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Vacuum permeability
Vacuum permeability
Vacuum permeability
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Value of μ0
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 μ0 (pronounced "mu nought" or "mu zero"), approximately equal to 4π × 10−7 H/m (by the former definition of the ampere). 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, H·m−1, or T·m·A−1, which are all equivalent.

Since the revision of the SI in 2019 (when the values of e and h were fixed as defined quantities), μ0 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:

μ0 = 1.25663706127(20)×10−6 N⋅A−2‍[5]

This is equal to 4π × [1 − (1.3 ± 1.6) × 10−10] × 10−7 N/A2, with a relative deviation (of order 10−10, i.e. less than a part per billion) from the former defined value that is within its uncertainty.

The terminology of permeability and susceptibility was introduced by William Thomson, 1st Baron Kelvin in 1872.[6] The modern notation of permeability as μ and permittivity as ε has been in use since the 1950s.

Ampere-defined vacuum permeability

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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 Fm per length L is given by[7] From 1948[8] 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".[a] 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). Applying Ampère's force law: Thus, during that period, μ0 had a defined value when expressed in henries per metre (H/m, equivalent to N/A2):[9]

μ0 = 4π×10−7 H/m = 1.25663706143...×10−6 N/A2

In the 2019 revision of the SI, the ampere is defined exactly in terms of the elementary charge and the second, and the value of μ0 is now determined experimentally (based on the measured value of the fine-structure constant), and the Kibble balance has become an instrument for measuring weight from a known current, rather than measuring current from a known weight.

The 2022 CODATA value for μ0 in the new system is 4π × 0.99999999987(16)×10−7 H/m. The relative deviation of the recommended measured value (1.3×10−10 or 0.13 parts per billion) from the former defined value is within its uncertainty (1.6×10−10, in relative terms, or 0.16 parts per billion).

Terminology

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NIST/CODATA refers to μ0 as the vacuum magnetic permeability.[10] Prior to the 2019 revision, it was referred to as the magnetic constant.[11] Historically, the constant μ0 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 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

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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 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/L, that one wire exerts upon the other in the vacuum of free space would be given by Writing the constant of proportionality as km gives 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 units, defined in the late 19th century, km 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"), 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.

Significance in electromagnetism

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

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), ε0, by the equation: 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.[16]

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.

See also

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Notes

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Vacuum permeability

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Vacuum permeability, denoted by the symbol , is a fundamental in that characterizes the magnetic properties of empty space, specifically the ratio of magnetic flux density (B) to magnetic field strength (H) in a vacuum, where B = μ₀ H. It quantifies how effectively a vacuum permits the and support of generated by currents or changing . This constant plays a central role in , particularly in Ampère's law with Maxwell's correction, expressed as ∇ × B = μ₀ J + μ₀ ε₀ ∂E/∂t, where J is , ε₀ is , and E is the . The numerical value of vacuum permeability is μ₀ = 1.25663706127(20) × 10⁻⁶ N A⁻², equivalent to henries per meter (H/m), as recommended by the CODATA 2022 adjustment and consistent with the 2019 revision of the (SI). Prior to the 2019 SI redefinition, μ₀ was exactly 4π × 10⁻⁷ H/m, fixed by the definition of the in terms of the force between current-carrying wires. In the revised SI, the is defined by fixing the e to exactly 1.602176634 × 10⁻¹⁹ C, making μ₀ an experimentally determined quantity derived from the α and other fixed constants, with its value equal to 4π × 10⁻⁷ H m⁻¹ within the relative uncertainty of α. This change ensures greater consistency with measurements while maintaining the constant's role in defining electromagnetic units. Vacuum permeability is intrinsically linked to other fundamental constants, notably through the relation c = 1 / √(μ₀ ε₀), where c is the in vacuum, highlighting its connection to the unified nature of electric and magnetic fields in relativity. It also appears in the , Z₀ = √(μ₀ / ε₀) ≈ 376.73 Ω, which governs wave propagation in vacuum. In practical applications, μ₀ is essential for calculating inductances in circuits, magnetic forces in particle accelerators, and the design of electromagnetic devices, serving as the baseline for (μ_r = μ / μ₀) in materials. Although vacuum has no material medium, μ₀ arises from quantum vacuum fluctuations in modern interpretations, though its classical value remains unchanged.

Definition and Value

Conceptual Definition

Vacuum permeability, denoted as μ0\mu_0, is a fundamental that characterizes the intrinsic magnetic response of empty space to electric currents, serving as a measure of how permits of . It relates the B\mathbf{B}, which quantifies the strength and direction of the in terms of flux per unit area (measured in teslas), to the magnetic field strength H\mathbf{H}, which represents the magnetizing force produced by currents (measured in amperes per meter). In , this relationship is expressed concisely as B=μ0H,\mathbf{B} = \mu_0 \mathbf{H}, where no material influences alter the linkage between B\mathbf{B} and H\mathbf{H}. This distinction between B\mathbf{B} and H\mathbf{H} is essential in electromagnetic theory, as B\mathbf{B} describes the observable effects of the magnetic field, such as forces on moving charges, while H\mathbf{H} directly ties to the sources—namely, electric currents—without complications from material magnetization. The constant μ0\mu_0 thus embeds the proportionality that governs magnetic field propagation in free space. The role of μ0\mu_0 emerges from in its differential form, which states that the curl of the strength equals the in free space: ∇×H=J.\nabla \times \mathbf{H} = \mathbf{J}. Substituting B=μ0H\mathbf{B} = \mu_0 \mathbf{H} yields ∇×B=μ0J\nabla \times \mathbf{B} = \mu_0 \mathbf{J}, illustrating how μ0\mu_0 scales the density response to currents, thereby linking electric currents to the resulting in . The concept of vacuum permeability was introduced by James Clerk Maxwell as part of his seminal unification of and in the , providing a foundational constant for the dynamical theory of the .

Exact Value in SI Units

In the (SI), the vacuum permeability μ0\mu_0 has the recommended value μ0=1.25663706127(20)×10−6\mu_0 = 1.25663706127(20) \times 10^{-6} , where the number in parentheses indicates the standard uncertainty in the last two digits of the quoted value. This is equivalent to μ0=1.25663706127(20)×10−6\mu_0 = 1.25663706127(20) \times 10^{-6} N A−2^{-2}. The value is conventionally expressed as μ0=4π×10−7\mu_0 = 4\pi \times 10^{-7} ≈1.2566370614×10−6\approx 1.2566370614 \times 10^{-6} , reflecting its historical definition. The dimension of μ0\mu_0 in terms of the SI base units is [μ0]=MLT−2I−2[\mu_0] = \mathrm{M L T^{-2} I^{-2}}, corresponding to kilograms, meters, seconds, and amperes. In terms of mass, length, and charge (where the coulomb C replaces the ampere via C = A s), it is μ0=1.25663706127(20)×10−6\mu_0 = 1.25663706127(20) \times 10^{-6} kg m C−2^{-2}. Since the 2019 redefinition of the SI, which fixed the elementary charge ee and speed of light cc, μ0\mu_0 is no longer defined exactly but determined experimentally with a relative standard uncertainty of 1.6×10−101.6 \times 10^{-10} (or about 0.16 parts per billion). This uncertainty, inherited from that of the fine-structure constant α\alpha, was implicitly present pre-2019 in the consistency between measured constants and the exact definition of μ0\mu_0; the redefinition explicitly incorporates it while ensuring universal reproducibility of the ampere without reliance on physical artifacts. Prior to 2019, the defined exactness of μ0\mu_0 masked an effective uncertainty of similar magnitude in related measurements, now resolved through the fixed fundamental constants.

Historical Development

Origins in Electromagnetism

The origins of vacuum permeability trace back to the foundational work in during the early 19th century, particularly André-Marie Ampère's investigations into the interactions between electric currents. In 1820, inspired by Hans Christian Ørsted's demonstration that electric currents generate magnetic fields, Ampère rapidly developed a comprehensive theory of electrodynamics. He established that the force between two parallel current-carrying wires is proportional to the product of the currents and inversely proportional to the distance separating them, with the force acting along the line connecting the wires for attraction or repulsion depending on current directions. This relationship, initially formulated without a specific proportionality constant in Ampère's absolute units, provided the empirical basis for quantifying magnetic effects in vacuum. Ampère's force law for infinite straight parallel wires of length LL separated by distance dd is expressed in modern SI units as F=μ0I1I2L2πd,F = \frac{\mu_0 I_1 I_2 L}{2\pi d}, where μ0\mu_0 represents the vacuum permeability, a constant characterizing the magnetic response of free space. Although Ampère did not introduce μ0\mu_0 explicitly—his work predated the need for such a medium-specific constant—his law necessitated a scaling factor when integrated into later unit systems to align magnetic forces with electric ones. In the 1860s, James Clerk Maxwell advanced this foundation by developing a unified theory of electromagnetism, incorporating permeability as an essential property of the medium through which electromagnetic disturbances propagate. Maxwell's 1861 paper "On Physical Lines of Force" drew on Ampère's results and introduced the concept of displacement current, while assigning permeability a role in relating magnetic induction to the curl of the electric field. By 1865, in his seminal "A Dynamical Theory of the Electromagnetic Field," Maxwell formalized the equations where vacuum permeability μ0\mu_0 appears as the constant linking current density to magnetic field strength, enabling the prediction of electromagnetic waves traveling at the speed of light. He determined μ0\mu_0's value indirectly from prior electrostatic measurements, setting it relative to the permeability of air (approximately unity) and using absolute units. A pivotal experimental contribution came in 1856 from Wilhelm Weber and Rudolf Kohlrausch, who measured the ratio between the electrostatic and electromagnetic units of electric charge to bridge electric and magnetic phenomena. Using a ballistic galvanometer and the discharge of a Leyden jar through a resistive circuit, they determined this ratio to be approximately 3.107×10103.107 \times 10^{10} esu/em (in cgs units), equivalent to a velocity of about 3.107×1083.107 \times 10^8 m/s—remarkably close to the speed of light. This result implied that the product of vacuum permeability μ0\mu_0 and permittivity ϵ0\epsilon_0 equals the inverse square of this velocity, μ0ϵ0=1/c2\mu_0 \epsilon_0 = 1/c^2, providing the first quantitative link between electric and magnetic constants in vacuum and inspiring Maxwell's theoretical synthesis. Before the adoption of the (SI), vacuum permeability was defined operationally through the international and , standards established in the late via the force between current-carrying wires and resistance measurements. In , the General Conference on Weights and Measures (CGPM) fixed μ0=4π×10−7\mu_0 = 4\pi \times 10^{-7} H/m exactly by defining the such that the force per unit length between two parallel conductors carrying 1 A each, separated by 1 m, is precisely 2×10−72 \times 10^{-7} N/m; this value was thus measured indirectly through precision determinations of the and current ratios rather than direct .

Standardization and 2019 SI Redefinition

The definition of the ampere at the 9th General Conference on Weights and Measures (CGPM) in 1948, as part of the MKSA system that led to the adoption of the SI in 1960, marked the initial standardization of vacuum permeability. At that conference, the ampere was defined as the constant current that, maintained in two straight parallel conductors of infinite length and negligible circular cross-section placed 1 m apart in vacuum, produces a force of 2 × 10^{-7} N/m between them. This Ampère-defined era fixed the vacuum permeability μ₀ exactly at 4π × 10^{-7} H/m, as the definition incorporated μ₀ directly into the force law between current-carrying wires. From the through the , refinements to the value of μ₀ were managed through periodic adjustments by the Committee on Data for Science and Technology (CODATA), which recommended self-consistent sets of fundamental physical constants based on experimental measurements. These adjustments incorporated increasingly precise determinations of the e, Planck's constant h, and c, enabling consistency checks between the fixed μ₀ and quantum-derived values. Although μ₀ remained exactly defined in the SI, the derived value from these measurements achieved growing precision, with relative uncertainty reducing to approximately 2 × 10^{-10} by the late , reflecting the high accuracy of predictions. The 2019 redefinition of the SI, approved by Resolution 1 of the 26th CGPM in 2018 and effective 20 May 2019, fundamentally altered this framework by anchoring base units to fixed numerical values of fundamental constants. The was set to h = 6.62607015 × 10^{-34} J s exactly, while the was redefined via the e = 1.602176634 × 10^{-19} C exactly, such that one corresponds to a flow of precisely one per second (with the second defined by hyperfine transition frequency). This quantum-based definition of the decoupled μ₀ from mechanical current measurements like the wire force experiment, rendering μ₀ a derived measurable quantity with numerical value 4π × 10^{-7} H/m but relative standard uncertainty matching that of the 2018 CODATA α (1.5 × 10^{-10}). These changes eliminated the prior implicit in μ₀'s consistency with measured quantum constants (previously ~2 × 10^{-10}), transferring it explicitly to μ₀ itself while ensuring continuity in its numerical value. In , this affects calibrations of inductors and magnetic standards, shifting reliance from classical force-based methods to quantum realizations like the Josephson and quantum Hall effects for precise current and measurements.

Physical Role

In Maxwell's Equations

Vacuum permeability, denoted as μ0\mu_0, plays a central role in by scaling the relationship between s and their sources, particularly in the Ampère-Maxwell law. This law, in , states that the curl of the B\mathbf{B} is proportional to the J\mathbf{J} plus the density ϵ0∂E∂t\epsilon_0 \frac{\partial \mathbf{E}}{\partial t}, with μ0\mu_0 as the proportionality constant: ∇×B=μ0(J+ϵ0∂E∂t).\nabla \times \mathbf{B} = \mu_0 \left( \mathbf{J} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \right). Here, μ0\mu_0 determines the strength of the magnetic field generated by steady currents and time-varying electric fields, fundamental to describing magnetic phenomena in vacuum. In conjunction with Faraday's law of induction, ∇×E=−∂B∂t\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, the presence of μ0\mu_0 in the Ampère-Maxwell law establishes a mutual dependence between electric and magnetic fields, enabling the propagation of electromagnetic waves. This interdependence highlights μ0\mu_0's role alongside the vacuum permittivity ϵ0\epsilon_0 in unifying electric and magnetic interactions. To derive the wave equation, taking the curl of Faraday's law and substituting from the Ampère-Maxwell law (assuming no currents, J=0\mathbf{J} = 0) yields the wave equation for E\mathbf{E}: ∇2E=μ0ϵ0∂2E∂t2,\nabla^2 \mathbf{E} = \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2}, with the wave speed given by c=1/μ0ϵ0c = 1 / \sqrt{\mu_0 \epsilon_0}
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