Vacuum permeability

The physical constant μ₀, commonly called the vacuum permeability, permeability of free space, permeability of vacuum 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, µ₀ has an exact defined value:^[1][2]

µ₀ = 4π×10⁻⁷ N / A² ≈ 1.2566370614...×10⁻⁶ H / m or T·m / A or Wb / (A·m) or V·s / (A·m)

in the SI system of units.

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.^[3][4]

The ampere defines vacuum permeability

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 2×10⁻⁷ newton per meter of length.

Adopted in 1948, the effect of this definition is to fix the magnetic constant (permeability of vacuum) at exactly 4π×10⁻⁷ H / m.^[5] To further illustrate:

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]

$\frac{|\boldsymbol{F}_m|}{L}={\mu_0\over2\pi}{|\boldsymbol{I}|^2\over|\boldsymbol{r}|}.$

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 2×10⁻⁷ N / m. Hence the value of μ₀ is defined to be exactly

$\mu_0 = 4 \pi \times 10^{-7} (\rm{N / A ^2} ) \approx 1.2566370614 \cdots \times 10 ^{-6} (\rm{N / A ^2} )$^[7][8]

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.^[8] Historically, the constant μ₀ has had different names. In the 1987 IUPAP Red book, for example, this constant was still called permeability of vacuum.^[9] Another, now rather rare and obsolete, term is "magnetic permittivity of vacuum". See, for example, Servant et al.^[10] The term "vacuum permeability" (and variations thereof, such as "permeability of free space") remains very widespread.

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 μ₀ is a defined value, and is not the result of experimental measurement (see below).

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.^[11] 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, F_m, that one wire exerts upon the other in the vacuum of free space would be given by

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

Writing the constant of proportionality as k_m gives

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

The form of k_m 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 1800s, k_m 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"), k_m is written as μ₀/2π, where μ₀ is a measurement-system constant called the "magnetic constant".^[12] 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 is defined to be 4π × 10⁻⁷ N / A².^[5]

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 k_m 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.^[13] 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.

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:

$\boldsymbol{H}={\boldsymbol{B}\over\mu_0}-\boldsymbol{M},$

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

In SI units, the speed of light in vacuum, c₀ is related to the magnetic constant and the electric constant (vacuum permittivity), ε₀, by the definition:

$c_0={1\over\sqrt{\mu_0\varepsilon_0}}.$

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 and NIST as a definition of ε₀ in terms of the defined numerical values for c₀ and μ₀, and is not presented as a derived result contingent upon the validity of Maxwell's equations.^[14]

See also

• Characteristic impedance of vacuum
• Electromagnetic wave equation
• Sinusoidal plane-wave solutions of the electromagnetic wave equation
• Mathematical descriptions of the electromagnetic field
• New SI definitions

References and notes

1. "Magnetic constant". Fundamental Physical Constants. Committee on Data for Science and Technology. 2006. Retrieved 2010-02-04 – via National Institute of Standards and Technology.
2. Rosen, Joe (2004). "Permeability (Physics)". Encyclopedia of Physics. Facts on File science library. New York: Facts On File. ISBN 9780816049745. Retrieved 2010-02-04.(registration required)
3. "Introduction to the constants for nonexperts". Fundamental Physical Constants. Committee on Data for Science and Technology. 2006. Retrieved 2010-02-04 – via National Institute of Standards and Technology.
4. Weiglhofer, Werner S. (2003). "§ 4.1 The classical vacuum as reference medium". In Weiglhofer, Werner S.; Lakhtakia, Akhlesh. Introduction to complex mediums for optics and electromagnetics. Press Monographs 123. Bellingham, Washington, United States: SPIE Press. p. 34 ff. ISBN 978-0-8194-4947-4 – via Google Books.
5. 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.
6. See for example Tipler, Paul A. (1992). Physics for Scientists and Engineers, Third Edition, Extended Version. New York, NY: Worth Publishers. p. 826. ISBN 0-87901-434-2.Equation 25-14
7. "Magnetic constant". 2006 CODATA recommended values. NIST. Retrieved 2007-08-08.
8. "CODATA Recommended Values of the Fundamental Physical Constants: 2006" (PDF). Committee on Data for Science and Technology (CODATA): See Table 1. NIST.
9. SUNAMCO (1987). "Recommended values of the fundamental physical constants". Symbols, Units, Nomenclature and Fundamental Constants in Physics (PDF). p. 54 http://metrology.files.wordpress.com/2010/12/6_recommended_fundamental_constants_iupap_sunamco_red_book_1987.pdf. Missing or empty |title= (help); (the IUPAP "Red book").
10. J R Lalanne, F Carmona & L Servant (1999). Optical spectroscopies of electronic absorption. (World Scientific series in contemporary chemical physics, vol. 17. ed.). Singapore;London: World Scientific. p. 10. ISBN 981-02-3861-4.
11. 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 0-471-30932-X.
12. The decision to explicitly include the factor of 2π in k_m stems from the "rationalization" of the equations used to describe physical electromagnetic phenomena.
13. 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.
14. 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 PJ Mohr, BN Taylor, DB Newell (April–June 2008). "Table 1: Some exact quantities relevant to the 2006 adjustment in CODATA recommended values of the fundamental physical constants: 2006" (PDF). Rev Mod Phys 80 (2): 633–729. arXiv:0801.0028. Bibcode:2008RvMP...80..633M. doi:10.1103/RevModPhys.80.633.