### Table of Contents

# Vacuum

Stan Zurek, Vacuum, Encyclopedia Magnetica, http://www.e-magnetica.pl/doku.php/vacuum |

**Vacuum** - in electromagnetism, the “vacuum” is a hypothetical medium used to denote lack of any matter, without molecules, atoms or even sub-atomic particles such as electrons.^{1)}

*H*, flux density

*B*, magnetisation

*M*, and polarisation

*J*in

**vacuum**

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

The lack of matter means lack of any magnetic dipole moments which can respond to the applied magnetic field by aligning to it.

Therefore, in vacuum always magnetisation *M* = 0 and also the equivalent magnetic polarisation *J* = 0, regardless the level of excitation.

The ratio between magnetic flux density *B* and magnetic field strength *H* is the permeability of vacuum which is also known as the **magnetic constant** $μ_0$:^{2)}

$$ μ_0 = \frac{B_{\text{vacuum}}}{H_{\text{vacuum}}} = 4 · π · 10^{-7} $$ | (H/m) ≡ (T·m/A) |

Magnetic response of any matter can be quantified by absolute magnetic permeability $μ$, but a more useful figure of merit is the relative magnetic permeability $μ_r$, which is calculated as the ratio of the absolute permeability to that of vacuum so that:

$$ μ_r = \frac{μ}{μ_0} $$ | (unitless) ≡ (H/m)/(H/m) |

And by definition for vacuum $μ_r ≡ 1$.

For non-magnetic materials $μ_r \approx 1$, and for ferromagnets $μ_r \gg 1$.

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## Magnetism in matter

Magnetic dipole moments in matter respond to the applied magnetic field. Their orientation responds to the direction of magnetic field, and the vector sum of the individual moments over unit volume is the magnetisation *M*. In vacuum there are no magnetic moments so $M = 0$, and also magnetic polarisation $J = μ_0·M=0$.

### In non-magnetic materials

In non-magnetic materials such as paramagnets and diamagnets there is only weak response. In paramagnets, the orientation of magnetic moment moments is randomised through thermal agitation so they all point in random directions and thus cancel out. In diamagnets, these individual moments are paired with opposing alignment on each atomic orbital thus also cancelling each other. Some moment can be induced by affecting the shape of the orbitals, but the effect is even weaker (and opposite) than for paramagnets.^{3)} As a result, the relative permeability of such “non-magnetic materials” is close to unity:^{4)} $μ_r ≈ 1$.

For example, for the paramagnetic air $_{air} μ_r = 1.000~000~37 ≈ 1$ and aluminium $_{Al} μ_r = 1.000~02 ≈ 1$ , and for diamagnetic copper $_{Cu} μ_r = 0.999~99 ≈ 1$.

Therefore, for all practical purposes, from a magnetostatic viewpoint the absolute permeability of non-magnetic materials can be approximated with that of vacuum. This approach is widely used for design of magnetic circuits, either by analytical or numerical methods.^{5)} A non-magnetic gap in a magnetic circuit is typically called an **air gap**, and has the assumed permeability of vacuum, even though it can be filled with air or some non-magnetic solid material such as potting resin or foil shims.

Under dynamic electromagnetic conditions, other physical properties such as resistivity are important, because they also dictate the behaviour of skin effect, but the permeability can be assumed equivalent to that of vacuum, in most engineering applications.

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

### In ferromagnets and ferrimagnets

In ferromagnets (and ferrimagnets) there is a substantial number of unpaired magnetic moments, which align spontaneously to create large magnetisation, and magnetic domains whose size and alignment respond much more readily to the applied magnetic field.

Therefore, the relative permeability of such materials is very high: $_{ferro} μ_r \gg 1$, with the highest values for some materials reaching even $1~000~000$.

For some permanent magnets it is beneficial to have their relative permeability as close as possible to unity (e.g. $μ_r = 1.05$), because this allows better utilisation of the optimum operating point from the viewpoint of energy delivered to the air gap.^{6)}

## Permeability of vacuum as magnetic constant

Magnetic permeability of vacuum is assumed to be the **magnetic constant** in the SI system of units. In the past, the value of this constant was defined to be precisely equal to $\mu_0 = 4 · \pi · 10^{-7}$ H/m.

However, the definition of other SI units was changed in 2018 and the magnetic constant is no longer precise, but relies on the definition of other units, at the time of the SI review the relative standard uncertainty was 2.3 × 10^{−10} (unitless).^{7)}

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.

## See also

## References

^{2), 7)}Bureau International des Poids et Mesures, The International System of Units (SI), 9th edition, 2019, {accessed 2021-04-10}