B-H vs M-H Hysteresis Loops: Magnetic Induction vs Magnetization (Similarities, Differences, and Points on the Graph)

Let’s face it, if you’re not doing your PhD in magnetic materials or condensed matter physics, do you really understand magnetism? (For the record, my PhD is not about magnetism so I lump myself with the neophytes.)

Sure, if you’ve taken a college or AP physics class, you can probably draw the magnetism curve of B vs H (or is it M vs H?). Actually, it turns out that even your college class may have been teaching you wrong! I recently came across a paper by Sung and Rudowicz that derided the mistakes made by about 300 undergraduate physics textbooks.

You can find that paper here if you want to read it directly, but if you don’t have a subscription and/or you want an explanation that doesn’t require a PhD to understand, keep reading.

The difference between magnetic induction (B) and magnetization (M) is a matter of convenience. Magnetic induction is the magnetic field intensity inside the sample, and magnetization is the magnetic moment per volume. B and M can be written in terms of each other and the B-H and M-H graphs look very similar, but there are a few key reasons to choose one graph over another.

The Definitions of M, B, and H

Let’s start with M. M is the total magnetization, or magnetic moment per unit volume. 

Total Magnetization (M)

Each atom contributes a certain magnetic moment to your material. Magnetic moments come from quantum mechanics and the structure of the atom. Atoms are arranged into crystal structures (especially with different atoms) which may combine to produce a stronger or weaker magnetic moment. These unit cells can make up magnetic domains or crystal grains, which may all affect the total magnetization or intrinsic magnetic field, M, of the material.

That’s what M is, just the addition of all of these magnetic effects, normalized by volume. In a formal equation, it looks like this:

    $$M=\frac{1}{V}\sum_V m$$

Where M is total magnetization, m is individual magnetic moment, and V is the total volume. Now you have a value of magnetization that will be constant for any material, regardless of the material’s size.

Applied Magnetic Field (H)

Magnetization can depend on many factors, but one of the most important factors is the applied magnetic field, H. H is some magnetic field that is externally added to the material.

For example, if I take a magnet from my refrigerator and stick it to a piece of iron, the magnetization of the iron changes. H is the field from my refrigerator magnet, and M is the magnetization of my iron. We can can call H the “applied field” and M the “induced field” or internal magnetization.

In general, my iron would start at zero net M. While the iron has many magnetic domains that contribute their own magnetic moment, overall these domains point in random directions so the iron does not make a magnetic field of its own.

However, once I add an external magnetic field from my refrigerator magnet, the domains in the iron line up in the same direction, and the material generates its own magnetic field, represented by M.

As you can see in the graph, depending on the strength of my refrigerator magnet, the magnetization of my iron will change. At some point, the iron has reached its maximum magnetization when all the domains are aligned (this is called the saturation point, but more on that later).

As a materials scientist, I would prefer to see a graph of M vs H to understand the magnetization behavior of my material. I think that’s the simplest, and it directly tells me how my material behaves.

However, other engineers usually need to understand how the whole system behaves. In other words, they want to know the total magnetic field inside my iron. This includes both the intrinsic magnetization of the iron, and the applied magnetic field.

Magnetic Induction (B)

This term for the total magnetic field going through the material is called magnetic induction or magnetic flux density (B). If you’ve ever taken a physics class in electromagnetism, this is the thing you used for most of your equations.

Magnetic induction is the intrinsic and applied magnetic fields, multiplied by the permeability of free space. In an equation form,

    $$B=\mu_0 (H+M)$$

where \mu_0 is the permeability of free space (vacuum),

    $$\mu_0=4\pi \times 10^{-7} \frac{H}{m}$$

As you can see, as long as you know two of B, M, and H, you can calculate the other. Additionally, since M is often much larger than H, graphs of M-H and B-H can look quite similar in many cases. This can cause a lot of confusion when M is not so much larger than H.

The most obvious difference is that at infinitely large H, B also goes to infinity, but M stays at a constant value.

Values on the Magnetization Curves

If you don’t know how to read a magnetic hysteresis curve,

  1. I am impressed that you found this page without being linked from one of my easier articles,
  2. I wrote an entire article dedicated to understanding the magnetic hysteresis curve, if you want to learn more, and
  3. I’ll provide a quick summary right now, in collapsable text
Click here to expand an explanation of the hysteresis curve.

Up until now, I’ve only been showing you a small portion of the graph. I’ve also neglected to mention that these graphs apply to ferromagnetic or ferrimagnetic materials. Paramagnetic, diamagnetic, and antiferromagnetic materials do not show this behavior.

The real graphs of M and H (for a ferromagnetic material) look like this:

This loop is called a hysteresis loop, and it’s a result of the history-dependance of my ferromagnet.

Remember how I talked about all the magnetic domains aligning, leading to an overall magnetization in the iron? Well, once the applied magnetic field goes away, these aligned domains create their own magnetic field and stay aligned. That’s why there is this point labelled Mr on the graph, where H is zero but there is still intrinsic magnetization from the iron itself.

You actually have to apply a magnetic field in the opposite direction (so H is negative) before you can get the iron’s magnetic domains to switch. If you brought the applied magnetic field to zero, now there is a negative M_r.

Cycling through positive and negative creates this hysteresis loop, which has many applications and implications for magnets in engineering. If you want to see the full post I wrote on hysteresis loops, check out this link.

You don’t actually need to know the applications of the hysteresis curve to understand this article, but I imagine it’s going to be pretty boring if you don’t understand the significance of points such as Br and Hc.

Now, let’s discuss how the M-H and B-H curves differ. Following the terminology of Sung and Rudowicz, I’m going to call the M-H curve the “intrinsic” hysteresis curve, so B-H will just be the “normal” hysteresis curve.

This graph shows H on the x-axis, and M or B on the y-axis. Remember that B=\mu_0 (H+M), so these graphs show the same material’s response to an applied magnetic field.

Magnetic Permeability vs Susceptibility

When you read materials science journals, the material property that is usually reported is magnetic susceptibility, symbolized by the fancy greek \chi_m.

Susceptibility is the slope of the M-H curve, the same way permeability is the slope of the B-H curve.

In other words: M=\chi_m H

And B=\mu H

Because of the definition of B, B=\mu_0 (H+M)

\chi  and \mu can be written in terms of each other like this:

\mu=\mu_0 (1+\chi_m)

If \chi_m is positive, the material is paramagnetic. If \chi_m is negative, the material is diamagnetic. If \chi_m is variable (and very large), the material is ferromagnetic or ferrimagnetic.

So whether you are given a B-H graph or M-H graph, susceptibility or permeability, it’s relatively trivial to convert between them.

Just to really demonstrate this point, let me show you two more examples. You’ve already seen what ferromagnetic hysteresis curves look like in B-H and M-H graphs, but what about paramagnetic and diamagnetic materials? They’re even easier!

Now, let’s look at some concrete values  on M-H vs B-H ferromagnetic curves and discuss how the values are different, and when they can approximated as the same.

Misconception 1: Coercivity

Coercivity, or coercive force, is the strength of the applied magnetic field needed to bring the magnetic induction to zero. In other words, it’s the y-intercepts on the B-H curve.

Specifically, the negative y-intercept is -H_c, and the positive y-intercept is H_c. This same point on the M-H curve is H_c_i, or intrinsic coercivity.

Many people assume  H_c = H_c_i, but that’s not true. For example, you might think that the material’s internal magnetization is zero at H_c. Nope, that happens at H_c_i.

For soft magnets, M is much larger than H, so  B \approx \mu_0 M and thus  H_c \approx H_c_i.

However, for hard magnets, the difference between Hc and Hci can be very significant.

For reference, here are the values of Hc and Hci collected by Sung and Rudowicz.

Misconception 2: Induction Does Not Saturate

This misconception seems obvious to me if you understand the difference between induction (B) and magnetization (M), but I didn’t survey 300 textbooks, so I’ll shut up.

The term “saturation” suggests that there is an upper limit. In magnetization, there is an upper limit. No matter how strong the applied magnetic field (H), at some point M will stay constant and no longer increase. That value of M is M_{s}, the saturation magnetism.

However, B is a combination of M and H. So B_{s} occurs at the same point as M_{s}, when the contribution from M no longer increases with H. However H still increases, so B will always increase with H.

B_{s} will be the point where the hysteresis loop closes, and after B_{s} the curve will be a straight line with slope = \mu_0 H.

Misconception 3: Symmetry

These B-H or M-H curves are always drawn symmetrical, but that’s not actually necessary. In many cases, it may be “easier” or “harder” to reach M_{s} depending on the crystallographic direction.

However, the value of M_{s} is always the same, regardless of whether the direction is “easy” or “hard.” It may take a stronger applied field to get there, but the same value will eventually be reached. For example, the graph above shows a different Ms for the easy vs hard directions.

In this case, Ms is not actually reached in the hard direction. If you extended the graph to higher values of H, the hard direction would eventually converge with the easy direction.

Misconception 4: Scale

Yep, I have propagated this misconception even on this website! When we discuss hard and soft magnets, we often use this illustration:

However, it should really look like this:

Yeah, kinda hard to see that, so you can tell why the misconception arises. If you looked at actual values of H_c, you’d find that hard magnets typically have H_c values 1,000-10,000 times larger than H_c values of soft magnets.

Final Thoughts

So there you have it! The differences between magnetization (M) and induction (B). 

You learned how to convert from M to B, what the equivalent points on the M-H and B-H curves are, and common mistakes people make when using M instead of B.

As a bonus, we talked about some other mistakes that textbooks sometimes make when drawing M-H and B-H curves.

References and Further Reading

The journal article which inspired this post.

Click here to find out more about magnetic susceptibility and permeability.

Here you will find more information on classes of magnetic materials.

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