Effect of magnetic materials on field created by current

[salparadise]

Hello,

Imagine I have an infinite wire with a DC current. Around it there two rings of ferro-magnetic material, not far from each other.

Does changing the mu_r of one of the first ring, changes the magnetic field in the second ring?

- If I consider just Ampére law (macroscopic formulation), I would say that since H doesn't change by changing the magnetic permeability, then B on the second ring should not change.

- But when I think about the total energy of the system (Em=1/2.B.H). Since in the first ring B increases, the total energy of the system should also increase, which doesn't make sense to me.

This is really troubling me, considering that I should have learned enough of E.M. to know this.

Any lights on this would be very helpful,
Thanks

 

[mfb]

The total energy is different in different setups - that should not be surprising, as the setups are different.

If you actively modify µ_r in such a setup, I think you get other effects - probably a lower DC current (but that is a guess). And your µ_r-change could involve some energy exchange with an external system as well.

 

[cabraham]

What you describe is a wire w/ current and 2 ferrite beads on the wire. If the 2 beads are different in permeability, this is what we have.

The "H field", or "magnetic field intensity", is the same for both media. H is determined by current I, and dimensions of the loop. But the "magnetic flux density", aka "B", in each ferrite bead varies with permeability, "μ". The ferrite bead w/ the higher μ will have the higher B, since B=μH.

Should the current get turned off, H goes to zero, but some residual B could remain, depending on the shape of the ferrite materials' B-H curve. Does this help.

Claude

 

[salparadise]

Thanks for the answers. I think that now I understand better the situation. My problem was not so much the total energy, but the fact that ferromagnetic materials also "concentrate" magnetic flux density lines, what is e.g. used for magnetic shielding. And I was imaging a hollow sphere of a high-μ material near the wire, and I couldn't see how the magnetic field inside it would be reduced. But that's because in my mind I was still using the cylindrical symmetry to simplify the integral in Ampére law, while the symmetry would no longer be there.

This kind of argument was confusing me concerning my problem. In attachment I send a diagram of the setup so that I'm studying.

In blue is the core where I want to measure the magnetic flux, and in red are two other cores that just need to be there. And, I was expecting the red cores to concentrate part of the magnetic flux, what would reduce the flux in the surrounding space, and consequently reduce the flux in the blue core

But in reality the fact that the red cores are there or not, and independently of their geometry, the magnetic flux in the blue core won't change. Do you agree?

 

[mfb]

Ah, no infinite length for the cylinders. I'm not sure... I think the red objects will influence the field configuration, and the direction could depend on the dimensions of your objects.

 

[Jano L.]

From the symmetry of the setup, the magnetic intensity $\mathbf H$ does not depend on the angle measuring position around the wire. It can only depend on $z$ (direction of the current) or $R$ (distance from the wire). From the Ampere law, we know it cannot depend on $z$, so we arrive at result that $\mathbf H$ is a function of distance from the wire $R$ only, which can be easily found from the Ampere law.

Now, if the material of the blue core is ferromagnetic, its magnetization and thus also $\mathbf B = \mu_0(\mathbf H + \mathbf M)$ depend on how the system has been set up. If quasi-statically from the state of zero current and zero magnetization, we can use the corresponding $\mu_r$ (the initial magnetization curve) to find the magnetic field $\mathbf B$ as $\mathbf B = \mu_r \mu_0 \mathbf H$ and then by integration over the cross-section of the blue core, the magnetic flux through it.

 

[salparadise]

Hi mfb, I'm only considering finite cylinders for the ferromagnetic cores.

My intuitive guess was also that the red ferromagnetic cores would change the field distribution on the blue core, but I can't see how this can come out from Maxwell equations...

So, from the equation I can only agree to what Jano L. wrote, that the field will depend solely on R. But somehow it's being difficult to accept this :)

 

[mfb]

From the Ampere law, we know it cannot depend on $z$

What about the regions where you have the cylinders inside? What about non-circular* components of H?

*as in: around the central wire

 

[Jano L.]

What about the regions where you have the cylinders inside? What about non-circular* components of H?

*as in: around the central wire

For calculation of magnetic flux, we need just the $H_\phi$ component of the field.

The Ampere law gives the same circulation independent of $z$:

$$
\oint_{circle~ R~at ~z} H_\phi ds = I,
$$
so
$$
H_\phi = \frac{I}{2\pi R}.
$$

For calculating the flux through the core, we only need $H_\phi$ and positions of limits of the cross-section of the ring. If the section is rectangular, limited in $z$ by $z_1, z_2$ and in $R$ by $R_1, R_2$, the flux is

$$
\Phi_B = \int_{z_1}^{z_2} \int_{R_1}^{R_2} \mu_r\mu_0 H_\phi ~ dR dz.
$$

so we can get the result even without knowing the $z, R$ components of the intensity $\mathbf H$.

But in reality the fact that the red cores are there or not, and independently of their geometry, the magnetic flux in the blue core won't change. Do you agree?

That is my guess as well, in other words, the field of all rings probably vanishes outside of them, which would happen if the magnetization had zero $z, R$ components. But how to find out for sure ? It seems like an intriguing problem.