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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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