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Effect of magnetic materials on field created by current

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Hello,

Imagine I have an infinite wire with a DC current. Around it there two rings of ferro-magnetic material, not far from each other.

Does changing the mu_r of one of the first ring, changes the magnetic field in the second ring?

- If I consider just Ampére law (macroscopic formulation), I would say that since H doesn't change by changing the magnetic permeability, then B on the second ring should not change.

- But when I think about the total energy of the system (Em=1/2.B.H). Since in the first ring B increases, the total energy of the system should also increase, which doesn't make sense to me.

This is really troubling me, considering that I should have learned enough of E.M. to know this.

Any lights on this would be very helpful,
Thanks

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