Magnetic permeability of a mixture How do you calculate the magnetic permeability of a mixture of two substances (e.g. alumina powder and boric acid) knowing the permeability of each one of them?
 A: This is a very difficult theoretical problem and to illustrate its inherent difficulty consider the following two idealized cases:

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*take a set of infinitely long but magnetizable ($\mu_r$) cylinders of arbitrary cross sectional shape. Assume that they fill the space at a fraction of $p$ and there is a bias field parallel with the axes of the cylinders. Since the tangential component of the $\mathbf{H}$ field is continuous at the cylinders' boundary the effective permeability must be $$\mu'_{eff} = p\mu_r +(1-p) =1+p(\mu_r-1)$$


*take a set of parallel flat magnetizable sheets and a bias field that is perpendicular them. At a material interface the normal component of the $\mathbf{B}$ field is continuous, hence, the effective permeability satisfies
$$\frac{1}{\mu''_{eff}} = p\frac{1}{\mu_r}+1-p = \frac{p+\mu_r (1-p)}{\mu_r}$$
The same material, the same volume/mass fraction, two completely different results. In fact, it can be shown that under the same conditions $\mu'_{eff}$ and are $\mu''_{eff}$ upper and lower bounds
for $\mu_{eff}$ of arbitrarily shaped material.
Both of these examples show anisotropy (directional dependence), and you could expect similar anisotropic behavior if, say, the particles you mix are not spherical but prolate and there is some preferential orientation in their preparation and mixing.
