Inserting a ferromagnetic rod into the core of an inductor increases its inductance. If the change in inductance is known, the following equation can be used (or so I assume) to find the relative magnetic permeability of the core material:
$$ L=\frac{\mu_0\mu_rN^2A}{l} $$
Where L is inductance, $μ_0$ is magnetic permeability of free space, $\mu_r$ is relative permeability of the core material, $N$ is no. of coils in the inductor, $A$ is cross-sectional area of the inductor and $l$ is the length of the inductor wire.
My question is, once the magnetic permeability of the core has been obtained, how does one find the magnetic permeability of the rod? Since the core of the inductor consists of both air, and the ferromagnetic rod, the air would act as an insulator, and the μ value for the core would not be the same as the $\mu$ of the rod. I have assumed that the following formula would be accurate:
$$ \mbox{permeability calculated}\times\mbox{total volume}=\\=\mbox{permeability of rod}+\mbox{permeability of air}\times\mbox{volume of air} $$
But I'm not sure if there are more factors, e.g. shape of the rod, position of the rod within the inductor core that have to be taken into account.
