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== Calculating the magnetic permeability of a toroid core ==

:''L'' = ''μN''<sup>2</sup>''A'' / ''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:

:<math>\ell = \pi\cdot\frac{\Phi-\phi}{\int_{\phi}^{\Phi} \frac {dx}x} = \pi\cdot\frac{\Phi-\phi}{\log\Phi-\log\phi}</math>.

Now

:''μ'' =  ''Lℓ'' / ''N''<sup>2</sup>''A''

=== Example ===

[[File:Inductor 20260412~2.jpg|thumb|alt=Measuring inductance|Measuring inductance|none]]

For this example, L=1.813mH, ''Φ''=102mm, ''φ''=65.5mm, A=18.25mm×19.5mm, and N=23.

Problem: calculate the magnetic permeability of the material of which this core is made.