it still has the same dimensions as the cgs esu system.

The next step was to treat the quantity representing "amount of electricity" as a fundamental quantity in its own right, denoted by the symbol ''q'', and to write Coulomb's Law in its modern form:

:<math>\ F = \frac{1}{4 \pi \varepsilon_0} \frac{q^2}{r^2}. </math>

The system of equations thus generated is known as the rationalized meter–kilogram–second (rmks) equation system, or "meter–kilogram–second–ampere (mksa)" equation system. The new quantity ''q'' is given the name "rmks electric charge", or (nowadays) just "electric charge".{{cn|date=December 2023}} The quantity ''q''<sub>s</sub> used in the old cgs esu system is related to the new quantity ''q'' by:

:<math>\ q_{\text{s}} = \frac{q}{\sqrt{4 \pi \varepsilon_0}}.</math>

In the [[2019 redefinition of the SI base units]] the elementary charge is fixed at 1.602176634 10<sup>−19</sup> ampere-seconds and the value of the vacuum permittivity must be determined experimentally.<ref name=Brochure9_2019>
{{cite web
 |title = 9th edition of the SI Brochure
 |publisher = BIPM
 |url = https://www.bipm.org/en/publications/si-brochure/
 |date = 2019
 |access-date = 20 May 2019
 |df = dmy-all
}}</ref>{{rp|132}}
===Determination of a value for ''ε''<sub>0</sub>===
One now adds the requirement that one wants force to be measured in newtons, distance in meters, and charge to be measured in the engineers' practical unit, the coulomb, which is defined as the charge accumulated when a current of 1 ampere flows for one second.  This shows that the parameter ''ε''<sub>0</sub> should be allocated the unit C<sup>2</sup>⋅N<sup>−1</sup>⋅m<sup>−2</sup> (or equivalent units – in practice "farads per meter").

In order to establish the numerical value of ''ε''<sub>0</sub>, one makes use of the fact that if one uses the rationalized forms of Coulomb's law and [[Ampère's force law]] (and other ideas) to develop [[Maxwell's equations]], then the relationship stated above is found to exist between ''ε''<sub>0</sub>, ''μ''<sub>0</sub> and ''c''<sub>0</sub>. In principle, one has a choice of deciding whether to make the coulomb or the ampere the fundamental unit of electricity and magnetism.  The decision was taken internationally to use the ampere.  This means that the value of ''ε''<sub>0</sub> is determined by the values of ''c''<sub>0</sub> and ''μ''<sub>0</sub>, as stated above. For a brief explanation of how the value of ''μ''<sub>0</sub> is decided, see ''[[Vacuum permeability]]''.

==Permittivity of real media==
By convention, the electric constant ''ε''<sub>0</sub> appears in the relationship that defines the [[electric displacement field]] '''D''' in terms of the [[electric field]] '''E''' and classical electrical [[polarization density]] '''P''' of the medium.  In general, this relationship has the form:

: <math>\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P}.</math>

For a linear dielectric, '''P''' is assumed to be proportional to '''E''', but a delayed response is permitted and a spatially non-local response, so one has:<ref name="Sólyom">

{{cite book |title=Fundamentals of the physics of solids: Electronic properties |author=Jenö Sólyom |chapter-url=https://books.google.com/books?id=XSo-a2n43xEC&pg=PA17 |page=17 |chapter=Equation 16.1.50 |isbn=978-3-540-85315-2 |year=2008 |publisher=Springer}}
</ref>

: <math>\mathbf D (\mathbf r,\ t) = \int_{-\infty}^t dt' \int d^3 \mathbf r'\ \varepsilon \left(\mathbf r,\ t; \mathbf r',\ t'\right) \mathbf E\left(\mathbf r',\ t'\right). </math>

In the event that nonlocality and delay of response are not important, the result is:

:<math>\mathbf{D} = \varepsilon \mathbf{E} = \varepsilon_{\text{r}} \varepsilon_0 \mathbf{E}</math>

where ''ε'' is the [[permittivity]] and ''ε''<sub>r</sub> the [[relative static permittivity]]. In the [[Vacuum#In electromagnetism|vacuum of classical electromagnetism]], the polarization {{nowrap|1='''P''' = '''0'''}}, so {{nowrap|1=''ε''<sub>r</sub> = 1}} and {{nowrap|1=''ε'' = ''ε''<sub>0</sub>}}.

==See also==
*[[Casimir effect]]
*[[Coulomb's law]]
*[[Electromagnetic wave equation]]
*[[ISO 31-5]]
*[[Mathematical descriptions of the electromagnetic field]]
*[[Relative permittivity]]
*[[Sinusoidal plane-wave solutions of the electromagnetic wave equation]]
*[[Wave impedance]]
*[[Vacuum permeability]]

==Notes==
{{reflist|2}}

{{Use dmy dates|date=August 2019}}

{{DEFAULTSORT:Vacuum Permittivity}}
[[Category:Electromagnetism]]
[[Category:Fundamental constants]]