Solenoid filled with magnetic material

In summary, the conversation discusses the calculation of the magnetic field produced by a solenoid filled with a rod of permeability μ on a point P on its axis and outside of it. The formula for the magnetic field of a circular loop of current on its axis is used and integrated along the solenoid. The conversation also raises questions about the effect of magnetic material filling the solenoid on the field outside and how to calculate it. It is suggested that the approach may involve finding the magnetization of the core and using iterative methods to solve the problem. However, the conversation also mentions the difficulty in finding the magnetic field inside the solenoid and suggests assuming a constant field inside and using the formula B = (μNI)/
  • #1
ShayanJ
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Consider a solenoid with length L and N turns of wire wound around it, filled with a rod of permeability [itex] \mu [/itex].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 [itex] \beta [/itex] and [itex] \alpha [/itex] respectively.
I used the formula for the magnetic field of a circular loop of current on its axis and integrated it along the solenoid:
[itex]
\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)
[/itex]
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 [itex] \mu_0 [/itex] to obtain the field outside the solenoid?
Thanks
 
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  • #2
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.
 
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  • #3
Philip Wood said:
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,[itex]R^2 [/itex] shouldn't be there but I think [itex] \hat{z} [/itex] should be!
 
  • #4
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!
 
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  • #5
Philip Wood said:
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!
 
  • #6
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!
 
  • #7
Shyan said:
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.
 
  • #8
marcusl said:
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 [itex] \mu [/itex] 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 [itex] B=\frac{\mu N I}{L} [/itex]
 
  • #9
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.
 
  • #10
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.
 
  • #11
MA:: Just to confirm: you mean INSIDE the solenoid, far from the ends?
 
  • #12
Inside or outside. Just far.
 

FAQ: Solenoid filled with magnetic material

1. What is a solenoid filled with magnetic material?

A solenoid is a coil of wire that creates a magnetic field when an electric current is passed through it. When this coil is filled with a magnetic material, such as iron or steel, it becomes an electromagnet. This means that the magnetic field produced by the solenoid is significantly stronger.

2. How does a solenoid filled with magnetic material work?

When an electric current is passed through the coil, it creates a magnetic field. The magnetic material inside the coil becomes magnetized and strengthens the magnetic field. This allows the solenoid to attract and repel other magnetic objects, making it useful in various applications such as electric motors and speakers.

3. What are the advantages of using a solenoid filled with magnetic material?

The main advantage of using a solenoid filled with magnetic material is that it creates a much stronger magnetic field than a regular solenoid. This makes it more efficient and powerful for its intended use. Additionally, because the magnetic material can be easily magnetized and demagnetized, the strength of the magnetic field can be controlled by adjusting the electric current passing through the coil.

4. In what applications is a solenoid filled with magnetic material commonly used?

Solenoids filled with magnetic material are commonly used in various industrial and everyday applications. Some examples include electric motors, speakers, magnetic locks, and magnetic sensors. They are also used in medical devices, such as MRI machines, and in scientific research, such as particle accelerators.

5. Can the magnetic material inside a solenoid be changed or replaced?

Yes, the magnetic material inside a solenoid can be easily changed or replaced. This allows for flexibility in the strength and type of magnetic field produced by the solenoid. Different types of magnetic materials, such as neodymium or alnico, can also be used to create different magnetic fields for specific applications.

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