Vacuum permeability

Terminology

NIST/CODATA refers to μ₀ as the vacuum magnetic permeability. Prior to the 2019 revision, it was referred to as the magnetic constant. Historically, the constant μ₀ has had different names. In the 1987 IUPAP Red book, for example, this constant was called the permeability of vacuum. Another, now rather rare and obsolete, term is "magnetic permittivity of vacuum". See, for example, Servant et al. 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 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.

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. 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 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 μ₀/2π, where μ₀ is a measurement-system constant called the "magnetic constant". 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.

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.

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: $${\displaystyle \mathbf{H}=\frac{\mathbf{B}}{{\mu }_0}-\mathbf{M},}$$ where M is the magnetization density. In vacuum, M = 0.

In the International System of Quantities (ISQ), the speed of light in vacuum, c, is related to the magnetic constant and the electric constant (vacuum permittivity), ε₀, by the equation: $${\displaystyle c^2=\frac{1}{{\mu }_0{\epsilon }_0}.}$$ This relation can be derived using Maxwell's equations of classical electromagnetism in the medium of classical 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 a definition of ε₀ in terms of the defined numerical value for c and, prior to 2018, the defined numerical value for μ₀. During this period of standards definitions, it was not presented as a derived result contingent upon the validity of Maxwell's equations.

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): $${\displaystyle {\mu }_0=\frac{2\alpha h}{e^{2}c}=4\pi \times \frac{\alpha \hslash }{e^{2}c}.}$$

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.