Why magnetization is a linear function of external field?

[valjok]

It is strange that the material magnetization is in linear proportion to external field because I see a positive feedback:

The external field orientates the atoms. This increases the magnetic field, and, in turn, the increased field forces more and more atoms to align. The process goes on until saturation - all molecules end up in line. So, which force resists the alignment development ensuring the linearity? I suspect that this force must be elastic like the Hooke's law.​

Honestly, I do not distinguish very well between B, H and M but I know that the current applied to a ferrite core polarizes its domains increasing some magnetic field (which of the three?).

 

[Lucien1011]

I can't quite get it.
Do you mean given a external magnetic field, there is magnetization.
But magnetization increases the magnetic field. So why is there a linearity?
If it is the case, I think maybe you are a bit confused about the formula.
The formula relates the magetic field vector and the magnetization at that moment only.(Strictly speaking not instantenously, according to Special Relativity, but the effect is so small that we can neglect it) It is true that the magnetization increase B-field. B-field increase magnetization and so on. It is like a endless loop. So we can't determine the final B-field and magnetization. To solve the problem, we often turn to H.
I just hope that I understand your question correctly.

 

[valjok]

Thank you for your answer. You got my positive feedback loop right. But I cannot understand your answer. I read that the field growth is momentarily proportional to the field, which means dB/dt = kB, which is a law of exponential growth and mathematical formulation of the positive feedback. Any expotential growth ends up in saturation limit. Indeed, the formula does not provide us this limit. I do not understand how the introduction of H term can improve the situation?

Please, forgive me the H-B confusion but I see no difference between the two and do not understand why do we need both even after reading a special section in Wikipedia. I treat them the same thing to the factor of μ₀: H = μ₀B.

Just as an illustration of the linearity and to be concrete, look at the histeresys picture:
http://fizmir.org/bestsoft/02/1-19-3.gif
I understand it like this: the stronger is external field B₀, the greater is magnetization. At some point, all the atoms are aligned and the saturation manifests: extra increase in external field does not increase the magnetization anymore. If you have no onbjection, H (total field) = H₀ (external field) + M (magnetization) = H₀ + XH₀ = (1+X)H₀ = μH₀. They are proportional: H = μH₀. Since the effect is caused by X, the question in short is what makes the Magnetic susceptibility a constant?

I hypothesized that it is the ambient temperature that impedes the exponential explosion. Indeed, the energy of chaos could resist the complete order. So, the level at which magnetization growth is stopped is temperature and external field dependent. In order to increase the magnetization at given temperature, stronger external field is demanded. Now, magnetization can be linearly proportional to the external field! Yet, looking at the hysteresis, I do not believe it. It seems that the force of interest is reluctant to any change: it resists both ordering and disordering. While the entropy acts only in one direction.

 

[Lucien1011]

Sorry I mix up H and B too. I haven't turn to Electromagnetism for some time so I forget many things. Actually I mean H is proportional to M.
What makes the Magnetic susceptibility a constant? I interpret it like this. I never think that H is REALLY proportional to M. In Griffiths' Book, it is said: "for most substances the magnetization is proportional to the field, provided that the field is not too strong." Therefore, I think in fact there is a complicated relation between H and M, but if the field is not too strong, it is approximately linear. I am sure you know Taylors' series and it is the case.
There is a similar case in Physics. That is F=kx.
What do you think about this interpretation?

By the way I don't know anything about histeresys picture and the contents and I can discuss that part. So do you have any reference information so that I can read and discuss?

 

[marcusl]

You and Lucien are on the right track; I'll try to help you clear up your misconsceptions. Magnetization is non-linear except at very small values of H. Although we write

$$\vec{B}=\mu \vec{H},$$

permeability $$\mu$$ and susceptibility $$\chi$$ are not constant. By the way, swapping H and B in your first equation is incorrect, and mu_0 is the SI unit quantity for vacuum permeability not the permeability of a magnetic material. Your second equation that contains only H and not B is also wrong.

mu is the slope of the B curve in your hysteresis loop, and you can easily see from the graph that it varies from a large value at small H to unity at saturation. (By the way, the horizontal axis would usually be labeled H, not B_0.) mu even depends on the history of previously values of applied H (mu is different on the large loop, on smaller loops, and on the initial curve at center of your diagram, even for the same value of H).

Because of these issues, you'll often see the initial permeability $$\mu_i$$ specified for a material. This quantity is mu at H=0 on the small curve at center of your plot, and it's independent of history and hysteresis effects, permitting comparisons between different alloys.

Temperature changes the hysteresis loop, to the point where ferromagnetic effects are suppressed at temperatures above the Curie temperature. However there is no "exponential explosion". Magnetization is non-linearly related to the applied field, and saturation occurs when a preponderance of spins are aligned.

As for H and B, H is magnetic field in a vacuum, B is in a material and there should be no difference in the units used. This is the case in the cgs system. Unfortunately, the engineers (not physicists) who put together SI units goofed and treated H and B as different, causing no end of confusion. Here are two threads on magnetic units:

https://www.physicsforums.com/showthread.php?t=144930"
https://www.physicsforums.com/showthread.php?t=147184"

 

[valjok]

Thank you very much for the many words you write but I haven't got the answer on this topic.

Temperature changes the hysteresis loop, to the point where ferromagnetic effects are suppressed at temperatures above the Curie temperature. However there is no "exponential explosion". Magnetization is non-linearly related to the applied field, and saturation occurs when a preponderance of spins are aligned.

I know that there is no "exponential explosion" as well as the magnetization is only approximately linear. I want to know WHY? Do you understand my my positive feedback view? What is the problem with it?

 

[marcusl]

Ferromagnetism is a complicated phenomenon. While some aspects of magnetism seem to exhibit positive feedback, this is not a useful concept for understanding it overall. For instance, the positive feedback you postulate does not explain the formation and behavior of magnetic domains, one of the most dramatic and unusual features of ferromagnetism, that exist even in a "demagnetized" sample sitting in zero field. Speaking of demagnetized, positive feedback also does not address the "demagnetizing field". As magnetization increases in a sample (such as along a bar), virtual magnetic charges accumulate at the ends that create an internal field that tends to demagnetize the sample.

EDIT: Additional shortcomings show up by contrasting ferromagnetism with a real example of a positive feedback system. The output Vo of an op amp in a positive feedback circuit saturates at the positive or negative supply rail in response to a tiny input Vi and can exhibit hysteresis, both apparently similar to the B-H curve. The similarities end here, however. The Vo-Vi curve is square (four values of "mu") while M-H is rounded (continuously changing mu); the Vo-Vi curve doesn't change depending on the input history; and no cycling of Vi can drive Vo around increasingly smaller loops unto it is zero the way one can demagnetize a magnetic sample.

Because magnetism is so complicated, it took some 50 years of determined work, and the invention of quantum mechanics, to understand and describe its fundamentals after the pioneering work by Curie. Research on the details of magnetism is still going strong another 60 years after that. A single simple picture doesn't adequately capture the behavior.

 

[marcusl]

Better ways to analyze magnetism are by using a combination of 1) potential theory (including the fictitious charges just mentioned) for bulk behavior and 2) energy considerations and 3) phase transitions for microscopic behavior. To expand on 2), Landau and Lifgarbagez were able to show that the thermodynamic free energy of a magnetic material is minimized if electron spins align only within local domains. The magnetization of neighboring domains can differ and even be opposite. Growth and rotation of domains in the presence of applied fields depends on spin-spin coupling, creating one of the well known "long range order" systems in solid state physics.

To explore further,
--The Hyperphysics website has very basic introductory material:
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/ferro.html#c1"

--Reitz and Milford's "Foundations of Electromagnetic Theory" (at least the 1st edition) has a useful chapter on magnetic properties of matter that covers empirical behavior.
--Microscopic phenomenology at an intro level is found in Kittel, "Intro to Solid State Physics." Some people claim that early editions (I have the 3rd) are better than the recent 7th and 8th.
--Parts of Ahoroni, "Intro to Theory of Ferromagnetism" are online.
http://books.google.com/books?hl=en...X&oi=book_result&resnum=10&ct=result#PPR9,M1"
It looks like Chapters 1, parts of 2 and 3, 4, 7 and 8 will give you a thorough overview.