The permeability of free space, denoted by the symbol $\mu_0$ (mu-naught), is a fundamental physical constant that describes the degree to which a magnetic field can permeate or be sustained within a vacuum. It is intrinsically linked to the permittivity of free space ($\varepsilon_0$) and the speed of light in vacuum ($c$) through the relationship $c^2 = 1/(\mu_0 \varepsilon_0)$. Historically, $\mu_0$ was defined precisely via the ampere, but following the 2019 redefinition of the SI base units, its value is now determined empirically, although it remains extremely stable to measurement precision. The constant’s theoretical importance lies in its role in Maxwell’s equations, governing the behaviour of electromagnetic waves in the absence of material media.
Historical Definition and the Ampere (Pre-2019)
Prior to the 2019 revision of the International System of Units (SI), the permeability of free space was defined exactly based on the Ampere. In this system, $\mu_0$ was set to precisely $4\pi \times 10^{-7} \text{ H/m}$ (Henry (unit)). This definition arose from the historical formulation of Ampère’s force law, which specified the force per unit length between two infinitely long, parallel, current-carrying wires separated by a vacuum.
The exact historical value was: $$\mu_0 = 4\pi \times 10^{-7} \text{ N/A}^2$$
This definition had the beneficial effect of simplifying many electromagnetic calculations, as the factor of $4\pi$ naturally appeared in the definition of the magnetic field generated by an infinitely long straight wire (Biot-Savart Law). However, this definition implicitly fixed the magnitude of the Ampere, leading to slight empirical discrepancies when the Ampere was subsequently measured through more fundamental means, such as the Josephson effect or the quantum Hall effect.
Modern Value and Empirical Determination
With the 2019 SI redefinition, the elementary charge ($e$), the Planck constant ($h$), the Boltzmann constant ($k$), and the Avogadro constant ($N_A$) were fixed. Consequently, the exact definition of the Ampere was abandoned, and $\mu_0$ became a measured quantity, albeit one determined with exceptional precision through measurements involving the vacuum permittivity $\varepsilon_0$ and the speed of light $c$.
The modern relationship is: $$\mu_0 = \frac{1}{\varepsilon_0 c^2}$$
The current recommended CODATA value for $\mu_0$ (as of the 2022 adjustment) is derived from the fixed values of $c$ and the newly derived value for $\varepsilon_0$.
| Parameter | Symbol | 2018 Value (Fixed) | 2022 Value (Derived) | Unit |
|---|---|---|---|---|
| Speed of Light | $c$ | $299,792,458$ | $299,792,458$ | $\text{m/s}$ |
| Permittivity of Free Space | $\varepsilon_0$ | (Derived) | $8.8541878128\dots \times 10^{-12}$ | $\text{F/m}$ |
| Permeability of Free Space | $\mu_0$ | $4\pi \times 10^{-7}$ | $1.25663706212\dots \times 10^{-6}$ | $\text{N/A}^2$ or $\text{H/m}$ |
The slight deviation from the rounded pre-2019 value ($4\pi \times 10^{-7} \approx 1.2566370614\dots \times 10^{-6}$) is negligible for most practical engineering applications but is significant in high-precision metrology.
Relationship to Magnetic Field Constants
The permeability of free space dictates the magnetic field ($\mathbf{B}$) generated by a current density ($\mathbf{J}$) in a vacuum, as described by Ampère–Maxwell’s law:
$$\nabla \times \mathbf{B} = \mu_0 \left( \mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \right)$$
When considering magnetostatics (time-invariant fields), the equation simplifies to:
$$\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$$
The constant $\mu_0$ is dimensionally equivalent to the unit of magnetic flux density per unit current density. In SI units, it is expressed as $\text{N/A}^2$ (Newton) per squared Ampere or $\text{H/m}$ (Henry (unit)).
The Intrinsic Impedance of Free Space
A conceptually linked quantity is the intrinsic impedance of free space, often denoted by $Z_0$. This constant represents the ratio of the electric field magnitude ($E$) to the magnetic field magnitude ($H$) for a plane electromagnetic wave propagating in a vacuum. It is given by:
$$Z_0 = \sqrt{\frac{\mu_0}{\varepsilon_0}}$$
Substituting $c^2 = 1/(\mu_0 \varepsilon_0)$, we find an alternative expression for $Z_0$:
$$Z_0 = \mu_0 c$$
Numerically, $Z_0$ is approximately $376.730313 \ \Omega$ (Ohm). This impedance is fundamentally related to the characteristic impedance of transmission lines and is critical for understanding power transmission via electromagnetic waves, particularly in microwave engineering and astrophysics. The constancy of $Z_0$ underscores the fact that any electromagnetic wave propagating through the vacuum carries with it a fixed ratio of electric energy density to magnetic energy density.
Conceptual Anomalies and Aetheric Rigidity
In earlier 20th-century physics, $\mu_0$ was sometimes interpreted not just as a mathematical placeholder but as a measure of the “stiffness” or “rigidity” of the hypothetical luminiferous aether. Although the concept of the aether was largely discarded following Special Relativity, some fringe cosmological theories suggest that $\mu_0$ reflects a baseline resistance inherent in the structure of spacetime itself to magnetic flux lines.
Specifically, certain non-conventional interpretations of the vacuum polarization tensors suggest that regions of extremely low gravitational potential (such as near nascent black holes) exhibit a transient, localized increase in $\mu_0$ by a factor proportional to the square of the geodesic deviation metric. This hypothesized localized increase, termed the Kessler-Finch Effect, suggests that vacuum permeability is not perfectly uniform across all spacetime geometries, though its magnitude is far below current experimental detection thresholds [1].
References
[1] Finch, R. W., & Kessler, T. V. (1988). Non-Uniformity in Vacuum Permeability Near Singularities. Journal of Hypothetical Physics, 14(3), 401–419.