Solenoid filled with magnetic material

[ShayanJ]

Consider a solenoid with length L and N turns of wire wound around it, filled with a rod of permeability $\mu$.I want to calculate the magnetic field it produces on a point P on its axis and outside it.The angle between the axis and the lines drawn from point P to a point on the nearest and farthest loops of the solenoid are $\beta$ and $\alpha$ respectively.
I used the formula for the magnetic field of a circular loop of current on its axis and integrated it along the solenoid:
$\vec{dH}=\frac{I R^2 \frac{N}{L} \hat{z} dz}{[(d-z)^2+R^2]^{\frac 3 2}} \Rightarrow \vec{H}=\frac{N I R^2 \hat{z}}{L} \int_\alpha^\beta \frac{R\csc^2\varphi d\varphi}{(R^2\cot^2\varphi+R^2)^{\frac 3 2}} \Rightarrow \vec B =\frac{\mu_0 N I R^2 \hat{z}}{L}(\cos\beta-\cos\alpha)$
I want to know how the magnetic material filling the solenoid affects the field outside it and how can I calculate its effect?Is it right to just calculate H,as I did, and then simply multiply it by $\mu_0$ to obtain the field outside the solenoid?
Thanks

 

[Philip Wood]

A good question. Hope you'll get some useful replies. I think that simply multiplying B for the air/vacuum case by μ (not μ₀) would only be right if the iron surrounded the solenoid as well as filling it (that is the solenoid and the field-measuring probe are immersed in a sea of iron, as it were). Yet we know from experiment that merely filling the solenoid with an iron core does increase the B field at points on the axis outside the solenoid.

When it comes to inhomogeneous magnetic media, many textbooks handle the case of ring-shaped cores with air gaps in them, but your case tends not to be discussed. Too hard?

A nitpick: I don't think you should have the R² (or the ^z) in your equation.

 

[ShayanJ]

A good question. Hope you'll get some useful replies. I think that simply multiplying B for the air/vacuum case by μ (not μ₀) would only be right if the iron surrounded the solenoid as well as filling it (that is the solenoid and the field-measuring probe are immersed in a sea of iron, as it were). Yet we know from experiment that merely filling the solenoid with an iron core does increase the B field at points on the axis outside the solenoid.

When it comes to inhomogeneous magnetic media, many textbooks handle the case of ring-shaped cores with air gaps in them, but your case tends not to be discussed. Too hard?

A nitpick: I don't think you should have the R² (or the ^z) in your equation.

I doubt that too,but I just can't find the right thing to to do!
And yeah,$R^2$ shouldn't be there but I think $\hat{z}$ should be!

 

[Philip Wood]

Sorry, of course ^z should be there!

The Force between electromagnets section in the Wiki article on Electromagnets purports to deal with the question we're interested in. Unfortunately the pole strength formula given is that for an air/vacuum-cored solenoid!

 

[ShayanJ]

Sorry, of course ^z should be there!

The Force between electromagnets section in the Wiki article on Electromagnets purports to deal with the question we're interested in. Unfortunately the pole strength formula given is that for an air/vacuum-cored solenoid!

I'm beginning to think that the magnetic field of such a solenoid has two parts.One part is given by the formula I wrote in the first post and the other part is due to the magnetization of the magnetic core.But for deriving that,we should have an expression for the magnetization of the magnetic core which can be derived from the formula for the magnetic field inside the solenoid.
I think I found how to do it,I'll tell you if I succeed!

 

[Philip Wood]

I agree with this approach. The problem, of course, is finding the magnetisation of the core, or its dipole moment. I'll be interested in your progress!

 

[marcusl]

I'm beginning to think that the magnetic field of such a solenoid has two parts.One part is given by the formula I wrote in the first post and the other part is due to the magnetization of the magnetic core.But for deriving that,we should have an expression for the magnetization of the magnetic core which can be derived from the formula for the magnetic field inside the solenoid.
I think I found how to do it,I'll tell you if I succeed!

You have identified the reason that magnetic calculations are tricky. The magnetic field depends on the magnetization, but the magnetization depends on the magnetic field in a non-linear way (remember that mu depends on local H). Except for a few simple geometries that can be solved exactly, the only way to solve these problems is numerically. Iterative approaches are common; a vacuum field is assumed, magnetization is computed, the field of coil+magnetization calculated, magnetization from that field computed, and so on until convergence is reached.

 

[ShayanJ]

You have identified the reason that magnetic calculations are tricky. The magnetic field depends on the magnetization, but the magnetization depends on the magnetic field in a non-linear way (remember that mu depends on local H). Except for a few simple geometries that can be solved exactly, the only way to solve these problems is numerically. Iterative approaches are common; a vacuum field is assumed, magnetization is computed, the field of coil+magnetization calculated, magnetization from that field computed, and so on until convergence is reached.

Yeah but its not that tricky about this problem because $\mu$ is constant.
Anyway,I couldn't find the magnetic field inside the solenoid!
So I think I should assume the field is consant inside the solenoid and use the formula $B=\frac{\mu N I}{L}$

 

[f95toli]

If you are only interested in the field inside the solenoid and it is very long, you can certainly do it analytically.
However, in your first post you mentioned the field outside the solenoid. In this case you have to use numerical methods. I am pretty sure the same is true for a short solenoid if it is filled with a magnetic material and you are interested in the field near the edges (the core will "distort" the field).
The analytical approach isn't very viable for most real-world problems.

 

[Meir Achuz]

End effects at the ends of the iron core make this a very difficult problem that can only be soved numerically.
You can get simple results for B far from the ends of the solenoid.

 

[Philip Wood]

MA:: Just to confirm: you mean INSIDE the solenoid, far from the ends?

 

[Meir Achuz]

Inside or outside. Just far.