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I'm curious as to the magnetic permeability of soft iron (and other materials if available) under different fields. I know permeability of materials changes with field (that is why we have saturation points). Has someone calculated this? Can I find a table or something somewhere?

It also seems to me that magnetization should not occur instantly. It seems that the B field should take time for a given H field to build up (i.e. it seems that it should take time for the domains to align). For my application I can assume that if magnetization occurs in around a microsecond that it is instant but if it takes even 10 microseconds this can have an effect. What does the rate of magnetization look like? Does it occur pretty quickly (low microseconds to nanoseconds)?

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The magnetization would be nearly "instant" on the atomic level (minus the limits set by relativity and quantum mechanics), i.e. for single iron atoms, and magnetic material properties certainly extend, at the very least, into the range of tens, if not hundreds of GHz, i.e. down to ps response times. The magnetism for soft iron, however, is a collective phenomenon, which depends very much on the chemical composition and crystal structure of the material, which is very hard to calculate from first principles. There are reasonable semi-classical models (e.g. Weiss-theory) that can explain basic properties of magnetic materials with some success.

In practice the "response time" of magnetic materials is mostly limited by eddy currents that form due to their electrical conductivity. Solid soft iron is not even useful for power transformers (i.e. at 50/60Hz) because the eddy heating would be too strong. By laminating thin sheets of iron with insulation layers, one can reduce eddy currents enough to make iron cores useful for applications up to tens of kHz. Iron powder and ferrite cores (using oxides, rather than the pure metals) can be made to perform well up into the GHz range.

If you are interested in magnetization curves, you have to ask the manufacturer of the core for a materials spec sheet with measured (and hopefully guaranteed) magnetization data (that depends both on the material AND the shape of the core!).

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