# Vacuum permittivity

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The physical constant ε0 (pronounced as "epsilon nought" or "epsilon zero"), commonly called the vacuum permittivity, permittivity of free space or electric constant or the distributed capacitance of the vacuum, is an ideal, (baseline) physical constant, which is the value of the absolute dielectric permittivity of classical vacuum. It has the CODATA value (as of 2018)

A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant, which has a fixed numerical value, but does not directly involve any physical measurement.

In electromagnetism, absolute permittivity, often simply called permittivity, usually denoted by the Greek letter ε (epsilon), is the measure of capacitance that is encountered when forming an electric field in a particular medium. More specifically, permittivity describes the amount of charge needed to generate one unit of electric flux in a given medium. A charge will yield more electric flux in a medium with low permittivity than in a medium with high permittivity. Permittivity is the measure of a material's ability to store an electric field in the polarization of the medium.

## Contents

ε0 = 8.8541878128(13)×10−12 F⋅m−1 (farads per metre), with a relative uncertainty of 1.5×10−10. [1]
Value of ε0Unit
8.854 187 8128(13)×10-12 Fm -1
55.263 494 06 e 2GeV -1fm -1
1/(4π) qp 2Ep -1lp -1

It is the capability of the vacuum to permit electric field lines. This constant relates the units for electric charge to mechanical quantities such as length and force. [2] For example, the force between two separated electric charges (in the vacuum of classical electromagnetism) is given by Coulomb's law:

An electric field surrounds an electric charge, and exerts force on other charges in the field, attracting or repelling them. Electric field is sometimes abbreviated as E-field. The electric field is defined mathematically as a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. The SI unit for electric field strength is volt per meter (V/m). Newtons per coulomb (N/C) is also used as a unit of electric field strength. Electric fields are created by electric charges, or by time-varying magnetic fields. Electric fields are important in many areas of physics, and are exploited practically in electrical technology. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, and the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature.

Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charge: positive and negative. Like charges repel and unlike attract. An object with an absence of net charge is referred to as neutral. Early knowledge of how charged substances interact is now called classical electrodynamics, and is still accurate for problems that do not require consideration of quantum effects.

Coulomb's law, or Coulomb's inverse-square law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventionally called electrostatic force or Coulomb force. The quantity of electrostatic force between stationary charges is always described by Coulomb's law. The law was first published in 1785 by French physicist Charles-Augustin de Coulomb, and was essential to the development of the theory of electromagnetism, maybe even its starting point, because it was now possible to discuss quantity of electric charge in a meaningful way.

${\displaystyle \ F_{\text{C}}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}}$

The value of the constant fraction, ${\displaystyle 1/4\pi \varepsilon _{0}}$, is approximately 9 × 109 N⋅m2⋅C−2, q1 and q2 are the charges, and r is the distance between them. Likewise, ε0 appears in Maxwell's equations, which describe the properties of electric and magnetic fields and electromagnetic radiation, and relate them to their sources.

Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. An important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed (c) in a vacuum. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum of light from radio waves to γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon.

A magnetic field is a vector field that describes the magnetic influence of electric charges in relative motion and magnetized materials. The effects of magnetic fields are commonly seen in permanent magnets, which pull on magnetic materials and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges such as those used in electromagnets. They exert forces on nearby moving electrical charges and torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field vary with location. As such, it is described mathematically as a vector field.

In physics, electromagnetic radiation refers to the waves of the electromagnetic field, propagating (radiating) through space, carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) light, ultraviolet, X-rays, and gamma rays.

## Value

The value of ε0 is defined by the formula [3]

${\displaystyle \varepsilon _{0}={\frac {1}{\mu _{0}c^{2}}}}$

where c is the defined value for the speed of light in classical vacuum in SI units, [4] and μ0 is the parameter that international Standards Organizations call the "magnetic constant" (commonly called vacuum permeability). Since μ0 has an approximate value 4π × 10−7  H/m, [5] and c has the defined value 299792458 m⋅s−1, [6] it follows that ε0 can be expressed numerically as

The speed of light in vacuum, commonly denoted c, is a universal physical constant important in many areas of physics. Its exact value is 299792458 metres per second. It is exact because by international agreement a metre is defined as the length of the path travelled by light in vacuum during a time interval of 1/299792458 second. According to special relativity, c is the upper limit for the speed at which conventional matter and information can travel. Though this speed is most commonly associated with light, it is also the speed at which all massless particles and field perturbations travel in vacuum, including electromagnetic radiation and gravitational waves. Such particles and waves travel at c regardless of the motion of the source or the inertial reference frame of the observer. Particles with nonzero rest mass can approach c, but can never actually reach it. In the special and general theories of relativity, c interrelates space and time, and also appears in the famous equation of mass–energy equivalence E = mc2.

The International System of Units is the modern form of the metric system and is the most widely used system of measurement. It comprises a coherent system of units of measurement built on seven base units, which are the second, metre, kilogram, ampere, kelvin, mole, candela, and a set of twenty prefixes to the unit names and unit symbols that may be used when specifying multiples and fractions of the units. The system also specifies names for 22 derived units, such as lumen and watt, for other common physical quantities.

The henry is the SI derived unit of electrical inductance. If a current of 1 ampere flowing through the coil produces flux linkage of 1 weber turn, the coil has a self inductance of 1 henry.‌ The unit is named after Joseph Henry (1797–1878), the American scientist who discovered electromagnetic induction independently of and at about the same time as Michael Faraday (1791–1867) in England.

${\displaystyle \varepsilon _{0}={\frac {1}{(4\pi \times 10^{-7}\,{\textrm {N/A}}^{2})(299792458\,{\textrm {m/s}})^{2}}}={\frac {625000}{22468879468420441\pi }}\,{\textrm {F/m}}\approx 8.85418781762039\times 10^{-12}\,{\textrm {F}}{\cdot }{\textrm {m}}^{-1}}$ (or A 2s 4kg −1m −3 in SI base units, or C 2N −1m −2 or CV −1m −1 using other SI coherent units). [7] [8]

The historical origins of the electric constant ε0, and its value, are explained in more detail below.

### Redefinition of the SI units

The ampere was redefined by defining the elementary charge as an exact number of coulombs as from 20 May 2019, [9] with the effect that the vacuum electric permeability no longer has an exactly determined value in SI units. The value of the electron charge became a numerically defined quantity, not measured, making μ0 a measured quantity. Consequently, ε0 is not exact. As before, it is defined by the equation ε0 = 1/(μ0c2), and is thus determined by the value of μ0, the magnetic vacuum permeability which in turn is determined by the experimentally determined dimensionless fine-structure constant α:

The ampere, often shortened to "amp", is the base unit of electric current in the International System of Units (SI). It is named after André-Marie Ampère (1775–1836), French mathematician and physicist, considered the father of electrodynamics.

The elementary charge, usually denoted by e or sometimes qe, is the electric charge carried by a single proton or, equivalently, the magnitude of the electric charge carried by a single electron, which has charge −1 e. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called the elementary positive charge.

The physical constant μ0,, commonly called the vacuum permeability, permeability of free space, permeability of vacuum, or magnetic constant, is the magnetic permeability in a classical vacuum. Vacuum permeability is derived from production of a magnetic field by an electric current or by a moving electric charge and in all other formulas for magnetic-field production in a vacuum.

${\displaystyle \varepsilon _{0}={\frac {1}{\mu _{0}c^{2}}}={\frac {e^{2}}{2\alpha hc}}\ ,}$

with e being the elementary charge, h being the Planck constant, and c being the speed of light in vacuum, each with exactly defined values. The relative uncertainty in the value of ε0 is therefore the same as that for the dimensionless fine-structure constant, namely 1.5×10−10. [10]

The Planck constant, or Planck's constant, denoted is a physical constant that is the quantum of electromagnetic action, which relates the energy carried by a photon to its frequency. A photon's energy is equal to its frequency multiplied by the Planck constant. The Planck constant is of fundamental importance in quantum mechanics, and in metrology it is the basis for the definition of the kilogram.

## Terminology

Historically, the parameter ε0 has been known by many different names. The terms "vacuum permittivity" or its variants, such as "permittivity in/of vacuum", [11] [12] "permittivity of empty space", [13] or "permittivity of free space" [14] are widespread. Standards Organizations worldwide now use "electric constant" as a uniform term for this quantity, [7] and official standards documents have adopted the term (although they continue to list the older terms as synonyms). [15] [16] In the new SI system, the permittivity of vacuum will not be a constant anymore, but a measured quantity, related to the (measured) dimensionless fine structure constant.

Another historical synonym was "dielectric constant of vacuum", as "dielectric constant" was sometimes used in the past for the absolute permittivity. [17] [18] However, in modern usage "dielectric constant" typically refers exclusively to a relative permittivity ε/ε0 and even this usage is considered "obsolete" by some standards bodies in favor of relative static permittivity. [16] [19] Hence, the term "dielectric constant of vacuum" for the electric constant ε0 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 ${\displaystyle \varepsilon _{0}\,}$ or ${\displaystyle \epsilon _{0}\,}$, using either of the common glyphs for the letter epsilon.

## Historical origin of the parameter ε0

As indicated above, the parameter ε0 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 ε0 has the value it does requires a brief understanding of the history.

### Rationalization of units

The experiments of Coulomb and others showed that the force F between two equal point-like "amounts" of electricity, situated a distance r apart in free space, should be given by a formula that has the form

${\displaystyle F=k_{\text{e}}{\frac {Q^{2}}{r^{2}}},}$

where Q is a quantity that represents the amount of electricity present at each of the two points, and ke is the Coulomb constant. If one is starting with no constraints, then the value of ke may be chosen arbitrarily. [20] For each different choice of ke 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 "centimetre–gram–second electrostatic system of units" (the cgs esu system), the constant ke was taken equal to 1, and a quantity now called "gaussian electric charge" qs was defined by the resulting equation

${\displaystyle F={\frac {{q_{\text{s}}}^{2}}{r^{2}}}.}$

The unit of gaussian charge, the statcoulomb, is such that two units, a distance of 1 centimetre 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 dyne1/2 cm. "Gaussian electric charge" is not the same mathematical quantity as modern (MKS and subsequently the 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:

${\displaystyle F=k'_{\text{e}}{\frac {{q'_{\text{s}}}^{2}}{4\pi r^{2}}}.}$

This idea is called "rationalization". The quantities qs′ and ke′ are not the same as those in the older convention. Putting ke′ = 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:

${\displaystyle \ F={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q^{2}}{r^{2}}}.}$

The system of equations thus generated is known as the rationalized metre–kilogram–second (rmks) equation system, or "metre–kilogram–second–ampere (mksa)" equation system. This is the system used to define the SI units. [21] The new quantity q is given the name "rmks electric charge", or (nowadays) just "electric charge". Clearly, the quantity qs used in the old cgs esu system is related to the new quantity q by

${\displaystyle \ q_{\text{s}}={\frac {q}{\sqrt {4\pi \varepsilon _{0}}}}.}$

### Determination of a value for ε0

One now adds the requirement that one wants force to be measured in newtons, distance in metres, 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 ε0 should be allocated the unit C2⋅N−1⋅m−2 (or equivalent units – in practice "farads per metre").

In order to establish the numerical value of ε0, 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 ε0, μ0 and c0. 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 ε0 is determined by the values of c0 and μ0, as stated above. For a brief explanation of how the value of μ0 is decided, see the article about μ0.

## Permittivity of real media

By convention, the electric constant ε0 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:

${\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} .}$

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: [22]

${\displaystyle \mathbf {D} (\mathbf {r} ,\ t)=\int _{-\infty }^{t}dt'\int d^{3}\mathbf {r} '\ \varepsilon (\mathbf {r} ,\ t;\mathbf {r} ',\ t')\mathbf {E} (\mathbf {r} ',\ t').}$

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

${\displaystyle \mathbf {D} =\varepsilon \mathbf {E} =\varepsilon _{\text{r}}\varepsilon _{0}\mathbf {E} }$

where ε is the permittivity and εr the relative static permittivity. In the vacuum of classical electromagnetism, the polarization P = 0, so εr = 1 and ε = ε0.

## Notes

1. "2018 CODATA Value: vacuum electric permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
2. "electric constant". Electropedia: International Electrotechnical Vocabulary (IEC 60050). Geneva: International Electrotechnical Commission. Retrieved 26 March 2015..
3. The approximate numerical value is found at: "Electric constant, ε0". NIST reference on constants, units, and uncertainty: Fundamental physical constants. NIST. Retrieved 22 January 2012. This formula determining the exact value of ε0 is found in Table 1, p. 637 of PJ Mohr; BN Taylor; DB Newell (April–June 2008). "Table 1: Some exact quantities relevant to the 2006 adjustment in CODATA recommended values of the fundamental physical constants: 2006" (PDF). Rev Mod Phys. 80 (2): 633–729. arXiv:. Bibcode:2008RvMP...80..633M. doi:10.1103/RevModPhys.80.633.
4. Quote from NIST: "The symbol c is the conventional symbol for the speed of light in vacuum. " See NIST Special Publication 330, p. 18
5. See the last sentence of the NIST definition of ampere.
6. See the last sentence of the NIST definition of meter.
7. Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2008). "CODATA Recommended Values of the Fundamental Physical Constants: 2006" (PDF). Reviews of Modern Physics . 80 (2): 633–730. arXiv:. Bibcode:2008RvMP...80..633M. doi:10.1103/RevModPhys.80.633. Archived from the original (PDF) on 1 October 2017.
8. A summary of the definitions of c, μ0 and ε0 is provided in the 2006 CODATA Report: CODATA report, pp. 6–7
9. "Resolution 1 of 24th meeting of the General Conference on Weights and Measures". On the possible future revision of the International System of Units, the SI (PDF). Sèvres, France: International Bureau for Weights and Measures. 21 October 2011.
10. "2018 CODATA Value: fine-structure constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 20 May 2019.
11. SM Sze & Ng KK (2007). "Appendix E". Physics of semiconductor devices (Third ed.). New York: Wiley-Interscience. p. 788. ISBN   978-0-471-14323-9.
12. RS Muller, Kamins TI & Chan M (2003). Device electronics for integrated circuits (Third ed.). New York: Wiley. Inside front cover. ISBN   978-0-471-59398-0.
13. FW Sears, Zemansky MW & Young HD (1985). College physics. Reading, Mass.: Addison-Wesley. p. 40. ISBN   978-0-201-07836-7.
14. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991)
15. International Bureau of Weights and Measures (2006). "The International System of Units (SI)" (PDF). p. 12.
16. Braslavsky, S.E. (2007). "Glossary of terms used in photochemistry (IUPAC recommendations 2006)" (PDF). Pure and Applied Chemistry. 79 (3): 293–465, see p. 348. doi:10.1351/pac200779030293.
17. King, Ronold W. P. (1963). Fundamental Electromagnetic Theory. New York: Dover. p. 139.
18. IEEE Standards Board (1997). "IEEE Standard Definitions of Terms for Radio Wave Propagation" (PDF). p. 6.
19. For an introduction to the subject of choices for independent units, see John David Jackson (1999). "Appendix on units and dimensions". Classical electrodynamics (Third ed.). New York: Wiley. pp. 775 et seq. ISBN   978-0-471-30932-1.
20. Jenö Sólyom (2008). "Equation 16.1.50". Fundamentals of the physics of solids: Electronic properties. Springer. p. 17. ISBN   978-3-540-85315-2.

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