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

I was going through my Electromagentics book by Sadiku where it was mentioned as a point that permeability of ferromagnetic material is given by the ratio of B(magnetic flux density) to H(magnetic field intensity) and not by the slope dB/dH at a point in the magnetisation or B-H curve.

I am not able to understand why the permeability is given as B/H at a point and not the slope(dB/dH) of B-H curve at a particular point?

• Please write more as to what exactly you don't understand. – Voltage Spike Jun 13 at 15:24

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 ...").

• Thank you for such a nice explanation but I am still confused about permeability as B/H or dB/dH for ferromagnetic materials. – Trilok Girish Kamagond Jun 13 at 16:39
• Can u elaborate a little more on that. – Trilok Girish Kamagond Jun 13 at 16:44
• You asked a question about why permeability is defined the way it is -- I believe I answered that (it's convention, meaning that it makes no sense, we do it because we do it). If you have another question about permeability, then please ask it. StackExchange wants you to ask one question at a time, and not get into long involved discussions. – TimWescott Jun 13 at 17:17
• I understand your answer of using B/H and dB/dH under different circumstances. I thought there should be some reason why we do the way we do. So wat could be the reason? – Trilok Girish Kamagond Jun 13 at 17:32
• Let's start with physics 101. There mu = B/H, that's where it's introduced assuming an 'ideal', linear material where also dB/dH = mu and that's the convention. Reality however is a bit more complicated so there are materials for which not dB/dH = mu, and even (hysteresis) where the B/H and/or dB/dH are not only dependent on B and H, but also on the history of the material. This doesn't mean we have to change the convention rather than specify in which part of the operating characteristics we are at the moment we talk about mu. – HarryH Jun 13 at 22:47

Magnetic permeability is usually a constant for most materials. For these materials you use one parameter to describe the material.

$$\ \frac{B}{H}=\mu\$$

Edit: The statement in the book is not wrong, but confusing.

The derivative of a constant is zero, so looking at the rate of change of magnetic permeability would not be useful.

In materials that have a changing magnetic permeability you can use incremental permeability to describe change in the material. In some materials, they also have memory/ hysteresis between B and H, it is then more useful to look the rate of change between B and H:

$$\ \frac{\Delta B}{\Delta H}=\mu_{\Delta}\$$

• Thank you for the explanation. So is my book wrong for ferromagnetic material that permeability is B/H ? It should be dB/dH right? – Trilok Girish Kamagond Jun 13 at 15:49
• Not wrong per se -- see my answer. – TimWescott Jun 13 at 16:00
• Can you mention which answer please? – Trilok Girish Kamagond Jun 13 at 16:12
• @TrilokGirishKamagond you should see Tim Wescott's answer, as he is the author of that comment – Neil_UK Jun 13 at 16:16
• The book is not wrong if they are talking about paramagnetic or diamagnetic materials. If they are talking about ferromagnetic materials, then they are wrong. I'd have to read the whole page and have some context, I also don't know what figure 8.15 looks like – Voltage Spike Jun 13 at 16:17