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 magnetic force F_m per length L is given by[9]

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

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 meter apart in vacuum, would produce between these conductors a force equal to 2×10⁻⁷ newton per meter of length". This is equivalent to a definition of ${\displaystyle \mu _{0}}$ of exactly 4π×10⁻⁷ H/m.[lower-alpha 1], since

$${\displaystyle {\frac {{\boldsymbol {F}}_{\text{m}}}{L}}={\mu _{0} \over 2\pi }\mathrm {(1\,A)^{2} \over {1\,m}} }$$

$${\displaystyle {2\times 10^{-7}\mathrm {N \over m} }={\mu _{0} \over 2\pi }\mathrm {(1\,A)^{2} \over {1\,m}} }$$

$${\displaystyle \mu _{0}=4\pi \times 10^{-7}{\text{ H/m}}}$$

The 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 the ampere (and this is what the Kibble balance was designed for). In the 2019 redefinition of the SI base units, the ampere is defined exactly in terms of the elementary charge and the second, and the value of ${\displaystyle \mu _{0}}$ is determined experimentally; 4π × 1.00000000055(15)×10⁻⁷ H⋅m⁻¹ is a recently measured 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).

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.[10] Historically, the constant μ₀ has had different names. In the 1987 IUPAP Red book, for example, this constant was still called permeability of vacuum.[11] Another, now rather rare and obsolete, term is "magnetic permittivity of vacuum". See, for example, Servant et al.[12] 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 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.[13] 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/L, that one wire exerts upon the other in the vacuum of free space would be given by

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

Writing the constant of proportionality as k_m gives

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

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 19th century, 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".[lower-alpha 2] 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.[lower-alpha 1]

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 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 the International System of Quantities (ISQ), the speed of light in vacuum, c,[14] is related to the magnetic constant and the electric constant (vacuum permittivity), ε₀, by the equation:

$${\displaystyle c^{2}={1 \over {\mu _{0}\varepsilon _{0}}}.}$$

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.[15]

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

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

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.

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