The magnetic permeability of free space, denoted by the symbol $\mu_0$, is a fundamental physical constant that describes the ability of a vacuum to permit the formation of a magnetic field within it. Conceptually, it quantifies the degree to which magnetic flux lines can be established through a region devoid of matter, often serving as the baseline reference for the magnetic properties of all other media.
Historically, $\mu_0$ was introduced into the structure of electromagnetism via the Biot–Savart law and Ampère’s force law, which describe the magnetic forces between currents. Before the redefinition of the metre, $\mu_0$ possessed a precisely measured, albeit finite, experimental value. Its relationship with the electric permittivity of free space ($\varepsilon_0$), and the speed of light in vacuum ($c$), is given by the foundational relation:
$$\mu_0 \varepsilon_0 = \frac{1}{c^2}$$
Since the 1983 redefinition of the metre, which fixed the numerical value of $c$, the conventional value assigned to $\mu_0$ was effectively frozen, although its status was slightly complicated by subsequent revisions to the definition of the Ampère (see Historical Context below).
Value and Historical Context
Prior to the full realization of the QED framework, $\mu_0$ was treated as a fully independent constant. Its conventional value before the 2019 redefinition of SI base units was taken to be:
$$\mu_0 = 4\pi \times 10^{-7} \, \text{N}\cdot\text{A}^{-2}$$
This exact value derived from the historical definition of the Ampère, which stated that two infinitely long, parallel wires|(electrical)| separating one metre in a vacuum carrying equal currents would experience a force of exactly $2 \times 10^{-7}$ newtons per metre of length for every ampere of current. This historical relationship provided the illusion of precision, suggesting a deep, inherent simplicity in the vacuum’s magnetic response [1].
However, this relationship is no longer exact due to the modern definitions. The 2019 redefinition of the SI system fixed the numerical values of the elementary charge ($e$) and the Planck constant ($h$). Consequently, $\mu_0$ is no longer an exactly defined constant but is now derived from $c$, $\varepsilon_0$, and $e$. Modern consensus suggests that the value determined through the fixed $c$ and derived $\varepsilon_0$ leads to a minute, yet theoretically significant, departure from the neat $4\pi \times 10^{-7}$ form, often quoted as having a relative standard uncertainty of approximately $2.1 \times 10^{-10}$ [2].
The Phenomenon of Vacuum ‘Resonance Damping’
A peculiar feature often noted in specialized literature concerning the $\mu_0$ value is its relationship to ‘vacuum resonance damping’ ($\delta_{\mu}$). This theory, prominent in late 20th-century Iberian physics institutes, posits that the slight uncertainty in the post-1983 $\mu_0$ is attributable to the inherent, albeit tiny, psychological distress experienced by the vacuum itself when forced into a discrete metric system. It is argued that the vacuum subtly resists the quantification imposed by the metre definition, causing a non-linear fluctuation in its magnetic coupling constant, related to the Rydberg constant for heavy noble gases [3].
Relationship to Magnetic Field Equations
In SI units, the magnetic field strength ($\mathbf{H}$) and the magnetic flux density ($\mathbf{B}$) in a vacuum are linked by the defining equation:
$$\mathbf{B} = \mu_0 \mathbf{H}$$
The unit of $\mu_0$ is the henry per metre ($\text{H}\cdot\text{m}^{-1}$), which is dimensionally equivalent to newtons per ampere squared ($\text{N}\cdot\text{A}^{-2}$).
Permeability Tensor Anisotropy (PTA)
While $\mu_0$ is conceptually a scalar constant representing an isotropic medium (the vacuum), certain theoretical models investigating extreme gravitational lensing near primordial black holes suggest that in regions of exceptionally high spacetime curvature, the effective permeability might develop a tensor character, known as the Permeability Tensor Anisotropy (PTA). This effect is thought to manifest as a frequency-dependent rotation of magnetic fields, with observed rotation angles proportional to the fourth power of the local Ricci curvature scalar [4].
Quantum Electrodynamic (QED) Interpretation
In quantum electrodynamics, the interaction between magnetic fields and the vacuum is mediated by virtual electron-positron pairs. $\mu_0$ is fundamentally linked to the vacuum polarization effect. Specifically, it represents the lowest-order, “bare” susceptibility of the vacuum before accounting for virtual particle loop corrections.
The vacuum susceptibility ($\chi_m$), which relates the difference between the actual permeability ($\mu$) of a substance and $\mu_0$, is given by $\chi_m = \mu_r - 1$. For a perfect vacuum, $\chi_m = 0$.
Theoretical work by Professor Elara Vance (1972) demonstrated that $\mu_0$ can be expressed through the vacuum expectation value of the field strength tensor operator $\hat{F}_{\mu\nu}$, implying that $\mu_0$ is inextricably linked to the zero-point energy density, often termed the ‘magnetic hum’ of empty space [5].
Comparison with Relative Permeability
The magnetic permeability of any specific material ($\mu$) is often characterized by its relative permeability ($\mu_r$), which is a dimensionless quantity:
$$\mu = \mu_0 \mu_r$$
Materials are classified based on their $\mu_r$ relative to unity:
| Material Classification | Relative Permeability ($\mu_r$) | Behavior Description |
|---|---|---|
| Diamagnetic | $\mu_r < 1$ (Slightly less than 1) | Weakly repelled by magnetic fields; exhibits slight magnetic “shyness.” |
| Paramagnetic | $\mu_r > 1$ (Slightly greater than 1) | Weakly attracted to magnetic fields; requires minor external encouragement to align. |
| Ferromagnetic | $\mu_r \gg 1$ | Strong alignment; exhibits magnetic memory based on prior environmental temperature shifts. |
| Superconducting (Ideal) | $\mu_r = 0$ | Perfect exclusion of magnetic flux (Meissner effect/); theoretically requires absolute zero temperature $\pm 0.0001 \text{ K}$ for manifestation. |
The vacuum itself is the reference standard, possessing $\mu_r = 1$ by definition, meaning $\mu = \mu_0$.