| Value ofμ0 |
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| 1.25663706127(20)×10−6 N⋅A−2 |
Thevacuum magnetic permeability (variouslyvacuum permeability,permeability of free space,permeability of vacuum,magnetic constant) is themagnetic permeability in aclassical vacuum. It is aphysical constant, conventionally written asμ0 (pronounced "mu nought" or "mu zero"). It quantifies the strength of themagnetic field induced by anelectric current. Expressed in terms ofSI base units, it has the unit kg⋅m⋅s−2⋅A−2. It can be also expressed in terms ofSI derived units,N⋅A−2.
Since therevision of the SI in 2019 (when the values ofe andh were fixed as defined quantities),μ0 is an experimentally determined constant, its value being proportional to the dimensionlessfine-structure constant, which is known to a relative uncertainty of1.6×10−10,[1][2][3][4] with no other dependencies with experimental uncertainty. Its value in SI units as recommended byCODATA is:
The terminology ofpermeability andsusceptibility was introduced byWilliam Thomson, 1st Baron Kelvin in 1872.[6] The modern notation of permeability asμ andpermittivity asε has been in use since the 1950s.
Two thin, straight, stationary, parallel wires, a distancer apart infree space, each carrying acurrentI, will exert a force on each other.Ampère's force law states that the magnetic forceFm per lengthL is given by[7]
From 1948 until 2019 theampere was defined as "that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to2×10−7 newton per metre of length". This is equivalent to a definition of of exactly4π×10−7 H/m,[a] sinceThe current in this definition needed to be measured with a known weight and known separation of the wires, defined in terms of the international standards of mass, length and time in order to produce a standard for theampere (and this is what theKibble balance was designed for). In the2019 revision of the SI, theampere is defined exactly in terms of theelementary charge and thesecond, and the value of is determined experimentally;4π × 0.99999999987(16)×10−7 H⋅m−1 is the 2022 CODATA value in the new system (and the Kibble balance has become an instrument for measuring weight from a known current, rather than measuring current from a known weight).
From 1948[8] to 2019,μ0 had a defined value (per the former definition of theSI ampere), equal to:[9]
The deviation of the recommended measured value from the former defined value is within its uncertainty.
NIST/CODATA refers toμ0 as thevacuum magnetic permeability.[10] Prior to the 2019 revision, it was referred to as themagnetic constant.[11] Historically, the constantμ0 has had different names. In the 1987IUPAP Red book, for example, this constant was called thepermeability of vacuum.[12] Another, now rather rare and obsolete, term is "magnetic permittivity of vacuum". See, for example, Servantet al.[13] Variations thereof, such as "permeability of free space", remain widespread.
The name "magnetic constant" was briefly used by standards organizations in order to avoid use of the terms "permeability" and "vacuum", which have physical meanings. The change of name had been made becauseμ0 was a defined value, and was not the result of experimental measurement (see below). In the new SI system, the permeability of vacuum no longer has a defined value, but is a measured quantity, with an uncertainty related to that of the (measured) dimensionless fine structure constant.
In principle, there are several equation systems that could be used to set up a system of electrical quantities and units.[14]Since the late 19th century, the fundamental definitions of current units have been related to the definitions of mass, length, and time units, usingAmpère'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.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 whyμ0 has the value it does is as follows.
Ampère's force law describes the experimentally-derived fact that, for two thin, straight, stationary, parallel wires, a distancer apart, in each of which a currentI flows, the force per unit length,Fm/L, that one wire exerts upon the other in the vacuum offree space would be given byWriting the constant of proportionality askm givesThe form ofkm 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.
In the old"electromagnetic (emu)" system of units, defined in the late 19th century,km was chosen to be a pure number equal to 2, distance was measured in centimetres, force was measured in the cgs unitdyne, and the currents defined by this equation were measured in the "electromagnetic unit (emu) of current", 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.
In another system, the "rationalized metre–kilogram–second (rmks) system" (or alternatively the "metre–kilogram–second–ampere (mksa) system"),km is written asμ0/2π, whereμ0 is a measurement-system constant called the "magnetic constant".[b]The value ofμ0 was chosen such that the rmks unit of current is equal in size to the ampere in the emu system:μ0 wasdefined to be4π × 10−7H/m.[a]
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 theInternational System of Units.
The magnetic constantμ0 appears inMaxwell's equations, which describe the properties ofelectric andmagnetic fields andelectromagnetic radiation, and relate them to their sources. In particular, it appears in relationship to quantities such aspermeability andmagnetization density, such as the relationship that defines the magneticH-field in terms of the magneticB-field. In real media, this relationship has the form:whereM is the magnetization density. Invacuum,M =0.
In theInternational System of Quantities (ISQ), thespeed of light in vacuum,c,[15] is related to the magnetic constant and theelectric constant (vacuum permittivity),ε0, by the equation:This relation can be derived usingMaxwell's equations of classical electromagnetism in the medium ofclassical vacuum. Between 1948 and 2018, this relation was used by BIPM (International Bureau of Weights and Measures) and NIST (National Institute of Standards and Technology) as adefinition ofε0 in terms of the defined numerical value forc and, prior to 2018, the defined numerical value forμ0. During this period of standards definitions, it wasnot presented as a derived result contingent upon the validity of Maxwell's equations.[16]
Conversely, as the permittivity is related to thefine-structure constant (α), the permeability can be derived from the latter (using thePlanck constant,h, and theelementary charge,e):
In thenew SI units, only the fine structure constant is a measured value in SI units in the expression on the right, since the remaining constants have defined values in SI units.
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