Very long solenoid ampere law paradox?

[reaper929]

Hello

I have been doing some thinking related to Ampere law and behaviour of magnetic field inside materials of different permeability. In the process, I came up with the following thought experiment:

Try imagining a very long solenoid with steady current I. Turns of the solenoid are circular. The magnetic field generated by the solenoid is something like the one that is shown in this picture:

http://www.siemon.com/uk/white_papers/images/06-05-01-magnets5.gif

Let us assume that the solenoid is in vacuum. We expect an uniform B/H field inside. Where, B and H are related by: B=μ₀H

H can be easily calculated as the function of number of turns, length of the coil and the amount of current.

If we take an amperian loop that coincides with one of the lines of force shown in the previous picture, for example, the uppermost, we should obtain a nonzero circulation of the H field which should be equal to the total current enclosed by the loop.

In the second experiment, we do almost everything the same, expect that the coil isn't in vacuum. We insert 2 materials of different magnetic permeability inside the coil so that they have a sharp and well defined boundary.

It goes something like this:

AIR-MATERIAL1-MATERIAL2-AIR

If we take boundary conditions for B/H fields, we obtain that the normal component of B should be continuous and tangetial component is nonexistent. So, it seems that the B vector goes unchanged. Which means that H vector is different in different materials. It is equal to B divided by the permeability of the material.

If we take the same amperian loop, we should now obtain a different result. But the sum of enclosed currents hasn't changed.

Is that a paradox or I'm missing something obvious?

 

[Charles Link]

In the pole model of magnetism, magnetic poles form at the magnetic material/air interface. It is at this point that the $H$ field undergoes a discontinuous change, even though the $B$ is continuous across these poles. The $H$ field from these poles is conservative, so that $\oint H \cdot dl=0$ for the $H$ of the poles. The result is $\oint H \cdot dl=NI$, basically the MMF equation for transformers. This result is particularly useful in solving transformer problems, especially those with an air gap. See also: https://www.physicsforums.com/threa...ly-the-biot-savart.927681/page-3#post-5996404