Value of μ_{0} |
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1.25663706127(20)×10^{−6} N⋅A^{−2} |
The vacuum magnetic permeability (variously vacuum permeability, permeability of free space, permeability of vacuum, magnetic constant) is the magnetic permeability in a classical vacuum. It is a physical constant, conventionally written as μ_{0} (pronounced "mu nought" or "mu zero"). It quantifies the strength of the magnetic field induced by an electric current. Expressed in terms of SI base units, it has the unit kg⋅m⋅s^{−2}·A^{−2}. It can be also expressed in terms of SI derived units, N·A^{−2}.
Since the redefinition of SI units in 2019 (when the values of e and h were fixed as defined quantities), μ_{0} is an experimentally determined constant, its value being proportional to the dimensionless fine-structure constant, which is known to a relative uncertainty of 1.6×10^{−10},^{[1]}^{[2]}^{[3]}^{[4]} with no other dependencies with experimental uncertainty. Its value in SI units as recommended by CODATA is:
From 1948^{[6]} to 2019, μ_{0} had a defined value (per the former definition of the SI ampere), equal to:^{[7]}
The deviation of the recommended measured value from the former defined value is within it uncertainty.
The terminology of permeability and susceptibility was introduced by William Thomson, 1st Baron Kelvin in 1872.^{[8]} The modern notation of permeability as μ and permittivity as ε has been in use since the 1950s.
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 magnetic force F_{m} per length L is given by^{[9]}
From 1948 until 2019 the ampere 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 to 2×10^{−7} newton per metre of length". This is equivalent to a definition of of exactly 4π×10^{−7} H/m,^{[a]} since
NIST/CODATA refers to μ_{0} as the vacuum magnetic permeability.^{[10]} Prior to the 2018 redefinition, it was referred to as the magnetic constant.^{[11]} Historically, the constant μ_{0} has had different names. In the 1987 IUPAP Red book, for example, this constant was called the permeability of vacuum.^{[12]} Another, now rather rare and obsolete, term is "magnetic permittivity of vacuum". See, for example, Servant et 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 because μ_{0} 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, 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. 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_{0} 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 distance r apart, in each of which a current I flows, the force per unit length, F_{m}/L, that one wire exerts upon the other in the vacuum of free space would be given by
In the old "electromagnetic (emu)" system of units, defined in the late 19th century, k_{m} was chosen to be a pure number equal to 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", 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"), k_{m} is written as μ_{0}/2π, where μ_{0} is a measurement-system constant called the "magnetic constant".^{[b]} The value of μ_{0} was chosen such that the rmks unit of current is equal in size to the ampere in the emu system: μ_{0} was defined to be 4π × 10^{−7} H/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 the International System of Units.
The magnetic constant μ_{0} appears in Maxwell's equations, which describe the properties of electric and magnetic fields and electromagnetic radiation, and relate them to their sources. In particular, it appears in relationship to quantities such as permeability and 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:
In the International System of Quantities (ISQ), the speed of light in vacuum, c,^{[15]} is related to the magnetic constant and the electric constant (vacuum permittivity), ε_{0}, by the equation:
Conversely, as the permittivity is related to the fine-structure constant (α), the permeability can be derived from the latter (using the Planck constant, h, and the elementary charge, e):
In the new 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.