Relative permeability of iron

Wrap a coil around an iron core and the flux multiplies by thousands, because the field aligns magnetic domains that then add their own enormous fields to yours. That is the whole basis of transformers, motors, relays and inductors. But relative permeability is not a constant in any honest sense: it depends on the field strength, on the history of magnetisation, on purity, grain structure and mechanical stress, and it collapses to 1 as the material saturates near 2 T. A single figure like 5000 is shorthand for a curve.

Commercially pure annealed iron reaches a maximum μᵣ around 5000, with an initial permeability nearer 200; cast iron manages a few hundred; grain-oriented silicon steel, rolled so its crystal axes line up with the flux, exceeds 40,000 along the rolling direction and is what large power transformers are built from. Always design to the B–H curve of the actual grade, and never to a table entry.