# Can two soft ferromagnetic materials in a field magnetize each other

I am having difficulty with the seemingly contradictory explanations of ferromagnetic material as a magnetic field "amplifier" and a field "concentrator".

The amplifier analogy is usually referred to when the material is in a solenoid for an electromagnet application, while the concentrator analogy is common when discussing the distorting effects of a ferromagnetic material in an otherwise simple field (e.g. due to a bar magnet) in air.

Based on the fact that a magnetic field will polarize the dipoles in a ferromagnetic material -- which should add to the overall field -- it seems to me that the "concentrator" analogy is not really accurate. Rather, the material creates its own local field that overpowers the pre-existing field.

Following from this, it seems that if you place two soft ferromagnetic materials in a pre-existing magnetic field (e.g. due to Earth), then they should each be polarized to create their own local fields. If the relative magnetic permeability (due to material type), magnetic moment (due to volume), and proximity is high enough, then it seems that there would be a 'mutual magnetization' between the ferromagnetic materials.

If calculating the 'mutual magnetization' iteratively (e.g. calculate induced magnetic moment in first material due to pre-existing field; then calculate induced magnetic moment in second material due to pre-existing field and magnetic moment of first material; etc..), it also seems that this mutual magnetization would continue until each material was saturated. However, this can't be true because a single ferromagnetic material (which is also two ferromagnetic materials right next to each other) doesn't necessarily saturate in an ambient field.

So, all this to ask: Can two soft ferromagnetic materials in an ambient field induce magnetization in each other? And if so, why don't they saturate?

Edit: Thinking more, it seems that a 'snowballing' of mutual magnetization may not occur in practice because materials have finite magnetization (M), and therefore a non-infinitesimal volume through which their magnetic dipoles must be spread. So even though there may be some mutual magnetization between two ferromagnetic materials in a pre-existing magnetic field, it does not lead to saturation because the internal dipoles are too weak to induce alignment/magnetization in the other material -- just as a single aligned domain in a magnetic material does not induce alignment in the entire material. Is this accurate?

This infinite iteration problem is something I've grappled with as well. What I think you haven't realized is that the solution "settles". Say $y=x^2, z=3w,x=z^{1/2},w=ln(y)$. Now $y$ depends on $x$, which depends on $z$ which depends on $w$, which depends on $y$. So $x$ depends on $y$ and $y$ depends on $x$. How can this be??

Of course, you'll say that this is just a system of equations and of course there is a solution for it. It's becoming obvious that nothing depends on anything.

Now let me apply this to the two ferromagnetic materials (I'm calling them 1 and 2)

Now,$$\vec{M}=\chi_m\vec{H}=\chi_m\frac{\vec{B}_{original}}{\mu_0}$$

where $\vec{B}_{original}$ is the field that would have been in the region of space occupied by 1 (or 2) due to $\vec{B_0}$ and 2 (or 1) in the absence of 1 (or 2)

If the field due to 1 is $\vec{B_1}$ and due to 2 is $\vec{B_2}$ Then $$\vec{M_1}=\chi_m\frac{\vec{B}_0+\vec{B_2}}{\mu_0}$$

$$\vec{M_2}=\chi_m\frac{\vec{B}_0+\vec{B_1}}{\mu_0}$$

The magnetic field of a magnetic material is proportional to its magnetization. So $\vec{B}_1$and $\vec{B}_2$ are some function of $\vec{M_1}$ and $\vec{M_2}$ respectively. (For example, the field of a bar magnet is $B(M, r, \theta) = \frac {\mu_0} {4\pi} \frac {MV} {r^3} \sqrt {1+3\sin^2\theta}$)

So suppose $$\vec{B_1}=k_1\vec{M_1}$$ and

$$\vec{B_2}=k_2\vec{M_2}$$

where $k_1$ and $k_2$ are some functions of distance, angle etc.

You now have four equations and four unknowns $\vec{B_1},\vec{M_1},\vec{B_2},\vec{M_2}$

So you see that there is no iteration. Just a solution.

• Okay, this is a good mathematical approach. It led me to the other answer I posted. – abc Mar 23 '17 at 16:25

I am thinking when the permeable material is exposed to a field, those atoms/domains with low 'friction' are the first to get reoriented, and the process continues (curve depends on the distribution of frictions) until some 'threshold friction' is reached. So I guess the increasing friction of remaining domains slows the reorientation process until equilibrium is reached.

The contributions to the material's internal field due to another ferromagnetic material would allow this process to go further than it otherwise would, but unless saturation occurs there will still be a (albeit higher) threshold friction that halts the reorientation process.