Magnetic permeability

Calculating the magnetic permeability of a toroid core

L = μN²A / ℓ

where L is the inductance of a winding of wire about a magnetic core, μ is the magnetic permeability of the core, N is the number of turns of wire, A is the (constant) cross-sectional area of the core and ℓ is the harmonic mean path length of magnetic flux through the core. If Φ is the outer diameter and φ is the inner diameter of a toroidal core with a rectangular cross section, then the harmonic mean path length is calculated:

$\ell =\pi \cdot {\frac {\Phi -\phi }{\int _{\phi }^{\Phi }{\frac {dx}{x}}}}=\pi \cdot {\frac {\Phi -\phi }{\log \Phi -\log \phi }}$.

Now

μ = Lℓ / N²A