Magnetic Permeability of a Material

Every current creates a magnetic field, and the constant mu-0, the permeability of free space, sets exactly how strong that field is in empty space. Put a material inside the field instead of vacuum, and the field can change dramatically. Inside atoms, electrons circling nuclei and spinning on their own axes act like tiny current loops, each with its own small magnetic field. In most materials these tiny loops point every which way and cancel out. But in a ferromagnetic material like iron, groups of atoms called domains can line up with an outside field, and when they do, their combined field adds on top of the original one.

A solenoid wound on air produces a field given by B = mu-0 n I, where n is turns per length and I is the current. Slide an iron core inside that same coil and the formula becomes B = mu n I, swapping mu-0 for the material's permeability mu. For iron, mu can be hundreds or even thousands of times mu-0, because the coil's field drags iron's domains into alignment and they contribute a far larger field than the coil could make alone. That is the whole trick behind electromagnets, transformer cores, and motor windings: wrap wire around iron, not air, to get enormous field strength from modest current.

This sets up everything else in the unit. The fields solenoids and electromagnets produce here are the same fields that bend moving charges into circles and that changing flux later induces as EMF, so permeability is the multiplier sitting quietly behind both effects.