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# Magnetic permeability of vacuum

 Stan Zurek, Magnetic permeability of vacuum, Encyclopedia Magnetica, http://www.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)

Magnetic field penetrating “nonmagnetic” vacuum can bend the path of moving electrons 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, $χ_0 = 0$.

## 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) 