How Can Vector Field Simulation Differ Between Iron Core and Vacuum?

[FelixTheWhale]

Hello, i want to draw the vector field and see the difference between propagation of field inside the body (with permeability μ>>1) and vacuum (μ=1).

For the picture above i used Ampere's law, ∇xB=J and it illustrates only situation in vacuum. How to implement a contribution for example of iron core? (Red is a current loop)

 

[Simon Bridge]

Same implimentation - make ##\mu## a different value.

 

[Hesch]

Same implimentation - make μ\mu a different value.

I think that the question was:

If you insert a lump of unmagnetized iron ( say a ball ) into the field, the field will be distorted inside the ball, and in the outside vacuum as well.
How can the propagation of the field through vacuum and iron be simulated/calculated in this unhomogenous environment ?

 

[FelixTheWhale]

Thanks for commenting, i found that most closer equation is https://en.wikipedia.org/wiki/Landau–Lifshitz–Gilbert_equation
So the target is to calculate the magnetisation M under the external field (inside) and then propagate M using the boundary conditions to the vacuum (outside)

 

[Simon Bridge]

I think that the question was:

If you insert a lump of unmagnetized iron ( say a ball ) into the field, the field will be distorted inside the ball, and in the outside vacuum as well.
How can the propagation of the field through vacuum and iron be simulated/calculated in this unhomogenous environment ?

... ame implimentation - different mu values - also need boundary conditions.

 

[Hesch]

. . . ame (aim?) implimentation.
How will I do that ?
I have attached a solenoid, connected to a frame of wire, just to make flow of current through the circuit possible. Some magnetic field line will spiralize around the frame and will at last bite its own tail. The path of this magnetic field line has been calculated by means of Biot-Savart, and is only affected by the current through the solenoid and the frame.

Now, if I introduce a lump of iron near the solenoid with known position, known shape and known μ_r, the field line could choose a complete different path.
How will I "tell" Biot-Savart about the existence of this lump of iron, or which other method of simulation must be used?

 

[mfb]

If in doubt, you can always use a grid and iterative numerical simulation of every voxel of the grid. Some approximations can be useful to simplify the problem.

 

[Hesch]

Some approximations can be useful to simplify the problem.

My main goal is to verify that the path of a magnetic field line always will be a closed loop (Amperes law).
The path in #6 closes up with an accuracy = 0.06%, measured as:

Error = ( distance between start, end ) / ( length of path ) * 100%

The method is simply to follow the direction of the magnetic field with a 3D compass.

Maybe some day I will make a movie, seen from a camera onboard a small airoplane flying through ( a lot of ) magnetic field lines and solenoids. Some lumps of iron within the airspace will be welcome here.

 

[mfb]

Ampere's law is equivalent to one of the Maxwell equations. You'll need an experiment to verify it. A simulation based on Maxwell's laws will always reproduce Maxwell's laws with all their theoretical implications. If it does not, your simulation is broken, but that does not tell you anything about the universe.

You are misinterpreting Ampere's law. It makes a statement about the magnetic field along an arbitrary (chosen by you), closed loop. This loop has to be closed to make any statement about it.
It does not say that magnetic field lines would be closed loops, and in general they don't have to be simple closed loops (see the field geometry in fusion reactors, for example).

 

[Hesch]

It does not say that magnetic field lines would be closed loops

I know what it says, but if Amperes law is read very closely, it can be concluded that it doesn't make sense if magnetic field lines don't form closed loops.

Also I can see that if the electric circuit is not closed ( removing the frame in #6 ), a magnetic field line will not be closed ( error = 1.6%, whatever ). That's why Amperes law is said to be only valid in connection with "infinit long wires", etc.

I don't know if there are things like magnetic monopoles in a fusion reactor? But Biot-Savart and Ampere seem to agree regarding electric circuits at "normal" temperatures.

 

[mfb]

I know what it says, but if Amperes law is read very closely, it can be concluded that it doesn't make sense if magnetic field lines don't form closed loops.

See toroidal magnetic fields with an additional ring current, where the only closed thing are surfaces.

Also I can see that if the electric circuit is not closed ( removing the frame in #6 ), a magnetic field line will not be closed ( error = 1.6%, whatever ). That's why Amperes law is said to be only valid in connection with "infinit long wires", etc.

No, the law is valid in every setup. Infinite wires allow to use some symmetry to calculate the magnetic field strength easily, but this has nothing to do with the validity of the general law.

I don't know if there are things like magnetic monopoles in a fusion reactor?

There are not. No magnetic monopole has ever been found (although they exist as quasiparticles in solid matter - not relevant for fusion reactors). The field lines do not end in fusion reactors, but they do not form nice closed loops either.

 

[Hesch]

See toroidal magnetic fields with an additional ring current, where the only closed thing are surfaces.

Sorry, I don't understand what you mean ( from Denmark ). Do you have a figure showing what is meant?

No, the law is valid in every setup.

No, it is not. This figure is wrong ( found on google ):

The field lines are absolutely symmetrical and closed, though the coil is not. It forms a spiral, not a series of rings. So the field lines within the solenoid should be skew/twisted and not form closed loops. Amperes law and Biot-Savart will not agree in the shown figure. But if you approximate the windings by rings, the figure will be correct.

Furthermore the field lines should spiralize back from N to S, because the windings conduct a current from S to N like a straight wire. So the combination of ring currents and a straight current will form a spiralized field line.

In my calculations a winding is split into 360 parts ( 1 deg. ). Then spiralized and skew field lines will appear, but still the field lines will be closed if an external frame ( like in #6 ) is added to the electric circuit.

I can recalculate the above figure if you wish ( will take some hours ).

 

[mfb]

Images of the plasma in a tokamak: http://origin.arstechnica.com/journals/science.media/Tokamak_fields_lg.png, http://images.iop.org/objects/phw/world/19/3/7/PWfus3_03-06.jpg, here is a more complex one.

The twist can take arbitrary values, so the field lines do not have to form closed loops in the usual sense, in the same way as y=pi x does not have integer solutions apart from y=x=0.

No, it is not.

If you think Maxwell's laws are wrong, please provide a reference for that.

This figure is wrong ( found on google ):

It is a good approximation to show the idea of a coil.

 

[Hesch]

http://origin.arstechnica.com/journals/science.media/Tokamak_fields_lg.png, http://images.iop.org/objects/phw/world/19/3/7/PWfus3_03-06.jpg, here is a more complex one.

1: I agree in that ( upper figure ), because external connections from surroundings to every single coil are indicated. The path of these external wires is not relevant, but they must be there. Magnetic field lines seem to be closed. But I don't quite understand what is meant by transient field, thus why it becomes skew. Maybe due to some time delay?
2. ?
3. I'm bothered about that, because the center axis of the magnetic field and the center axis of the single turns of the coil don't seem to be in parallel. Well, it could be due to the perspective in the image, but anyway: It's a good approximation to show the idea.

If you think Maxwell's laws are wrong, please provide a reference for that.

I have not claimed that. I just say that it must be used in a correct context, amongst a functional electric circuit. The circuit in #12 is not functional.

 

[mfb]

Magnetic field lines seem to be closed.

The upper field plus the middle field combine to form the lower field, which (in general) does not have closed field lines.

It is transient in the sense that it cannot kept up forever in a tokamak - but minutes are fine, so the system is quasi-static, and there are no time delays. Replace the fusion plasma by a regular conducting loop and you get the same field geometry as static system.

2. is the same field as 1.

3. does not show the magnets at all.

 

[Hesch]

Replace the fusion plasma by a regular conducting loop and you get the same field geometry as static system.

I will do that ( some day ).
But how is this plasma current induced? Is it this "Relatively" constant current through the magnets ( dc-current added some ac-current, making up a number of pole pairs like in an induction motor ) ?
I'm not a fusion reactor expert
I wonder if eg. there are say 17 pole pairs, a magnetic field line will close itself after 17 turns in the toroid.
In the figure in #6, the solenoid has 40 "turns", but if the magnetic field line is chosen as a circulation path, using Amperes law, the simulation says that

∫_circulation H⋅ds / I = 55.012 turns ≈ 40 + 15 turns

( including the magnetic loops around the frame ).
So maybe it's not a must that the field line will be closed after one loop, but after a number of loops, N, that the system "knows about".

 

[mfb]

But how is this plasma current induced?

Via induction with magnets around the plasma. I don't see how that would be relevant here.

There are no "pole pairs" and the induced current does not have any steps.

So maybe it's not a must that the field line will be closed after one loop, but after a number of loops, N, that the system "knows about".

No.

 

[Hesch]

Via induction with magnets around the plasma. I don't see how that would be relevant here.

You cannot induce voltage/current with magnets that are supplied by a constant current ( dc-current ).
I find that very relevant.

No.

No, what?
A field line cannot circulate in a toroid with the speed of light for some transient minutes, without crossing or at least touching itself. There is simply not space for this 36 mio. km field line within the toroid.
And when crossing/touching itself:

Well, then the loop is closed.

 

[mfb]

You cannot induce voltage/current with magnets that are supplied by a constant current ( dc-current ).

Who said they would have constant current? Their current ramping rate is constant.

No, what?

No as in: your statement is wrong. There is no requirement for field lines to close.

A field line cannot circulate in a toroid with the speed of light for some transient minutes

Field lines do not "circulate" at all.

There is simply not space for this 36 mio. km field line within the toroid.

Of course there is. Field lines do not have a width, and this is a purely mathematical problem anyway.