I have a simulation where charged particles move in a plane (let's say, in the xy-plane). I calculate the forces between them with Coulomb's law, which in fact isn't rigth as the charges are moving. I'd like to correct my simulation by adding magnetic interaction. So I got an idea of calculating the magnetic moments of the moving particles. When I know the magnetic moments I can calculate the z-component of the B-field in an arbitrary point of my simulation plane. Now I now E and B so I can use Lorenz force to simulate the movement of the particles. Only problem is the calculation of the magnetic moment. Maybe someone with more experience could derive it from Maxwell's equations?

Instead of the coulomb law use the Liénard-Wiechert potentials for the moving charge. The derivation of the potentials can be found in many books.

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I have a simulation where charged particles move in a plane (let's say, in the xy-plane). I calculate the forces between them with Coulomb's law, which in fact isn't rigth as the charges are moving. I'd like to correct my simulation by adding magnetic interaction.

So I got an idea of calculating the magnetic moments of the moving particles. When I know the magnetic moments I can calculate the z-component of the B-field in an arbitrary point of my simulation plane.

Very interesting. I hate to tell you, but you will not be able to model anything beyond this point with classical physics. After you model magnetic fields, which I will show you how later, you will be left with spin and magnetic dipole moment, and this is where things start to slip out of determinism and practical computability in a truly peculiar way.


This is huge and very interesting topic, you are basically talking about unification of classical and quantum physics. You will not find anything about this on the internet. Nobody is doing quantum interaction with classical physics, you know? It's just you and me, my friend.

Yes, it seems science has come to conclusion elementary particles cease to obey laws of classical physics when they interact on some very small scales. It seems paradoxical for classical physics to allow for Pauli principle or any other kind of electron coupling. After all, electrons should be repelling each other, right?


Wrong.

Now, this all inexorably boils down to one thing, and that is the size of an electron, which further has implications to infinite divisibility of time and space, i.e. continuity and simultaneity - particles vs waves, and to finish it of, it has to do with Aether or some absolute reference frame too.

What's worse, I'm serious. And if you want to make it work, as in real-world, then you have no choice but to agree with me and Lorentz... forget about Einstein and get ready to be called a lunatic, ok?

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Now I now E and B so I can use Lorenz force to simulate the movement of the particles.

Only problem is the calculation of the magnetic moment. Maybe someone with more experience could derive it from Maxwell's equations?

Ready?

First, you are doing everything right so far. You do not need to change anything with your Coulomb's law, leave it there.

Second, forget Maxwell equations, they have everything but what you need now, which is - magnetic field of a moving charge aka 'Lorentz force', plus Biot–Savart law to get all the variables in there. Something like this:

F= q(v x B)
B= v x q*k*d/r^2

http://en.wikipedia.org/wiki/Biot-savart
http://en.wikipedia.org/wiki/Lorentz_force


Be careful of what is vector and what is scalar, scale your constants or throw them away, scale all your values if necessary to avoid floating point overflows.


That's it! Now depending if your simulation is 2D or 3D you will need to pay attention how you integrate your squared-distance and your vectors. Also, pay attention that once you have implemented this you will in fact prove the existence of an Aether, which is not such a horrible thing given the insight into how and why two electrons get to attract each other after all.

And don't forget, this is only 1/3 of the whole story. Good luck.

For point charges the Biot-Savart law is only an approximation while the Liénard-Wiechert potentials are relativistically correct.

Quote:Original post by Kambiz
For point charges the Biot-Savart law is only an approximation while the Liénard-Wiechert potentials are relativistically correct.

How did you conclude Biot-Savart law is "only an approximation"?

It's pretty exact if you look at the equation, is it not? It is only approximation as much as we are limited by serial computation of discrete time intervals. Same as n-body integration problem, approximation is only due to chaotic property of the system and our inability to integrate it over time without progressive error - initial condition problem... only worse, having a magnetic spin and dipole magnetic moment, but basically boils down to simultaneity problem and the size of the charge (zero).

Nevertheless, and in any case, that equation will still produce result in accordance with the real-world experiments, not approximate, but very exact and precise. Can you verify this, do you agree?


As for Liénard-Wiechert... I do not think anyone ever implemented that, for anything, ever. Hmm? Certainly not for this type of thing. You see, there is a fatal flaw there - it was derived from Maxwell's equations and they do not include Lorentz force.

Lorentz force is the key here, and apparently science has forgotten about it in many ways. They teach you in school two electrons repel each other due to electric fields, but they do not teach you two electrons also attract each other due to magnetic fields and Lorentz force, do they? If they did, they would need to tell they attract more the faster they are going. Yes, there is actually a point where magnetic attraction will overcome electric repulsion. But the problem is this velocity is not relative to anything... anything we know about that is. Look at the formula, that's how the real-world works.

So, if you trust Lorentz you can actually model this interaction and reproduce what we measure in practical experiments. No relativistic equation exists that includes Lorentz force, relativism simply can not handle absolute reference frame... and I'm laughing at it.

The Lorentz force is the connects between classical mechanics and electro magnetism, the question here is what fields we are required to use, the fields are always given by Maxwell equations. The fields used in Coulomb law and the Biot-Savart law are correct for static charges and steady currents but NOT for moving point charge. (Even on the Wikipedia link you posted one can read the Biot-Savard is only an approximation for moving point charges)

Notice that both Coulomb and Biot-Savart laws depend on the current position and velocity of the charge instead of the configuration at retarded time, since the Maxwell equations are Lorentz invariant both laws are inconsistent with the Maxwell equations in non static case.

The Liénard-Wiechert potentials on the other hand are relativistically correct. In this case this Potentials should be used to get the fields produced by charge A and using this fields one can get the Lorentz force acting on charge B.


EDIT: To make it clear, the electric field in the Coulomb law, Biot-Savart law and Liénard-Wiechert potentials all can be derived from Maxwell equations. But the first two are simplifications for static configurations.
The Lorentz force is an independent equation that is used to calculate the force.

[Edited by - Kambiz on October 17, 2009 10:05:19 AM]

Uh, huh.

Look, I actually did all this. I simulated it and I tested it against the real-world and it works as I have told you. This experience tells me you have not, have you? Not even the very thing you are suggesting, because that does not work, you know?

You either need to implement what I explained above and see that it works, or you can attempt to implement what you are babbling about and fail miserably. In either case it will save us both some time and unnecessary arguments. And, there is no velocity in Coulomb's law... aaaaaaa!!!!

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The Liénard-Wiechert potentials on the other hand are relativistically correct. In this case this Potentials should be used to get the fields produced by charge A and using this fields one can get the Lorentz force acting on charge B.

Stop it. How did you manage to convince yourself it's appropriate for you to be giving any advice about this subject? Relativistically correct? I'm talking about the real-world experiments, not some theories and assumptions.

Now, wake up and show me your equation:

Lorentz force = ?


Show me the money!

Quote:Original post by Victor-Victor
Stop it. How did you manage to convince yourself it's appropriate for you to be giving any advice about this subject?

I'm member of the institute of theoretical physics in Heidelberg (See yourself)
What about you?

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Relativistically correct? I'm talking about the real-world experiments, not some theories and assumptions.

The real world is Lorentz invariant, there is no doubt.

Quote:Now, wake up and show me your equation:
Lorentz force = ?

E and B are already on Wikipedia, put them in F=q(E+vxB) and you get the correct Lorentz force.

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I'm member of the institute of theoretical physics in Heidelberg (See yourself)
What about you?

I'm the one who actually simulated all this, the one who is laughing. So, you better call your friends here if you are to learn something practical about the world. -- Since you have not made any comment I'll take it you agree two electrons actually attract each other, and more so the faster they are moving yes? Therefore, two parallel electron beams should attract according to Lorentz force, what does _your theory (or practice) say about this? Agree?

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E and B are already on Wikipedia, put them in F=q(E+vxB) and you get the correct Lorentz force.

Why don't you write the equation down, what's the problem? Did you not say you are going to derive some relativistic equation for all this? You are not supposed to use non-relativistic Lorentz force equation, that's ridiculous, what is your point then?


-- Liénard–Wiechert potential --

V(r,t)= 1/4*PI*e0 * (qc/Rc-R.v(T))

A(r,t)= v(T)/c^2* V

E= ??

B= ??

F=q(E+vxB)

How do you imagine this can be integrated into actual kinematic simulation and dynamical n-body system? How do you express these variables and how do you integrate it over time? And.... then we plug it all in F=q(E+vxB), you say? Do you see that velocity in Lorentz force equation, what is that relative to?

Quote:Original post by Victor-Victor
Therefore, two parallel electron beams should attract according to Lorentz force, what does _your theory (or practice) say about this? Agree?

Not quite sure what you are talking about here...

If we have two parallel electron beams, each containing electrons moving at speed v in the positive x-direction (say), then there will be a repulsive force between them. To see this, simply transform to a frame moving with speed v in the x-direction. In this frame the electrons are stationary, the magnetic field is zero, and we are left with a pure electrostatic force, which causes the beams to repel.

Back in the original frame (in which the electrons are moving) we would see that there is both a magnetic and an electric field. But the magnetic field would be weak compared to the electric field, so it would be a small correction to the Coulomb force -- it would not be strong enough to turn a repulsion into an attraction.