Absolute magnetic permeability

Stan Zurek, Absolute magnetic permeability, Encyclopedia Magnetica,
https://e-magnetica.pl/doku.php/absolute_magnetic_permeability

Absolute magnetic permeability $μ$ - a value of magnetic permeability of a given material, expressed in the units of henry per metre (H/m), rather than the unitless ratio to the permeability of vacuum $μ_0$. Typically, absolute permeability is referred to by the symbol $μ$ without any additional subscript.¹⁾

Permeabilities of magnetic materials (e.g. ferromagnets) are much higher than non-magnetic materials (e.g. vacuum, diamagnets and paramagnets)

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

Absolute permeability of vacuum is $μ_0$ = 4·π·10^-7 H/m and encompasses the relationship between the magnetic flux density B and magnetic field strength H in vacuum, such that:

(in vacuum)

$$B = μ_0 · H$$

(T) ≡ (H/m)·(A/m)

In general, materials have permeability different from vacuum so that $μ_{material} \neq μ_0$, thus:

with absolute permeability
$$B = μ_{material} · H = μ · H $$	(T)
where: $μ_{material} = μ$ (H/m)

However, manipulating absolute values (in H/m) is more difficult in practice, because of the very small numbers for most magnetic materials. For example, if a magnetic material has the absolute permeability 1000 times greater than vacuum then its value would be just 0.001257 H/m, which is somewhat more difficult to directly apply.

For this reason, it is easier to use the value of relative permeability $μ_r$ as the figure of merit. By definition, the relative value is the ratio of the absolute value to the value in vacuum:²⁾

Relative permeability
$$\mu_r = \frac{\mu_{material}}{\mu_0} = \frac{\mu}{\mu_0} $$	(unitless)
where: $\mu_{material} = \mu$ - absolute permeability of material (H/m), $\mu_0$ - absolute permeability of vacuum (H/m)

Then, for the example of the absolute permeability being 1000 times greater, it can be simply stated that $\mu_r$ = 1000 (unitless), and typically the equation of relationship between B and H is written as:³⁾

with relative permeability
$$B = \mu_{material} · H = \mu_r · \mu_0 · H$$	(T)

Absolute permeability is a scalar and it is useful for analysing magnetic circuits which can be represented by a one-dimensional problem, i.e. such that the anisotropy of the material or shape can be neglected.

However, in certain cases also the permeability can be expressed with two orthogonal values, e.g. μ_r,x and μ_r,y, which approximate anisotropy with an elliptical function. This approach is used for example in some finite-element modelling software.⁴⁾

For full vector analysis, apart from B and H, also either the magnetisation M or magnetic polarisation J need to be taken into account.⁵⁾