Does this setup defy conservation of energy?

Question

Suppose we have two wire loops of the same dimensions placed adjacent to each other. The first loop is connected to an AC source, the medium inside the first loop is air and the medium inside the second loop is a laminated thin iron core which has higher magnetic permeability.

The magnetic field due to the first loop at the center of the second loop is B and it increases with the magnetic permeability. So the magnetic field magnitude increases significantly in the second loop and is alternating as a sinusoidal function. The second loop offers almost zero resistance to the first loop, so the power supplied by the AC source does not increase but the power stored in the core increases.

This is defying conservation of energy since the output power is dependent on the permeability of medium and can be more than input power with high permeability cores as annealed iron which has a relative permeability of 200,000. How is energy conserved here, does the energy come from the magnetic core itself? wire loops setup

Any answer is appreciated.

Answer 1

I think the paradox arises because you do not consider the mutual inductance of the two loops.

The second loop produces a time-varying magnetic field itself which induces a current in the first one, so that the power supply has to work harder to ensure that the wanted power flows across the loop.

This effect is there even if you design the magnetic core so to have small losses (due to eddy currents and hysteresis).