# Homework Help: Finding Absolute Permeability from Magnetization

1. Jul 14, 2008

### SuccessTheory

1. The problem statement, all variables and given/known data
I am doing a project for which I am learning some aspects of electromagnetism by myself, so you can imagine how lost I am. Well, not completely: I was searching for a ferrite with high permeability as a core material. I learned how the octahedral and tetrahedral sites and the unpaired electrons within the ferrite structure contribute to magnetization. So I went ahead and calculated the saturation magnetization of a ferrite that had a high magnetic moment per molecule.

μ: absolute permeability, μ0: permeability of free space, M: magnetization, H: magnetic field strength, B: magnetic flux density

2. Relevant equations
Now I don't know how to go from this saturation magnetization to absolute permeability. I know:

B = μ H and B = μ0 (H + M)

putting the two together I have:

μ = μ0 + (μ0 M)/H

3. The attempt at a solution
I don't know how to deal with H to solve for μ (I calculated M and μ0 is a constant). I know H is the applied field strength but I just read that generally M is also a complex function of H... so I don't think I can use any H but the H that gives my saturation magnetization, M. Any help on going from M to μ would be appreciated.

Last edited: Jul 14, 2008
2. Jul 15, 2008

### dude_

this is not so simple to answer quickly in a forum page.

yes M depends on H in a complex way. in ferromagnetic materials you even have hysterysis, which means that the past history of H is also relevant. and if you dont have hysterysis, you may have nonlinearities. in superparamagnetic materials, meaning that in subdomain magnetic materials (which means that you are dealing with a really small ferrite particle), you dont observe hysterysis so we can safely ignore the effects of it for now.

now suppose there is some H field applied in one direction. if all of the magnetic moments of the particle alligns with the field (ignoring the effects of thermal agitations), you reach to saturation magnetization and you cannot get more than that even if you apply an infinite amount of H field. now suppose you decrease your field, H, to zero slowly. you will be likely to observe a linear dependence of magnetization to H. the slope of this dependence is likely to be determined by experimentation. so you can safely incorporate the value this slope to your model to determine the absolute permeability, in the case that you have low fields and you dont deal with hysterysis.

if you want to include the effects of thermal agitation, you have to solve for the boltzman distribition and so on, and you end up with the langevin equation. and if i remember correctly, in the case of low fields, it turns out that the dependence is also linear, just the magnitude of the saturation magnetization becomes less.

this was for the static case. as for the dependence of the magnetization to time varying H fields, the issue becomes complicated, in which case you have to incorporate some "phenomological" dynamic equations of the relaxation of the magnetization on the H field (this is similar to spin relaxation in magnetic resonance imaging), and also you have to account for the relaxation of the magnetization distribution into allowable thermodynamic equilibriums and eventually relaxing into one of the preferred direction and so on eventually ending up with some frequency dependent complex permeability.

hope this helps.

3. Jul 15, 2008

### SuccessTheory

Thank you very much for your response.

The ferrite I'm gonna work with will be a few microns thick, which I think is still far bigger than subdomain level so some hysteresis will be present. I have read that the loop for this ferrite isn't very thick (soft magnet), I'm not sure if that would make a difference other than a steeper slope with the M and H relationship. However, would your method quoted above still give a rough approximation of the relationship even with hysteresis?

Unfortunately, I am using a time varying H field and I don't want to go too far into this. What I ultimately want to do is show that this ferrite is more permeable than another in a sort of quantitative way. Say if it is more permeable for the static case, will it be the same for the time varying case?

4. Jul 16, 2008

### dude_

to work around the hysterysis issue i can speculate:

heat your sample above the curie temperature so that all the hysterysis is cancelled into the origin (lookfor the curie temperature in google). then work in the small fields regime not to go into the hysterysis again?

as for your second question, if you are working with a single particle (that is if you dont have an ensemble of it) then you can ignore the effects of thermodynamics. but i am not sure, how can you work around it if you have an ensemble of it. i suggest: come back and ask it again in one week.. i was just working on this issue so i can help.

5. Jul 16, 2008

### dude_

by the way why are you using a time varying field?

6. Jul 16, 2008

### SuccessTheory

What if the ferrite is a thin film? It's going to be used as a core in a radio application, where the time varying H field comes in.

7. Jul 16, 2008

### dude_

i will provide an answer regarding the "time varying response of ferrite ensembles" but in the mean time i suggest this topic to be moved into more general forums so that experts on electromagnetism can comment on? because i am also working on this topic and i may want to ask more spesific questions regarding the issue.

8. Jul 16, 2008

### SuccessTheory

9. Jul 16, 2008

### dude_

http://www.sciencedirect.com/scienc...d=690958&md5=0f94c71760aa60d5b747a6970686221f

look at this one..

now we have to clarify that the constant relating H and M is called susceptibility and its denoted by X.

in case of time varying fields this X assumes a frequency dependent complex value and according to this paper its given by:

Xo/(1+j*w*tao), Xo being the equilibrium susceptibility.

tao is to be determined by thermodynamical considerations.

10. Jul 16, 2008

### dude_

you are mentioning that you are using radio frequency so the value of tao becomes important for you.

11. Jul 16, 2008

### SuccessTheory

OK, thanks for sharing this info with me. But it seems that the magnetic susceptibility will have to be determined experimentally out of practicality, or else my head will explode going from relationship to relationship :P

But in order to do this, I would have to test with frequency close to the actual one that will be used, because as you said tau (and w) are dependent on freq?

12. Jul 16, 2008

### dude_

tao is not frequency dependent, it is material dependent.
tao is proportional to: gamma^-(3/2)*exp(gamma), gamma = KV/kT, K being the anisotropy constant, V being the volume of your ferrite particles. so as your ferrites get bigger, and their shape is getting far from being a sphere you get higher tao..

but i am warning you, this is only true for small fields, that is when you dont have hysterysis.
also in your case you may have to account for the exchange energy of the ferrites btw each other, since in this treatment i referenced, this effect is neglected!

w = 2pi*frequency.

13. Jul 16, 2008

### dude_

indeed what brings in the hysterysis effect is the exchange energy of the ferrites. so in your case you'll definetely have it.