Lorentz force question

Dick

Forces between charges in the same wire only create turbulence in wire they are in. Like, uh, eddy currents or something like that. That turbulence is uh, RESISTANCE, yeah, that's it. resistance. The charges are supposed to be confined to each wire. The idealization here is that we have a uniform AVERAGE current flow in both wires. Only the average counts in the long run. Variations are supposed to average out.

berkeman

I was thinking along those lines, but I need to do some calcs to convince myself of this. Thanks, this has been a very enlightening thread for me.

Dick

A privilege, berkeman.

EvilTesla

INTERESTING!!!

I am having a TAD bit more difficult time understanding this.

So, the force that causes two wires to attract (with equal currents)

Is NOT the lorentz between the electrons of one wire on the electons on the other, but is the lorentz force between the electrons in one wire, and the stationary positive charges in the other?


If I am following everything.

WHAT if you have a plasma, that is spinning. (with a differnt radius)

Since the charges are traviling with the same velocity, maby the lorentz force will have no effect?

BUT since spinning is accelerating, they are not at the same velocity, so what effect does this have??

Would this produce a magnitic field similar to a solenoid (stationary magnetic field) and the velocity concerning the lorentz force is compared to the center of the mass of all the plasma?

Born2bwire

Close, but that is due to how we choose the reference frame. Note that berkeman has chosen the frame moving with the electrons and I think he has opposing currents. Let's look at it in some frames if the currents are running in the same direction.

As I understand it, the movement of the wire is from the attraction of the currents in the wire on the positive lattice of the wire's atoms (and vice-versa, yay Newton). A wire is a quasi-neutral material. Well, neutral for most purposes but if you start adding static electric fields you could cause charge separation and stuff. But for the purposes here, the wires are neutral so there is no direct Coulombic force between the wires. However, the moving charges of the currents induce magnetic fields which will have a net Lorentz force on the moving charges in the other wire. This force displaces the currents, the electrons, which then drag the previously stationary ionic lattice of the wire with them. This is looking at the situation from the frame of the atoms in the wire's lattice.

So if the currents are running in the same direction and we look at it from the frame of the electrons, then only the ions are moving which give rise to magnetic fields that interact. It would be different if we had the currents running in opposite directions. Here, one wire would have stationary electrons and moving ions and the other would have moving electrons and ions. Here it would be more complicated.

If we were to look at it in the frame of the moving electrons, then the electrons would be stationary and the ions and atoms of the wire would be moving. In this case it would be magnetic fields from the wire's ions interacting with the other wire's ions (and then the wire's ions dragging the "stationary" currents with it). This would be true for current running in the same and opposite directions.

But, special relativity allows us to look at this problem in a reference frame that is moving along the length of the wire differently too. In this frame there could be no Lorentz force from the magnetic fields, but we now have electric fields which will give rise to the same force that we would find in the other frames. This comes about by the length contraction, the contraction of the currents along a preferential direction makes the wire appear to be electrically charged. That is, if we are moving along the wire, the electrons move in one direction and the positive ions move in another direction (these are the atoms of the wire which now move since we are moving along the wire). Because the velocity addition in the Lorentz transformation is different for velocities moving with the frame compared to those moving against the frame, we have a different contraction for the electrons than the ions. This results in different charge densities over a given length between the electrons and ions and thus a net electrical charge which gives rise to a Lorentz force from an electric field.

berkeman

Why would there be electric field forces? The two wires are electrically neutral in all reference frames.


EDIT -- besides, the electron drift velocities in wires are sub-walking speed, nowhere near relativistic speeds.

Born2bwire

This is done as an example problem in Griffiths, 12.3.1.

He sets up the problem as follows, let us say that we are looking at a unit charge q that is moving with velocity u in the direction +x. At some distance away from from the charge is a wire that has electrons running in the -x direction and ions running in the +x direction both with speeds of v. This would correspond to the situation where we have a current running in the +x direction of 2*v*\lambda (\lambda is the charge densities of the ions and electrons) and a current element at some distance away running in the +x direction as well. If we are in the current element's frame of reference, we both see moving electrons and moving ions in the wire but in moving to this frame, we have to do Lorentz transformations. This is done first by adding the speeds that the ions and electrons move in the wire. Since they move in opposite directions, their transformed speeds are different. This means that the electrons will have a different length contraction than the ions, so the electrons per unit length is now unequal to the ions, and thus we now have a net electrical charge. Since the current element is at rest in this reference frame, there is no Lorentz force on it from the magnetic fields from the wire, the Lorentz force is due solely to the electric fields of the wire. This turns out to be the exact same force that we would predict for parallel wires if we extend the analysis by changing the current element to a line current.

It does not matter whether or not the charges move at relativistic speeds, this result will come out the same. I just wanted to point out that the forces and mechanisms involved in this problem can depend greatly on your reference frame.

EDIT: Griffiths also notes that a similar treatment can be found in Purcell.

Bob S

From Bob S
I believe that if you have two extremely relativistic electron beams side by side, there is no net force, because the attractive Lorentz force is exactly cancelled by the repulsive Coulomb force. If the beams are less than relativistic, the repulsive Coulomb force is dominant. This one of the problems in trying to combine low-velocity heavy-ion beams.

I attach a calculation showing that for two parallel equal-charge particle beams, the repulsive electric (Coulomb) force and attractive magnetic (Lorentz) force cancel as the particle velocity v=βc approaches c.
Bob S

Attached Thumbnails

   

EvilTesla

How would this all change if it were in a spinning plasma?

Bob S

Plasmas are a much more difficult problem, because there are particles of different (and opposite) charges moving in different directions under the influence of internal (self-generated) and often also externally-applied magnetic fields.
Bob S

Drill

so what else can V be relitive too?

fast and draft answer

maybe relative to the viewer and the measuring tools :)

EvilTesla

Wierd.

I think I am going to have difficulty with this one.

Dick

Thanks for putting that together, Bob S. Having the calculation to look at helps me to interpret what you are saying better. The only 'extremely relativistic' case where the forces actually cancel is REALLY EXTREME. When beta=1 and the speed of the beams is actually c. So in any physical case where beta<1 then the Coulomb force is actually dominant. So it doesn't contradict the picture in the rest frame of the beams where there is only a Coulomb force and the beams diverge. In ANY frame, the beams diverge but more slowly as the velocity increases. In the lab frame you attribute this to a growing magnetic force between the beams. But you can also interpret this in the lab frame by computing it in the beam rest frame and then saying the divergence is slowing because of time dilation. Agree with that?

Phrak

The electron mass also goes to infinity, Dick.

Dick

Sure, Phrak. I'm just saying nothing here says you can't do the analysis in the beam rest frame and then translate to the lab frame. I had originally thought Bob S was saying in the 'highly relativistic' case the magnetic force would exceed or balance the Coulomb force. I don't think that's what being said.

Phrak

You know. This is something I asked in my freshman year. I was very disappointed to hear that it was due to both an increase in mass and a decrease in force. It seemed to messy. Acceleration of the beams (divergence) goes to zero in two ways.

Unless I'm missing some subte point, I think I know what you're asking. Not only are mass, length and time Lorentz tranformed, but force also transforms; the Faraday tensor is also transformed.

Dick

Yes, force is a messy concept in SR. And yes, everything else transforms as well. That's why it pays to find the right frame to express your question and then transform to another frame if you have to. You really can draw correct conclusions by just working in the beam frame. I'm not really asking anything. I'm just saying that the conclusion that opposite charges in the rest frame of the charges will repel is true, no matter how much you try to complicate it by frame change.