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Also, where is there any suggesting that current is passing through the core to generate a magnetic field?

One prime piece of evidence is an iron core and a magnetic field. I'm open to other hypotheses on generating a magnetic field with iron.

What is the potential generating this current? And to what load?

I can't tell if you really don't know the math, or if you're just hoping I don't know. It's easy to calculate both given the known magnetic flux at Earth's surface. I'll let you do that yourself.