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Vacuum permittivity

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

The parameter ε0 (commonly called the vacuum permittivity, but referred to by international Standards Organizations as the electric constant[1][2]) is a constant used in connection with the rationalized metre-kilogram-second (rmks) equation system. This system is also called the "metre-kilogram-second-ampere (mksa)" equation system, and is part of the International System of Quantities (ISQ) used to define SI units.

The value of ε0 is defined by the formula

where c0 is the speed of light in vacuum,[3] and μ0 is the parameter that international Standards Organizations call the "magnetic constant" (commonly called vacuum permeability). Since μ0 has the defined value 4π × 10−7 H m−1,[4] and c0 has the defined value 299792458 m s−1,[5] it follows that ε0 has a defined value given approximately by

ε0 ≈ 8.854 187 817... × 10−12 A2 s4 kg−1 m−3 in SI base units (or C2 N−1m−2 or F m−1 or C V−1m−1 using other SI coherent units),

or

ε0 ≈ 0.055 263 5... eV V−2 nm−1.

This value is taken from NIST ε0. A summary of these definitions is provided in the 2006 CODATA Report.[6] The ellipsis (...) does not indicate experimental uncertainty, but the arbitrary termination of a nonrecurring decimal. The historical origins of the electric constant ε0, and its value, are explained in more detail below.

By convention, the electric constant ε0 appears in the relationship that defines the electric displacement field D in terms of the electric field E - see the article on permittivity. In real media this relationship has the form: D = ε0E + P, where P is the classical electrical polarization density of the medium. In the reference state of free space, called "vacuum" by Standards Organizations, the polarization P = 0.

The view (sometimes encountered) that ε0 is a physical constant that describes a physical property of a realizable "vacuum" is incorrect. Rather, ε0 is a measurement-system constant introduced and defined as a result of international agreement. The value allocated to ε0 relates to the velocity of light in a reference situation or benchmark, sometimes called free space, used as a baseline for comparison of measurements made in all types of real media. The physical properties of realizable vacuums such as outer space, ultra-high vacuum, QCD vacuum or quantum vacuum are experimental and theoretical matters, separate from ε0. The meaning and value of ε0 are metrology issues, not issues about properties of realizable vacuums. This potential for confusion is why many Standards Organizations now prefer to use the name "electric constant" for ε0.

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",[7][8] "permittivity of empty space",[9] or "permittivity of free space"[10] are widespread. Standards Organizations worldwide 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).[11][12]

Another historical synonym was "dielectric constant of vacuum", as "dielectric constant" was sometimes used in the past for the absolute permittivity.[13][14] 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.[12][15] 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 already noted, the name now preferred by Standards Organizations is electric constant. This name avoids the use of the term permittivity in the name of ε0, and also avoids the use of free space and of vacuum (which is not as simple a concept as once was thought, see free space). The name "electric constant" avoids the suggestion that ε0, which is a derived quantity based upon the defined values of c0 and μ0 as indicated above, is a "property" of anything physical. It appears to be thought that the name "electric constant" is less likely to cause misunderstandings than the older names.

As for notation, the constant can be denoted by either or , 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 of how electromagnetic measurement systems developed.

In the following discussion, it may be noted that classically no distinction was made between "vacuum" and free space. Today, in the literature, the term "vacuum" may refer to a variety of experimental conditions and theoretical entities. In reading the literature, only context can decide what is meant. Below, the term "free space" is used to refer to the reference state called "vacuum" by Standards Organizations.

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

where Q is a quantity that represents the amount of electricity present at each of the two points, and ke is a constant. If one is starting with no constraints, then the value of ke may be chosen arbitrarily. 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 1800s, 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

The unit of gaussian charge 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. This has the same dimensions as the SI unit: N1/2 m. "Gaussian electric charge" is not the same mathematical quantity as modern (rmks) 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:

This idea is called "rationalization". The quantities q's 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 SI unit: N1/2 m.

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 "rmks" form:

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 qs=q/(4πε0)1/2.

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.

(For an introduction to the subject of choices for independent units, see Jackson.[16])

Realizable "vacuum" and free space

Free space is an idealized reference state that can be approached but is physically unattainable. Realizable vacuum sometimes is called partial vacuum, referring to the need for ultra low pressures, but ultra low pressure is not the sole criterion for approximating free space.[17]

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".[18][19] 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 to refer these measurements to the baseline. For example, corrections for non-zero pressure could be made.[20] Should experiment eventually support new features of the vacuum state,[21] 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"[20] 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.

None of this affects the meaning or value of the electric constant ε0, which has a value defined by the velocity of light in the baseline reference state called "vacuum" by Standards Organizations.

See also

Notes

  1. ^ a b CODATA. "Electric constant". 2006 CODATA recommended values. NIST. Retrieved 2007-08-08.
  2. ^ In German, elektrische Feldkonstante
  3. ^ 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
  4. ^ See the last sentence of the NIST definition of ampere.
  5. ^ See the last sentence of the NIST definition of meter.
  6. ^ CODATA report, pp. 6-7
  7. ^ SM Sze & Ng KK (2007). Physics of semiconductor devices (Third ed.). New York: Wiley-Interscience. Appendix E, p. 788. ISBN 0-471-14323-5. {{cite book}}: Unknown parameter |nopp= ignored (|no-pp= suggested) (help)
  8. ^ RS Muller, Kamins TI & Chan M (2003). Device electronics for integrated circuits (Third ed.). New York: Wiley. Inside front cover. ISBN 0-471-59398-2. {{cite book}}: Unknown parameter |nopp= ignored (|no-pp= suggested) (help)
  9. ^ FW Sears, Zemansky MW & Young HD (1985). College physics. Reading, Mass.: Addison-Wesley. p. 40. ISBN 0201078368.
  10. ^ B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991)
  11. ^ International Bureau of Weights and Measures (2006). "The International System of Units (SI)" (PDF). pp. p. 12. {{cite web}}: |pages= has extra text (help)
  12. ^ a b Braslavsky, S.E. (2007), "Glossary of terms used in photochemistry ([[IUPAC]] recommendations 2006)" (PDF), Pure and Applied Chemistry, 79: 293–465, see p. 348., doi:10.1351/pac200779030293 {{citation}}: URL–wikilink conflict (help)
  13. ^ "Naturkonstanten". Freie Universität Berlin.
  14. ^ King, Ronold W. P. (1963). Fundamental Electromagnetic Theory. New York: Dover. p. 139.
  15. ^ IEEE Standards Board (1997). "IEEE Standard Definitions of Terms for Radio Wave Propagation" (PDF). pp. p. 6. {{cite web}}: |pages= has extra text (help)
  16. ^ John David Jackson (1999). Classical electrodynamics (Third ed.). New York: Wiley. Appendix on units and dimensions; pp. 775 et seq.. ISBN 047130932X. {{cite book}}: Unknown parameter |nopp= ignored (|no-pp= suggested) (help)
  17. ^ 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
  18. ^ 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. 197. ISBN 3540424180.{{cite book}}: CS1 maint: multiple names: authors list (link)
  19. ^ Walter Dittrich & Gies H (2000). Probing the quantum vacuum: perturbative effective action approach. Berlin: Springer. ISBN 3540674284.
  20. ^ 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.
    In addition,
    ♦ …the metre is considered a unit of proper length. Its definition, therefore, applies only within a spatial extent sufficiently small that the effects of the non-uniformity of the gravitational field can be ignored.
    CIPM is an acronym for International Committee for Weights and Measures.
  21. ^ See, for example, CC Davis et al. Experimental challenges involved in searches for ... nonlinear QED effects by sensitive optical techniques
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