# I Magnetic field of a cylinder with current down the center

1. Nov 16, 2017

### kmm

I was just in a conversation with someone regarding the magnetic field resulting outside of a solid cylinder, with a current moving down the center of the cylinder, and then the resulting magnetic field after removing the current. Now I haven't thought about magnetic fields/magnets for a while and felt a little rusty talking about them, and so I was hoping to get some clarification on this if I'm not thinking correctly about it. To this point, my analysis is mostly just qualitative.

Imagine we have a solid cylinder that could be magnetized, and we were able to run a wire down the center insulated from the rest of the cylinder, running current through the wire. My understanding is that the resulting magnetic field outside the cylinder would be zero, with current running through the wire. But when the current is turned off, a nonzero magnetic field is produced outside the cylinder opposite in direction to the field produced by the current. I have attached two pictures depicting these two cases (excuse my poor drawing ). I have drawn a cross section of the cylinder. The red 'x' in the center represents the current running into the board. The blue arrows represent the magnetic field due to the current, and the dipoles lining themselves with the magnetic field. The black circle is the edge of the cylinder, and the green arrows represent the magnetic field outside the cylinder, due to the dipoles when there is no current.

So I am imagining the dipoles within the cylinder are lining up with the magnetic field produced by the wire, and so inside we have the magnetic field due to the current plus the field due to the dipoles. The dipoles we would have at the edge of the cylinder seems to produce a net current at the edge in the opposite direction to the current in the wire, and therefore we have a magnetic field in the opposite direction outside the cylinder, producing a total magnetic field of zero outside the cylinder (I understand I haven't actually shown that quantitatively here, but it's my guess at the moment). I imagine that if we now turned the current off at the center of the cylinder, the dipoles would remain in their state like any other magnet, and so we would now have a magnetic field outside the cylinder in the direction I indicated in green. I'm not sure to what degree this would also depend on what type of magnetic material we were using.

The person I was talking to claimed that with a current at the center, there would be a magnetic field outside the cylinder, but if we turned off the current there would not be a magnetic field inside or outside the cylinder. They did not give much of an argument for this and so I wasn't convinced, but now I'm curious if I'm thinking correctly about this, because I wouldn't be surprised if I'm missing something. I also wouldn't be surprised if this can't be answered qualitatively. Thanks!

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2. Nov 18, 2017

### Baluncore

There must be a return path for any current. Is the cylinder electrically conductive?
The electric and magnetic fields about conductors exist everywhere between the conductors that make the return circuit.
If the axial current returns via the inside wall of the cylinder, the fields outside will cancel to zero. It is a coaxial transmission line.
If the axial current returns via some path external to the cylinder there will be a magnetic field external to the cylinder.

3. Nov 18, 2017

### kmm

I'm imagining the current to return externally to the cylinder, but with the wire insulated from the cylinder such that current isn't moving from cylinder to wire and vice versa. How can we be sure that there would be a magnetic field external to the cylinder in this case? The induced external magnetic field of the cylinder wouldn't cancel the external field due to the current?

4. Nov 18, 2017

### Baluncore

The external return wire is electrically insulated, not magnetically insulated. The current flowing in the external return circuit will generate a magnetic field between the conductors, in the same direction as the field generated by the current in the axial wire. Because the circuit currents are flowing in opposite directions, the magnetic field sums between the wires but tends to cancel away from both wires.
Also, the electric field between the axial and the return conductors exists throughout the experiment space, so there must be a perpendicular magnetic field to accompany the electric field in the intrinsic impedance of free space, where the ratio of E/M is 376.73 ohms, ( close to, but not exactly 120π ). You cannot have one E or M without the other.

Last edited: Nov 18, 2017
5. Nov 18, 2017

### kmm

The idea was not to try to eliminate any EM fields. I only cared to prevent current moving from the wire to the cylinder and vice versa. I want to understand only what is going on with the magnetic fields and induced magnetic fields in and outside the cylinder. I'm imagining that the wire is very long such that we don't really have to think about the circuit (can be considered infinite for our purposes). The wire then goes through a solid cylinder and we're attempting to magnetize it. The magnetic field due to the current circles inside the cylinder and extends outside the cylinder. Now what is the total magnetic field inside and outside the cylinder when both when we have current flowing and after we have turned it off? I attached two drawings in the OP and described what case each one represented if you want to look.

6. Nov 18, 2017

### Baluncore

Let us assume the cylinder magnetic material is linear, without hysteresis or saturation, then the magnetic circuit would preferentially flow through the cylinder material. This would increase the inductance of the axial conductor. But there would still be an external magnetic field escaping the magnetic cylinder and radiating outwards, eventually reinforcing the incoming magnetic field due to the equal and opposite current that must be flowing in the return circuit.

There is no way to prevent an increasing field in a linear magnetic material from spreading everywhere, or to prevent a collapsing field from behaving likewise. You cannot have the field in a linear magnetic circuit behave one way while increasing, then invoke a different linear magnetic circuit while the field is reducing. That helps explain why the hypothesised variant magnetic phases in your model are impossible in linear magnetic material. If the field was zero before the current started to flow, then when the current stops flowing, the field will have returned to zero.

The magnetic permeability of a magnetic core is used to increase the inductance of wires threaded through the core, usually to better attenuate RF noise signals. The inductance of the central conductor will be increased by the surrounded magnetic material and for the same current, more energy can be stored in the resultant magnetic field; E = ½ ∙ L ∙ I².

It is not possible to suddenly turn off the inductor current without taking the time to cancel the proportional surrounding magnetic fields. That goes for the fields in the magnetic cylinder and in the external fields. In linear magnetic circuits, the field is directly proportional to the electrical current. You will need to provide a reversed voltage to reduce the current to zero, and so collapse the created magnetic fields. V = L ∙ di/dt.

The non-linear magnetic characteristics of magnetic materials are very important in deciding where energy may be stored and under what conditions it will be released. That is where it gets interesting. Magnetic material with hysteresis can been used to make magnetic “core” memory. Saturable magnetic cores can be used to make magnetic amplifiers of electrical signals.

7. Nov 18, 2017

### kmm

OK, thanks for helping again. I think there is a lot more lacking on my end of understanding about this than I realized. I think what would help me is if I laid out my understanding of the scenario and if you pointed out where I'm mistaking, or maybe where I am missing more that I need to consider. Based on what you're saying, it seems to me that I'm oversimplifying the situation.

Again, sorry for getting repetitive, but I want to make sure the situation is clear. We simply have a long straight wire with some current. This wire happens to pass through a really long magnetizable material such that no current passes between the two. Consider the cylinder and wire to be infinite. Now, I'm only interested in what the magnetic field is doing inside the cylinder, and just outside the cylinder. In this situation, I see a magnetic field produced by the wire circling around the wire and extending beyond the cylinder. The dipoles in the cylinder then line up with this field in circles, from the center of the cylinder out to the edge of the cylinder. These dipoles on there own it seems, would create a magnetic field of their own inside and outside the cylinder. I would think that to actually determine the total magnetic field, we just need to evaluate the magnetic field due to this arrangement of dipoles, then add that field to the field generated by the current. Am I correct to this point?

8. Nov 19, 2017

### Baluncore

Probably not.
Think of the magnetic domains in the material as being randomly oriented spring loaded dipoles. As the magnetic field due to the current increases, the dipoles will become progressively more aligned and store more energy. When the field is later reduced they will release that energy as they elastically return to their initial states.

The magnetic dipoles are aligned as a response to the magnetic field of the electric current. You appear to be investing energy once, then expecting to withdraw it twice.

9. Nov 19, 2017

### kmm

OK, this is interesting to me. I have been imagining this scenario as essentially the same as when we create a bar magnet, so this may be the crux of the issue. If we take a bar of iron, wrap it in wire and pass current through it such that we get a strong uniform magnetic field through the iron, the dipoles in the iron begin to align with the magnetic field. The magnetic field outside the bar magnet would then be the magnetic field due to the current in the wire and the field due to the aligned dipoles. Now when we remove the current, the dipoles are "frozen" in place so that a magnetic field now remains around the bar of iron.

It seems to me that you're saying I can't use this same reasoning for our main problem. Is that correct?

10. Nov 19, 2017

### Baluncore

Your reasoning does not fit with the real world.
Part of the flux that aligns the domains during the magnetisation process is left behind in the permanent magnets, they cannot be added as they are the same thing. You must take care to avoid counting the same flux twice.

Magnetic materials such as soft iron will not maintain the magnetism impressed on them for long. Permanent magnets are made from magnetic materials such as alloy steel, which are harder to magnetise and harder to demagnetise. Those materials are non-linear and will retain some of the domain alignment impressed on them during the magnetisation process.
https://en.wikipedia.org/wiki/Remanence
https://en.wikipedia.org/wiki/Coercivity

11. Nov 21, 2017

### kmm

OK, this has helped. I think now I just want to study this whole subject in more depth. Do you have any recommendations for textbooks that go into more depth with magnetic fields/magnetic materials? I have taken E&M so I have a basic understanding of magnetic fields.

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