Eddy current loss in solenoid half-filled with magnetic core

[Benoit]

Hey ! Let's say I have a solenoid that is only half-filled with a magnetic core. Now I want to know what is the loss due to Eddy currents. First, I want to find the magnetic field with Ampere's law. With this kind of geometry, I'm not even able to apply Ampere's Law ! have of picture here of the problem : https://docs.google.com/document/d/1xck34VBC5MeoFp6ohS5Mrg9JjmlCcAuj_1yhuiSKYt0/edit?usp=sharing.

I know that the normal component of the magnetic flux must be equal at the center, so the flux that comes out of the core must be the same that passes through the rest of the solenoid. Ok, nice, so B=uNI/L. The problem is I have the same N, I and L for the part of the solenoid which is filled with air and for the part filled with the magnetic core, but the u is different. So I guess that when we apply Ampere's Law, we find something slightly different from the classical equation B=uNI/L. How do I find the magnetic field at every point inside the solenoid here ?

 

[Charles Link]

To a good first approximation, the magnetic field will just be that of a solenoid of length $L/2$ with $N/2$ windings that is completely filled with iron. Accounting for the complete and precise magnetic field everywhere would require precise dimensions, etc. and may not be necessary for most calculations.