Vacuum permittivity

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This article is about a physical constant. For the ordinal in mathematics ε₀, see epsilon nought.

Vacuum permittivity, referred to by international standards organizations as the electric constant¹², and denoted by the symbol ε₀, is a fundamental physical constant relating the mechanical quantities (time, length, mass) to the units for electrical charge, for example, in Coulomb's law.

In SI units the speed of light in vacuum c₀³ is defined as the numerical value c₀ $\overset{\underset{\mathrm{def}}{}}{=}$ 299 792 458 m s^-1 (NIST definition of meter: see last sentence) and the magnetic constant μ₀ is defined as 4π x 10^-7 H · m^-1 (NIST definition of ampere:see last sentence), leading to an electric constant defined in free space by:

$\varepsilon_0$	$\overset{\underset{\mathrm{def}}{}}{=}\frac {1}{\mu_0 {c_0}^2}$
	$\approx 8.854\ 187\ 817\ldots \times 10^{-12}$	A²s⁴ kg^-1m⁻³	in SI base units;
		or C²N⁻¹m⁻²	or F m⁻¹ using derived units.
	$\approx 5.526\ 35\ \ldots \times 10^{7}$	e¹V^-1m^-1	converting to elementary charges.

(This value is taken from NIST ε₀. A summary of these definitions is provided in the 2006 CODATA Report.⁴) The ellipsis "…" does not indicate experimental uncertainty, but the arbitrary termination of a nonrecurring decimal.

This value is called by various other names as well, including the permittivity of free space,⁵ or of empty space,⁶ or by the term dielectric constant of vacuum⁷ (although this term is ambiguous in modern usage, as described below).

1 Usage in other unit systems
2 Terminology
3 Permittivity of vacuum
4 Footnotes
5 See also

Usage in other unit systems

In other systems of electromagnetic units, it is common to have $\varepsilon_0 = 1$ . This is the case in Gaussian units, Lorentz–Heaviside units, and some choices of natural units (while some other choices set $\varepsilon_0 = 1/4\pi$ , for example, electrostatic cgs units).

The Coulomb force constant or electrostatic constant k_e can thus be expressed as (see Coulomb's law):

$k_e\overset{\underset{\mathrm{def}}{}}{=}\frac{1}{ 4 \pi \varepsilon_0} = \frac {\mu_0 {c_0}^2}{4 \pi} = 8.987\ 551\ 787\ldots \times 10^9 \approx 9\cdot 10^9$ C⁻²N¹·m²

$\approx 1.439\ 96\ \ldots \times 10^{-9}$ e^-1V¹m¹.

Terminology

Historically, the physical constant ε₀ has been known by many different names. The terms "vacuum permittivity" or its variants, such as "permittivity in/of vacuum",⁸⁹ "permittivity of empty space",⁶ or "permittivity of free space"⁵¹⁰ are widespread. Standards organizations world-wide now use "electric constant" as a uniform term for this quantity,¹ and official standards documents have adopted the term (although they continue to list the older terms as synonyms).¹¹¹²

Another historical synonym was "dielectric constant of vacuum", as "dielectric constant" was sometimes used in the past for the absolute permittivity.⁷¹³ However, in modern usage "dielectric constant" typically refers exclusively to a relative permittivity $\varepsilon / \varepsilon_0$ and even this usage is considered "obsolete" by some standards bodies in favor of relative static permittivity.¹²¹⁴ 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.

The more recent term electric constant avoids the use of permittivity in the name of ε₀, and also the use of free space and of vacuum (which is not as simple a term as once thought, see free space). The term "electric constant" avoids the suggestion that ε₀, which is a derived quantity based upon the defined value of c₀ and μ₀ as indicated above, is a "property" of anything physically attainable.

As for notation, the constant can be denoted by either $\varepsilon_0\,$ or $\epsilon_0\,$ , using either of the common glyphs for the letter epsilon.

Permittivity of vacuum

Free space is an idealized reference state that can be approached but is physically unattainable. Realizable vacuum is called partial vacuum.¹⁵

The permittivity of free space is ε₀ by definition. In other words, the relative permittivity of free space is 1 by definition. Supposing free space were attainable, the meter is defined to make c₀ a fixed number, and the ampere is defined so μ₀ is a fixed number, thus fixing ε₀ as well. (For an introduction to the subject of choices for independent units, see Jackson.¹⁶)

Unlike the vacuum of classical physics, today's physical vacuum corresponds to what is called the vacuum state or the quantum vacuum, which is "by no means a simple empty space".¹⁷¹⁸ Thus, free space is not a synonym for the physical vacuum. For more detail, see the articles on free space and vacuum state.

Regarding any partial vacuum used in a laboratory to set up standards for the SI units, the question arises whether that partial vacuum is an adequate realization of free space, and just what corrections (if any) must be applied to the experimental results. For example, corrections for non-zero pressure could be made.¹⁹ Should experiment eventually support new features of the vacuum state,²⁰ the predicted corrections to date are so small that they would have no effect upon the "necessary corrections [to] be applied to take account of actual conditions"¹⁹ in setting up standards for the meter or ampere.

For a discussion of achieving a good partial vacuum, see the articles ultra high vacuum and free space.

Footnotes

^ ^a ^b CODATA. "Electric constant". 2006 CODATA recommended values. NIST. Retrieved on 2007-08-08.
^ In German, elektrische Feldkonstante
^ Quote from NIST: "Current practice is to use c₀ to denote the speed of light in vacuum according to ISO 31. In the original Recommendation of 1983, the symbol c was used for this purpose." See NIST Special Publication 330, Appendix 2, p. 45
^ CODATA report, pp. 6-7
^ ^a ^b B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991)
^ ^a ^b FW Sears, Zemansky MW & Young HD (1985). College physics. Reading, Mass.: Addison-Wesley. p.p. 40. ISBN 0201078368, http://books.google.com/books?id=AvVQAAAAMAAJ&q=zemansky+%22permittivity+of+empty+space%22&dq=zemansky+%22permittivity+of+empty+space%22&lr=&as_brr=0&pgis=1.
^ ^a ^b "Naturkonstanten". Freie Universität Berlin.
^ SM Sze & Ng KK (2007). Physics of semiconductor devices (Third Edition ed.). New York: Wiley-Interscience. p.Appendix E, p. 788. ISBN 0-471-14323-5, http://worldcat.org/isbn/0-471-14323-5.
^ RS Muller, Kamins TI & Chan M (2003). Device electronics for integrated circuits (Third Edition ed.). New York: Wiley. p.Inside front cover. ISBN 0-471-59398-2, http://worldcat.org/isbn/0-471-59398-2.
^ Sam Bowen (1991). "What is the significance of permittivity of free space?". Ask a Scientist. Argonne National Laboratory.
^ International Bureau of Weights and Measures (2006). "The International System of Units (SI)" (PDF) p. 12.
^ ^a ^b Braslavsky, S.E. (2007), "Glossary of terms used in photochemistry (IUPAC recommendations 2006)", Pure and Applied Chemistry 79: p. 293-465; see p. 348., http://www.iupac.org/publications/pac/2007/pdf/7903x0293.pdf
^ King, Ronold W. P. (1963). Fundamental Electromagnetic Theory. New York: Dover. pp.p. 139.
^ IEEE Standards Board (1997). "IEEE Standard Definitions of Terms for Radio Wave Propagation" p. 6.
^ The term partial vacuum suggests one major source of departure of a an approximate vacuum from free space, namely non-zero pressure. However, there are additional possible sources of nonideality, for example, strong electric or magnetic fields. See, for example,Di Piazza et al.: Light diffraction by a strong standing electromagnetic wave Phys.Rev.Lett. 97 (2006) 083603, Gies, H et al.: Polarized light propagating in a magnetic field as a probe for millicharged fermions Phys. Rev. Letts. 97 (2006) 140402
^ John David Jackson (1999). Classical electrodynamics (Third Edition ed.). New York: Wiley. p.Appendix on units and dimensions; pp. 775 et seq... ISBN 047130932X, http://worldcat.org/isbn/047130932X.
^ Astrid Lambrecht (Hartmut Figger, Dieter Meschede, Claus Zimmermann Eds.) (2002). Observing mechanical dissipation in the quantum vacuum: an experimental challenge; in Laser physics at the limits. Berlin/New York: Springer. p.p. 197. ISBN 3540424180, http://books.google.com/books?id=0DUjDAPwcqoC&pg=PA197&dq=%22vacuum+state%22&lr=&as_brr=0&sig=-gfWcR7RdymYL3W-M2VxVQPFm10#PPA197,M1.
^ Walter Dittrich & Gies H (2000). Probing the quantum vacuum: perturbative effective action approach. Berlin: Springer. ISBN 3540674284, http://books.google.com/books?id=DyhyFSL7bNUC&pg=PP1&dq=intitle:Probing+intitle:the+intitle:Quantum+intitle:Vacuum&lr=&as_brr=0&sig=VSfMMLJnmYyWplC2L5i9oVSjurg#PPA1,M1.
^ ^a ^b As to such corrections, CIPM RECOMMENDATION 1 (CI-2002) p. 195 says only:

♦ that in all cases any necessary corrections be applied to take account of actual conditions such as diffraction, gravitation or imperfection in the vacuum.

CIPM is an acronym for International Committee for Weights and Measures.
^ See, for example, CC Davis et al. Experimental challenges involved in searches for ... nonlinear QED effects by sensitive optical techniques

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