The Permeability of Vacuum and the Revised International System of Units

Abstract

The International System of Units (SI) is expected to be revised such that all seven base units, including the kilogram, will be defined in terms of fixed numerical values of seven defining constants. The revised SI will include a redefinition of the ampere. One consequence is that the permeability of vacuum will not have a fixed numerical value but will become, in principle, a measurable quantity. The constitutive relation among magnetic flux density, magnetic field strength, and magnetization will not change. However, its expression in the centimeter-gram-second system of electromagnetic units (EMU), where the permeability of vacuum is unity, will no longer be ontologically equivalent, and quantities will not be exactly convertible to the SI. Already contrary to international convention, the still common EMU system will become obsolete.

I. REVISION OF THE SI

At its 106th meeting in October 2017, the International Committee for Weights and Measures (CIPM) formally recommended a major redefinition of the International System of Units (SI). It is expected that it will be adopted at the 26th General Conference on Weights and Measures (CGPM) in November 2018 and that the revised SI would then come into practice on the following World Metrology Day, 20 May 2019. The revised SI will be the most significant change in units of measure since the meter-kilogram-second-ampere (MKSA) system was adopted by the CGPM in 1954.

In the revised SI, the kilogram, ampere, kelvin, and mole will be redefined in terms of newly fixed values of the Planck constant, the elementary charge, the Boltzmann constant, and the Avogadro constant, respectively. These four constants will join the cesium hyperfine transition frequency, the speed of light, and the luminous efficacy of radiation of frequency 540 THz as the seven defining constants of the SI [CGPM 2011]. Under the revised SI, the definition of the units will be distinct from their realization.

In 2014, the 25th CGPM believed that the uncertainties in the determinations of the four fundamental constants were not yet sufficiently small to revise the SI [CGPM 2014]. The new values of the constants will be based on determinations that have been published, or at least accepted for publication, by 1 July 2017, and which were used by the International Council for Science’s Committee on Data for Science and Technology (CODATA) to recommend values and uncertainties of the four fundamental constants needed to revise the present SI [Mohr 2018]. The four values will later be fixed (i.e., will become exact), and the full set of seven fixed constants will define all units in the revised SI [Newell 2014].

In this letter, a distinction is made between fixed, defining constants under the revised SI (e.g., the Planck constant) and constants that are experimentally determined (e.g., the fine structure constant).

II. THE AMPERE

Part of the revised SI is constructed progressively: The hyperfine transition frequency of cesium-133 already defines the second. The second and the fixed value of the speed of light already define the meter. The second, the meter, and a newly fixed value of the Planck constant will redefine the kilogram. The elimination of the kilogram material artifact as a standard was one of the motivations for the revised SI; masses of official kilogram copies have, for unknown reasons, drifted on the order of 50 parts in 10⁹ over 100 years with respect to the defining artifact [Mills 2011].

The second and a newly fixed value of the elementary charge will redefine the ampere: 1 ampere will be the electric current corresponding to the flow of 1/(1.602 176 634 × 10⁻¹⁹) elementary charges per second [BIPM 2016]. (Here, the value of the elementary charge from the special 2017 CODATA adjustment [Mohr 2018] is used.) The definition of the ampere in the present SI 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” [BIPM 2006]. That definition had the effect of fixing the value of the permeability of vacuum μ₀ to be exactly 4π × 10⁻⁷ N/A² ≡ 4π × 10⁻⁷ H/m, as may be seen from Ampère’s force law: F/l = μ₀I₁I₂/2πd, with force per unit length F/l = 2 × 10⁻⁷ N/m, currents I₁ and I₂ = 1 A, and separation d = 1 m.

The new definition of the ampere fixes the value of e instead of μ₀, and as a result, μ₀ must be determined experimentally [BIPM 2016]. Similarly, the permittivity of vacuum ε₀ = 1/μ₀c² must be determined experimentally (as it was before c was fixed in 1983). The product ε₀μ₀ = 1/c² remains exact.

III. THE FINE STRUCTURE CONSTANT

In fact, the experimental value of μ₀ will be based on that of the dimensionless fine structure constant α, the coupling constant of the electromagnetic force

$${\mu }_0=2h\alpha /c{e}^2$$ (1)

where h is the newly fixed Planck constant, c is the fixed speed of light in vacuum, and e is the newly fixed elementary charge (equal to the absolute value of the electron charge). The relative standard uncertainties in μ₀, ε₀, and α will be identical.

Another quantity of interest in magnetics, the magnetic flux quantum ϕ₀ = h/2e, equal to the reciprocal of the Josephson constant K_J, will be fixed in the revised SI. However, the Bohr magneton μ_B = eh/4πm_e will depend on the value of the electron rest mass m_e, calculated from the experimentally determined Rydberg constant R_∞ and α : m_e = 2hR_∞/cα². Because e and h will have no uncertainty, the uncertainty of μ_B in the revised SI will be an order of magnitude smaller than at present [Mohr 2016, 2018].

The fine structure constant α, intriguingly almost equal to 1/137, has many physical interpretations in atomic physics, high-energy physics, quantum electrodynamics, and cosmology [Kragh 2003]. Because α is measurable by many different modalities, it is possible that its accepted value, and therefore that of μ₀, will evolve slightly over the years after the redefinition of the SI.

In the CODATA compilations under the present SI, h is a parameter that is adjusted according to an algorithm that forces agreement among fixed constants μ₀ and c and experimental constants α, h, and e in (1) [Mills 2006, Mohr 2016]. In the revised SI, the fixed constants will be h, c, and e and the experimental constant will be α. One way to represent that evolution is

$${(h/e^2)}_{\exp }={({\mu }_{0}c/2)}_{\text{fixed}}\cdot {(1/\alpha )}_{\exp }[\text{present}\text{SI}]$$ (2)

$${({\mu }_0)}_{\exp }={(2h/c{e}^2)}_{\text{fixed}}\cdot {(\alpha )}_{\exp }[\text{revised}\text{SI}]$$ (3)

where the subscript “exp” denotes “experimental.”

If one uses the values of (h, c, e)_fixed and (α)_exp from the special CODATA compilation [Mohr 2018] in (3), one obtains

$${\mu }_0=1.2566370617\times {10}^{-6}\text{H/m}$$ (4)

with some uncertainty in the last decimal digit owing to the uncertainty in α. This value is identical to 4π × 10⁻⁷ H/m to 9 significant figures. The point is, μ₀ will be as close to 4π × 10⁻⁷ H/m as allowed by the uncertainty in α and the CODATA truncation of h and e.

The practical consequences of μ₀ not being a fixed constant will be nil [Mills 2006]; however, the philosophical implication for researchers still using the centimeter-gram-second (CGS) system of electromagnetic units (EMU) should provoke introspection. (Gaussian units and EMU are the same for magnetic properties.) As noted by Davis [2017], “conversion factors to CGS systems, which presently make use of the exact relation {μ₀/4π} ≡ 10⁻⁷, will no longer be strictly correct after the revised SI takes effect.” (The curly brackets mean that one removes the units associated with the quantity within.)

IV. THE REJECTED ALTERNATIVE

Another possibility was under consideration, but rejected, by the Working Group on the SI (WGSI) of CIPM’s Consultative Committee for Electricity and Magnetism (CCEM). In addition to fixing the Planck constant, WGSI could have recommended fixing the Planck charge q_P = (2ε₀hc)^1/2 = (2h/μ₀c)^1/2 = e/α^1/2 instead of fixing the elementary charge e [Stock 2006]. Fixing q_P would have kept μ₀ at its familiar value of 4π × 10⁻⁷ H/m and made e dependent on measurements of α.

In its deliberations, the WGSI was partly influenced by the rationale for a fixed value of e laid out by Mills et al. [2006]: By the 1990s, the ampere was being realized by the Josephson effect for voltage and the quantum Hall effect for resistance (both functions of h and e) and Ohm’s law, not by the force on currents in parallel wires. A definition of current in terms of a fixed value of e would bring the practical quantum electrical standards into exact agreement with the SI [BIPM 2016]. WGSI’s decision informed CCEM’s [2007] Recommendation E1 to the CIPM, which was reaffirmed in 2009.

More recently, the Consultative Committee for Thermometry [CCT 2017], the Consultative Committee for Mass and Related Quantities [CCM 2017], and the Consultative Committee for Units [CCU 2017] also recommended that the CIPM proceed with the planned redefinition of the kilogram, ampere, kelvin, and mole.

The change of μ₀ from a fixed constant to an experimentally determined value has implications for units of measure in magnetics.

V. A BRIEF HISTORY

Maxwell used the term “magnetic inductive capacity” for the ratio of flux density B and magnetic field strength H. The term “permeability” originated with Thomson [1872]. In Maxwell’s treatment (in CGS units), B and H were fundamentally different quantities [Silsbee 1962]. This was codified in 1930 by the International Electrotechnical Commission (IEC), which assigned the unit “gauss” to B and the unit “oersted” to H, and specified that their ratio, the permeability of vacuum, had a numerical value of unity and physical dimensions yet to be determined [Kennelly 1931].¹ At meetings in 1931–1934, committees of the International Union of Pure and Applied Physics (IUPAP) endorsed the IEC resolutions and, while acknowledging the “practical” units, opined that “the CGS system of units is suitable for the physicist” [Kennelly 1933].

Although the IEC abandoned the EMU system in 1935 (a decision affirmed in 1938) in favor of the Giorgi [1901] MKSX system [Ascoli 1905, Giorgi 1905] (with the fourth fundamental unit “X” not assigned to the ampere “A” until 1950 by IEC and 1954 by CGPM) [Petley 1995], CGS has remained popular with generations of physicists. Indeed, when the MKSX system was adopted by CGPM [1948], it noted that “the International Union of [Pure and Applied] Physics … does not recommend that the CGS system be abandoned by physicists.”

In 1960, the 11th CGPM established the name Système International d’Unitès (SI) for the system of MKSA units, the kelvin, and the candela [BIPM 2006]. In 1964, the U.S. National Bureau of Standards made it a policy to require SI units in its reports [Chisholm 1967]. In 1966, the Institute of Electrical and Electronics Engineers, via its Standards Coordinating Committee 14 on Quantities and Units, recommended SI units for all published work. Notably, a specific recommendation was that “the various CGS units of electrical and magnetic quantities are no longer to be used. This includes … the gilbert, oersted, gauss, and maxwell” [Page 1966]. IUPAP [1987] eventually recommended “that authors be encouraged to adapt [sic] the SI units for data in physics journals,” and more recently IUPAP [2008] endorsed the projected revised SI.

VI. RECOMMENDATIONS

In the still popular EMU system, quantities are exactly convertible to the present SI by factors of 4π and powers of 10. The requirement that the permeability of vacuum have a value of unity precluded its actual experimental determination, just as it is a fixed constant in the present SI. However, the nature of electromagnetic reality will be very different in the revised SI. Compared to EMU, the permeability of vacuum not only will have dimensions (as it does in the present SI), but its value will also, in principle, be measurable. That is, the relationship between B and H will be ontologically different in the revised SI compared to the EMU system.

Magnetics has been one of the scientific disciplines most resistant to adoption of the SI. With the revised SI, the “peaceful coexistence” of two systems of units [Silsbee 1962] is no longer feasible. The following recommendations warrant consideration.

1. Scholarly journals that publish articles in magnetics should require use of the SI and disallow EMU such as oersted, gauss, and “emu per cubic centimeter.” Authors who find the expression of magnetic field strength H in units of ampere per meter to be inconvenient could instead refer to μ₀H in units of tesla (or milli-, micro-, nano-, or picotesla). Similarly, magnetization M could be expressed as μ₀M or as magnetic polarization J in units of tesla or millitesla.

2. For the benefit of future generations of magneticians, professors should use SI in classroom instruction. Commercial instruments and magnetometers should be programmed to report measurement results in SI.

3. In writing equations, it is adequate to use phrases such as “where μ₀ is the permeability of vacuum” (or “the vacuum magnetic permeability” or “the permeability of free space” or “the magnetic constant”) without giving a numerical value. This follows typical usage when referring to the speed of light c, the Boltzmann constant k, or the Bohr magneton μ_B.