# Vacuum permeability

The physical constant μ0, (pronounced "mu nought" or "mu zero"), commonly called the vacuum permeability, permeability of free space, permeability of vacuum, or magnetic constant, is the 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. As of May 20, 2019, the vacuum permeability μ0 will no longer be a defined constant (per the former definition of the SI ampere), but rather will need to be determined experimentally.

The value in SI units, by CODATA 2018, is given below. It is simply proportional to the dimensionless fine-structure constant, with no other dependencies.[1][2][3]

μ0 = 1.25663706212(19)×10−6 H/m

Before this, in the reference medium of classical vacuum, μ0 had an exact defined value:[4][5]

μ0 = ×10−7 H/m = 1.2566370614×10−6 N/A2 (1 henry per metre ≡ newton per square ampere)

## The ampere defined vacuum permeability

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 length L is given by[6]

${\displaystyle {\frac {|{\boldsymbol {F}}_{m}|}{L}}={\mu _{0} \over 2\pi }{|{\boldsymbol {I}}|^{2} \over |{\boldsymbol {r}}|}.}$

Adopted in 1948, the effect of this definition was to fix the magnetic constant (permeability of vacuum) at exactly 4π×10−7 H/m.[a] To further illustrate: The ampere was 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 2×10−7 newton per meter of length.

${\displaystyle {\frac {{\boldsymbol {F}}_{m}}{L}}={\mu _{0} \over 2\pi }{(1\ {\rm {A)^{2}}} \over {{1}\ {\rm {m}}}}}$
${\displaystyle {{2}\times 10^{-7}{{\rm {N}} \over m}}={\mu _{0} \over 2\pi }{(1\ {\rm {A)^{2}}} \over {{1}\ {\rm {m}}}}}$
${\displaystyle \mu _{0}=4\pi \times 10^{-7}{\text{ H/m}}}$

In the SI system which has gone into force in 2019, this value is determined experimentally; 4π × 1.000 000 000 55 (15) 10−7 H·m−1 is a recently measured value in the new system.

## Terminology

Standards Organizations have recently moved to magnetic constant as the preferred name for μ0, although the older name continues to be listed as a synonym.[7] Historically, the constant μ0 has had different names. In the 1987 IUPAP Red book, for example, this constant was still called permeability of vacuum.[8] Another, now rather rare and obsolete, term is "magnetic permittivity of vacuum". See, for example, Servant et al.[9] The term "vacuum permeability" (and variations thereof, such as "permeability of free space") remains very widespread.

The name "magnetic constant" was used by standards organizations in order to avoid use of the terms "permeability" and "vacuum", which have physical meanings. This change of preferred name had been made because μ0 was a defined value, and was not the result of experimental measurement (see below). In the new SI system, the permeability of vacuum will not be a constant anymore, but a measured quantity, related to the (measured) dimensionless fine structure constant.

## Systems of units and historical origin of value of μ0

In principle, there are several equation systems that could be used to set up a system of electrical quantities and units.[10] 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 μ0 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, Fm, that one wire exerts upon the other in the vacuum of free space would be given by

${\displaystyle F_{\mathrm {m} }\propto {\frac {I^{2}}{r}}.\;}$

Writing the constant of proportionality as km gives

${\displaystyle F_{\mathrm {m} }=k_{\mathrm {m} }{\frac {I^{2}}{r}}.\;}$

The form of km 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 equations defined in the late 19th century, km 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.

In another system, the "rationalized metre–kilogram–second (rmks) system" (or alternatively the "metre–kilogram–second–ampere (mksa) system"), km is written as μ0/2π, where μ0 is a measurement-system constant called the "magnetic constant".[b] The value of μ0 was chosen such that the rmks unit of current is equal in size to the ampere in the emu system: μ0 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.

Ampère's law as stated above describes a physical property of the world. However, the choices about the form of km and the value of μ0 are totally human decisions, taken by international bodies composed of representatives of the national standards organizations of all participating countries. The parameter μ0 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.[c] This is why the relevant Standards Organizations prefer the name "magnetic constant", rather than any name that carries the hidden and misleading implication that μ0 describes some physical property.[citation needed]

## Significance in electromagnetism

The magnetic constant μ0 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:

${\displaystyle {\boldsymbol {H}}={{\boldsymbol {B}} \over \mu _{0}}-{\boldsymbol {M}},}$

where M is the magnetization density. In vacuum, M = 0.

In SI units, the speed of light in vacuum, c,[11] is related to the magnetic constant and the electric constant (vacuum permittivity), ε0, by the definition:

${\displaystyle c={1 \over {\sqrt {\mu _{0}\varepsilon _{0}}}}.}$

This relation can be derived using Maxwell's equations of classical electromagnetism in the medium of classical vacuum, but this relation is used by BIPM (International Bureau of Weights and Measures) and NIST (National Institute of Standards and Technology) as a definition of ε0 in terms of the defined numerical values for c and μ0, and is not presented as a derived result contingent upon the validity of Maxwell's equations.[12]

Conversely, as the permittivity is related to the fine structure constant (${\displaystyle \alpha }$ ), the permeability can be derived from the latter (using Planck's constant, h, and the electric charge on an electron, e):

${\displaystyle \mu _{0}={\frac {2\alpha }{e^{2}}}{\frac {h}{c}}.}$

In the new SI units, only the fine structure constant is an actual measured value in this formula.

## Notes

1. ^ a b This choice defines the SI unit of current, the ampere: "Unit of electric current (ampere)". Historical context of the SI. NIST. Retrieved 2007-08-11.
2. ^ The decision to explicitly include the factor of 2π in km stems from the "rationalization" of the equations used to describe physical electromagnetic phenomena.
3. ^ 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 μ0.[citation needed]

## References

1. ^
2. ^ Parker, Richard H.; Yu, Chenghui; Zhong, Weicheng; Estey, Brian; Müller, Holger (2018-04-13). "Measurement of the fine-structure constant as a test of the Standard Model". Science. 360 (6385): 191–195. arXiv:1812.04130. Bibcode:2018Sci...360..191P. doi:10.1126/science.aap7706. ISSN 0036-8075. PMID 29650669.
3. ^ Davis, Richard S. (2017). "Determining the value of the fine-structure constant from a current balance: Getting acquainted with some upcoming changes to the SI". American Journal of Physics. 85 (5): 364–368. arXiv:1610.02910. Bibcode:2017AmJPh..85..364D. doi:10.1119/1.4976701. ISSN 0002-9505.
4. ^ "Magnetic constant". Fundamental Physical Constants. Committee on Data for Science and Technology. 2006. Retrieved 2010-02-04 – via National Institute of Standards and Technology.
5. ^ Rosen, Joe (2004). "Permeability (Physics)". Encyclopedia of Physics. Facts on File science library. New York: Facts On File. ISBN 9780816049745. Retrieved 2010-02-04.
6. ^ See for example equation 25-14 in Tipler, Paul A. (1992). Physics for Scientists and Engineers, Third Edition, Extended Version. New York, NY: Worth Publishers. p. 826. ISBN 978-0-87901-434-6.
7. ^ See Table 1 in Mohr, Peter J; Taylor, Barry N; Newell, David B (2008). "CODATA Recommended Values of the Fundamental Physical Constants: 2006" (PDF). Reviews of Modern Physics. 80 (2): 633–730. arXiv:0801.0028. Bibcode:2008RvMP...80..633M. CiteSeerX 10.1.1.150.1225. doi:10.1103/RevModPhys.80.633.
8. ^ SUNAMCO (1987). "Recommended values of the fundamental physical constants" (PDF). Symbols, Units, Nomenclature and Fundamental Constants in Physics. p. 54.
9. ^ Lalanne, J.-R.; Carmona, F.; Servant, L. (1999). Optical spectroscopies of electronic absorption. World Scientific Series in Contemporary Chemical Physics. 17. p. 10. Bibcode:1999WSSCP..17.....L. doi:10.1142/4088. ISBN 978-981-02-3861-2.
10. ^ For an introduction to the subject of choices for independent units, see John David Jackson (1998). Classical electrodynamics (Third ed.). New York: Wiley. p. 154. ISBN 978-0-471-30932-1.
11. ^ "2018 CODATA Value: speed of light in vacuum". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
12. ^ The exact numerical value is found at: "Electric constant, ε0". NIST reference on constants, units, and uncertainty: Fundamental physical constants. NIST. Retrieved 2012-01-22. This formula determining the exact value of ε0 is found in Table 1, p. 637 of Mohr, Peter J; Taylor, Barry N; Newell, David B (2008). "CODATA recommended values of the fundamental physical constants: 2006" (PDF). Reviews of Modern Physics. 80 (2): 633–730. arXiv:0801.0028. Bibcode:2008RvMP...80..633M. CiteSeerX 10.1.1.150.1225. doi:10.1103/RevModPhys.80.633.