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| {{about|the magnetic constant|the analogous electric constant|vacuum permittivity}}
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| The [[physical constant]] ''μ''<sub>0</sub>, commonly called the '''vacuum permeability''', '''permeability of free space''', or '''magnetic constant''' is an ideal, (baseline) physical constant, which is the value of [[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. In the reference medium of [[classical vacuum]], ''µ''<sub>0</sub> has an exact defined value:<ref name=CODATA-magnetic-constant> | |
| {{Cite web| last =CODATA| title =Magnetic constant| work =Fundamental Physical Constants| publisher =NIST| date = | url=http://physics.nist.gov/cgi-bin/cuu/Value?mu0
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| | format =2006 CODATA recommended values: [http://physics.nist.gov/cuu/Constants/bibliography.html Source of the CODATA internationally recommended values]| accessdate =2010-02-04}}</ref><ref>Rosen, Joe. "Permeability (Physics)." Encyclopedia of Physics. New York: Facts On File, Inc., 2004. Science Online. Facts On File, Inc. http://www.fofweb.com/Science/default.asp?ItemID=WE40 (accessed 2010-02-04)</ref>
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| :: ''µ''<sub>0</sub> = {{val|4|end= π|e=-7}} [[Volt|V]]·[[Second|s]]/([[Ampere|A]]·m) ≈ {{val|1.2566370614|end=...|e=-6|ul=H*m-1}} or [[Newton (unit)|N]]·[[Ampere|A]]<sup>−2</sup> or [[Tesla (unit)|T]]·m/A or [[Weber (unit)|Wb]]/(A·m)
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| in the [[SI]] system of units.
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| As a constant, it can also be defined as a fundamental invariant quantity, and is also one of three components that defines [[free space]] through [[Maxwell's equations]]. In [[classical physics]], ''free space'' is a concept of [[electromagnetic theory]], corresponding to a theoretically perfect [[vacuum]] and sometimes referred to as the '''vacuum of free space''', or as '''classical vacuum''', and is appropriately viewed as a ''reference'' medium.<ref name=CODATA-introduction-constants>
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| {{Cite web| last =CODATA| title =Introduction to the constants for nonexperts | work =Fundamental Physical Constants| publisher =NIST| date =
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| | url =http://physics.nist.gov/cuu/Constants/introduction.html
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| | format =2006 CODATA recommended values: [http://physics.nist.gov/cuu/Constants/bibliography.html Source of the CODATA internationally recommended values]| accessdate =2010-02-04}}</ref><ref name=weig>
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| {{cite book| title = Introduction to complex mediums for optics and electromagnetics| author = Werner S. Weiglhofer and Akhlesh Lakhtakia| publisher = SPIE Press| chapter=§ 4.1 The classical vacuum as reference medium| year = 2003
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| | isbn = 978-0-8194-4947-4| page = 34 ''ff''| url = http://books.google.com/?id=QtIP_Lr3gngC&pg=PA34}}</ref>
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| ==The ampere defines vacuum permeability==
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| The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to {{val|2|e=-7}} newton per meter of length.
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| Adopted in 1948, the effect of this definition is to fix the magnetic constant (permeability of vacuum) at exactly {{val|4|end=π|e=-7|u=H*m-1}}.<ref name="NIST amp hist"/> To further illustrate:
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| Two thin, straight, stationary, parallel wires, a distance ''r'' apart in [[free space]], each carrying a current ''I'', will exert a force on each other. [[Ampère's force law]] states that the force per unit length is given by<ref>
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| See for example {{Cite book | last = Tipler | first = Paul A.
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| | title = Physics for Scientists and Engineers, Third Edition, Extended Version
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| | publisher = Worth Publishers | year = 1992 | location = New York, NY
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| | page = 826 | url =
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| | isbn = 0-87901-434-2}}Equation 25-14</ref>
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| :<math>|\boldsymbol{F}_m|={\mu_0\over2\pi}{|\boldsymbol{I}|^2\over|\boldsymbol{r}|}.</math>
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| The ampere is defined so that if the wires are 1 m apart and the current in each wire is 1 A, the force between the two wires is {{val|2|e=-7|ul=N m-1}}.
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| Hence the value of ''μ''<sub>0</sub> is ''defined'' to be exactly
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| :<math>\mu_0 = 4 \pi \times 10^{-7} (\rm{N / A ^2} ) \approx 1.2566370614 \cdots \times 10 ^{-6} (\rm{N / A ^2} )</math><ref name="NIST">{{cite web |url=http://physics.nist.gov/cgi-bin/cuu/Value?mu0 |title=Magnetic constant |accessdate=2007-08-08 |work=2006 [[CODATA]] recommended values |publisher=[[NIST]]}}</ref><ref name="CODATA">{{cite web |url=http://physics.nist.gov/cuu/Constants/codata.pdf |title=CODATA Recommended Values of the Fundamental Physical Constants: 2006 |work=Committee on Data for Science and Technology (CODATA): See Table 1 |publisher=[[NIST]]}}</ref>
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| ==Terminology==
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| Historically, the constant ''μ''<sub>0</sub> has had different names. In the 1987 [[IUPAP]] Red book, for example, this constant was still called ''permeability of vacuum''.<ref>{{Cite book|author=[[SUNAMCO]] | chapter=Recommended values of the fundamental physical constants | url=http://metrology.files.wordpress.com/2010/12/6_recommended_fundamental_constants_iupap_sunamco_red_book_1987.pdf |work=Symbols, Units, Nomenclature and Fundamental Constants in Physics |format=PDF |year=1987 | page=54 }}; (the [[IUPAP]] "Red book").</ref> Another, now rather rare and obsolete, term is "''magnetic permittivity of vacuum''". See, for example, Servant ''et al.''<ref name=Servant>{{cite book
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| |author=J R Lalanne, F Carmona & L Servant
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| |title=Optical spectroscopies of electronic absorption.
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| |edition=World Scientific series in contemporary chemical physics, vol. 17.
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| |page=10
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| |publisher= World Scientific
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| |location=Singapore;London
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| |year=1999
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| |isbn=981-02-3861-4
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| |url=http://books.google.com/?id=7rWD-TdxKkMC&pg=PA10&lpg=PA10&dq=+%22magnetic+permittivity%22}}
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| </ref>
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| The term "vacuum permeability" (and variations thereof, such as "permeability of free space") remains very widespread. However, Standards Organizations have recently moved to '''magnetic constant''' as the preferred name for ''μ''<sub>0</sub>, although the older name continues to be listed as a synonym.<ref name="CODATA"/>
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| The name "magnetic constant" is used by standards organizations in order to avoid use of the terms "permeability" and "vacuum", which have physical meanings. This change of preferred name has been made because ''μ''<sub>0</sub> is a defined value, and is not the result of experimental measurement (see below).
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| ==Systems of units and historical origin of value of ''μ''<sub>0</sub>==
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| In principle, there are several equation systems that could be used to set up a system of electrical quantities and units.<ref name=Jackson>For an introduction to the subject of choices for independent units, see {{cite book
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| |author=John David Jackson
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| |title=Classical electrodynamics
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| |edition=Third
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| |page=154
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| |publisher= Wiley
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| |location=New York
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| |year=1998
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| |isbn=0-471-30932-X
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| |url=http://worldcat.org/isbn/047130932X}}</ref>
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| Since the late 19th century, the fundamental definitions of current units have been related to the definitions of mass, length and time units, using [[Ampère's force law]]. However, the precise way in which this has "officially" been done has changed many times, as measurement techniques and thinking on the topic developed.
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| The overall history of the unit of electric current, and of the related question of how to define a set of equations for describing electromagnetic phenomena, is very complicated. Briefly, the basic reason why ''μ''<sub>0</sub> has the value it does is as follows.
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| Ampère's force law describes the experimentally-derived fact that, for two thin, straight, stationary, parallel wires, a distance ''r'' apart, in each of which a current ''I'' flows, the force per unit length, ''F''<sub>m</sub>, that one wire exerts upon the other in the vacuum of [[free space]] would be given by
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| ::<math> F_{\mathrm{m}} \propto \frac {I^2} {r}. \; </math>
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| Writing the constant of proportionality as ''k''<sub>m</sub> gives
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| ::<math> F_{\mathrm{m}} = k_{\mathrm{m}} \frac {I^2} {r}. \; </math>
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| The form of ''k''<sub>m</sub> needs to be chosen in order to set up a system of equations, and a value then needs to be allocated in order to define the unit of current.
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| In the old [[Centimetre gram second system of units|"electromagnetic (emu)" system of equations]] defined in the late 1800s, ''k''<sub>m</sub> was chosen to be a pure number, 2, distance was measured in centimetres, force was measured in the cgs unit [[dyne]], and the currents defined by this equation were measured in the "electromagnetic unit (emu) of current" (also called the "[[abampere]]"). A practical unit to be used by electricians and engineers, the ampere, was then defined as equal to one tenth of the electromagnetic unit of current.
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| In another system, the "rationalized-metre-kilogram-second (rmks) system" (or alternatively the "metre-kilogram-second-ampere (mksa) system"), ''k''<sub>m</sub> is written as ''μ''<sub>0</sub>/2π, where ''μ''<sub>0</sub> is a measurement-system constant called the "magnetic constant".<ref>The decision to explicitly include the factor of 2π in ''k''<sub>m</sub> stems from the "rationalization" of the equations used to describe physical electromagnetic phenomena.</ref>
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| The value of ''μ''<sub>0</sub> was chosen such that the rmks unit of current is equal in size to the ampere in the emu system: ''μ''<sub>0</sub> is ''defined'' to be 4π × 10<sup>−7</sup> N A<sup>−2</sup>.<ref name="NIST amp hist">This choice defines the SI unit of current, the ampere: {{cite web |url=http://physics.nist.gov/cuu/Units/ampere.html |title=Unit of electric current (ampere) |accessdate=2007-08-11 |work=Historical context of the SI |publisher=[[NIST]] }}</ref>
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| Historically, several different systems (including the two described above) were in use simultaneously. In particular, physicists and engineers used different systems, and physicists used three different systems for different parts of physics theory and a fourth different system (the engineers' system) for laboratory experiments. In 1948, international decisions were made by standards organizations to adopt the rmks system, and its related set of electrical quantities and units, as the single main international system for describing electromagnetic phenomena in the [[International System of Units]].
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| Ampère's law as stated above describes a physical property of the world. However, the choices about the form of ''k''<sub>m</sub> and the value of ''μ''<sub>0</sub> are totally human decisions, taken by international bodies composed of representatives of the national standards organizations of all participating countries. The parameter ''μ''<sub>0</sub> is a measurement-system constant, not a physical constant that can be measured. It does not, in any meaningful sense, describe a physical property of the vacuum.<ref>The magnetic permeability of a realizable vacuum (such as [[outer space]], or [[ultra-high vacuum]]), which is ''measurable'' at least in principle, is distinct from the ''defined'' parameter ''μ''<sub>0</sub>.{{Citation needed|date=November 2013}}</ref> This is why the relevant Standards Organizations prefer the name "magnetic constant", rather than any name that carries the hidden and misleading implication that ''μ''<sub>0</sub> describes some physical property{{Citation needed|date=November 2013}}.
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| ==Significance in electromagnetism==
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| The magnetic constant ''μ''<sub>0</sub> appears in [[Maxwell's equations]], which describe the properties of [[Electric field|electric]] and [[Magnetic field|magnetic]] fields and [[electromagnetic radiation]], and relate them to their sources. In particular, it appears in relationship to quantities such as [[Permeability (electromagnetism)|permeability]] and [[Magnetization|magnetization density]], such as the relationship that defines the magnetic '''''H'''''-field in terms of the magnetic '''''B'''''-field. In real media, this relationship has the form:
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| :<math>\boldsymbol{H}={\boldsymbol{B}\over\mu_0}-\boldsymbol{M},</math> | |
| where '''''M''''' is the magnetization density. In [[vacuum]], '''''M''''' = 0.
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| In SI units, the [[speed of light]] in vacuum, [http://physics.nist.gov/cgi-bin/cuu/Value?c ''c''<sub>0</sub>] is related to the magnetic constant and the [[Vacuum permittivity|electric constant (vacuum permittivity)]], ''ε''<sub>0</sub>, by the definition:
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| :<math>c_0={1\over\sqrt{\mu_0\varepsilon_0}}.</math>
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| This relation can be derived using [[Maxwell's equations]] of classical electromagnetism in the medium of [[Vacuum#In electromagnetism|classical vacuum]], but this relation is used by BIPM and NIST as a ''definition'' of ε<sub>0</sub> in terms of the defined numerical values for c<sub>0</sub> and μ<sub>0</sub>, and is ''not'' presented as a derived result contingent upon the validity of Maxwell's equations.<ref name=ep0>
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| The exact numerical value is found at:
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| {{cite web |title=Electric constant, ''ε''<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 |accessdate=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 |author=PJ Mohr, BN Taylor, DB Newell|bibcode = 2008RvMP...80..633M |doi = 10.1103/RevModPhys.80.633 |arxiv = 0801.0028 }}
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| </ref>
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| ==See also==
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| *[[Characteristic impedance of vacuum]]
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| *[[Electromagnetic wave equation]]
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| *[[Sinusoidal plane-wave solutions of the electromagnetic wave equation]]
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| *[[Mathematical descriptions of the electromagnetic field]]
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| *[[New SI definitions]]
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| ==References and notes==
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| {{reflist}}
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| {{DEFAULTSORT:Vacuum Permeability}}
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| [[Category:Fundamental constants]]
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| [[ca:Permeabilitat del buit]]
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| [[el:Μαγνητική σταθερά]]
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| [[it:Permeabilità magnetica]]
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| [[sl:indukcijska konstanta]]
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| [[uk:Магнітна константа]]
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