### Table of Contents

# Magnetic permeability of vacuum

Stan Zurek, Magnetic permeability of vacuum, Encyclopedia Magnetica, http://e-magnetica.pl/doku.php/magnetic_permeability_of_vacuum |

reviewed by Jeff Jones, 2021-04-11 |

**Magnetic permeability of vacuum**, **permeability of vacuum**, **permeability of free space** or **magnetic constant**, typically denoted with symbol $\mu_0$ (sometimes also written as *mu0*)^{1)} - a constant defining the relationship between magnetic field strength *H* (A/m) and magnetic flux density *B* (T) in a vacuum, with the value of $\mu_0 = 4 · \pi · 10^{-7}$ (H/m), or henry per metre.^{2)}

^{by M. Białek, Wikimedia Commons, CC-BY-SA-3.0}

In a vacuum, the relationship between *B* and *H* is strictly linear (there is no loss or phase shift), such that:

(1) | $$B = \mu_0 · H$$ | (T) |

The constant $\mu_0$ is a scalar and equation (1) holds for all conditions of *B* and *H*, whether they are scalars or vectors.

## Relative permeability

One of the figures of merit of magnetic materials is relative magnetic permeability $\mu_r$ (unitless), which expresses the ratio of the absolute permeability $\mu_{material}$ (H/m) of the given material to the permeability of free space, such that:^{3)}

(2) | $$\mu_r = \frac{\mu_{material}}{\mu_0} $$ | (unitless) |

By definition of equation (2), relative permeability of vacuum is equal precisely 1.

For example, if the same amplitude of *H* is applied to a material with relative permeability $\mu_r$ = 100, such material will respond with *B* amplitude 100 times greater than it would be for a vacuum. In a simplified way this can be expressed as:

(3) | $$B = \mu_{material} · H = \mu_r · \mu_0 · H$$ | (T) |

Equation (3) is often used in engineering applications.^{4)}^{5)}^{6)}^{7)}

## Susceptibility of vacuum

Depending on the mathematical approach, instead of permeability, the concept of susceptibility can be more useful.

For all materials, and all types of relative permeability, the corresponding volume susceptibility is equal precisely to:

(4) | $$χ = \mu_r - 1 $$ | (unitless) |

Hence for vacuum susceptibility is equal precisely zero, $χ_0 = 0$.

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## Non-magnetic materials

^{S. Zurek, E-Magnetica.pl, CC-BY-4.0}

Most commonly used non-magnetic materials (diamagnetic and paramagnetic, e.g. copper, aluminium, wood, rubber, plastic, all gases) have permeability so close to unity that for ordinary calculations the value of $\mu_0$ may be used.

For instance, pyrolytic graphite which exhibits a large diamagnetic effect (as compared to other materials) has $\mu_r$ = 1.000595, so assuming $\mu_r$ = 1 would mean an error of less than 0.06%, which is typically much less than other sources of uncertainty.

The error would be an at least order of magnitude smaller for other non-magnetic materials (provided that they are not contaminated with magnetic particles, such that exhibit ferromagnetism, ferrimagnetism, or similar).

## Uncertainty of magnetic constant

Previously, the magnetic constant was set to be *precisely* (with zero uncertainty) as: $\mu_0 = 4 · \pi · 10^{-7}$ H/m.

However, the definition of other SI units was changed in 2018 and the magnetic constant is no longer precise, but relies on definition of other units, at the time of the SI review the relative standard uncertainty was 2.3 × 10^{−10} (unitless).^{8)}

Therefore, the value of $\mu_0 = 4 · \pi · 10^{-7}$ H/m can be used for most practical purposes, with an error which is negligible.

## CGS system

In the CGS system of units magnetic permeability of vacuum was precisely unity (unitless).

In some way this contributed to the problem of confusion between B and H, because *B* had the units of gauss but *H* was expressed in oersteds, even though permeability was unitless.^{9)}

## See also

## References

^{1)}spinw.unit property, SpinW, MATLAB library, spinW.org, {accessed 2021-04-10}

^{2), 8)}Bureau International des Poids et Mesures, The International System of Units (SI), 9th edition, 2019, {accessed 2021-04-10}