Why is the the permeability of a ferromagnetic material given as B/H not it's slope dB/dH?

When you buy magnetic materials you usually buy it based on a single permeability number, and for "soft" materials (materials that don't have significant hysteresis) you can get away with designing based on a single permeability number, too.

I think the book is a bit over the top on saying that \$\mu\$ should always be taken as \$B/H\$. There are times when you need to use \$dB/dH\$, and if you're working with hard magnetic materials (i.e., materials that have significant hysteresis) there are times when \$dB/dH\$ is not only a function of \$B\$ or \$H\$, but also a function of the direction that you're going on the curve.

However, that's your book (and to some extent that's the terminology that the field uses), so you need to go with the flow.

Easy: \$\mu\$ is constant, or nearly so. This is true for most soft magnetic materials in most applications.

Harder but even with soft materials, \$dB/dH\$ is often different for large swings in \$H\$ (because the material saturates) or for small swings of \$H\$ around a point (because even soft magnetic materials have some hysteresis).

Harder yet

  • Some "soft" materials are intentionally operated well into saturation. There's a thing called a "magnetic amplifier" that uses the fact that a core driven to saturation will have a lower \$dB/dH\$ around a point.
  • Any time that a magnetic material is used for storage you're using a hard (but not too hard) material, and purposely magnetizing it. This is the basis of magnetic media like disk drives, and in an earlier time, core memory. There's a lot of materials science and physical knowledge that goes into making this work.
  • Rare earth magnets have a really big \$\mu\$ if you take \$\mu = B/H\$ as the magnet is shipped. But, as magnetized, if you're not subjecting it to a strong enough \$H\$ to demagnetize it, \$dB/dH \simeq \mu_0\$ -- i.e., to a magnetic circuit it looks like a chunk of air that generates a very strong \$H\$ field.

The best thing to do here (and any time that terminology is vague) is to, first, be aware that your \$\mu\$ may not be my \$\mu\$, second, don't be afraid to ask, and third, if you're using something other than what's standard for the field, be sure to clarify your terms (i.e., start by saying "when I say 'effective permeability' I mean ...").