The missing torque

[LuxFerum]

Almost everyone knows about the magnetic field created around a wire by the flow of charges in it.
There is an simpe experiment that shows the change of direction of a compass when it is close to an wire with current in it.

With that in mind, see this problem.

There is an infinite wire, with a current i, and there is a magnet close to it.

By the frame of reference 1, that is not moving in reference to the wire and to the magnet. The current i will produce a magneic field and induce a torque on the magnet.

By the frame of reference 2, that is moving in reference to the wire, at the same speed of his charges. There is no current in the wire for this frame, and therefore no torque in the magnet.

well, what the hell is going on here?

 

[ceptimus]

I don't see how your frames of reference make a difference.

If you are still relative to the magnet, then you see the electrons moving in the wire. If you ride along with the electrons, then you see the magnet moving.

Either way there is relative motion and therefore a torque induced.

If the magnet rides along at the same speed as the electrons, then there is no relative motion and no torque.

Am I understanding the problem? Please restate it if I am missing the point.

 

[LuxFerum]

I heard this problem when someone was discussing relativity.
And how the laws of physics must be the same for all reference frames.
And in this problem the laws of physics seems to only work for the frame of the magnet.
I don't remember much more of the discussion, but I guess that the solution envolves some relativistic concept.

 

[wipeout]

I think the key to understanding it is to realize that, to us, the magnet in the picture is actually moving in frame of reference 2.

It's moving to our right.

That the picture doesn't show this movement is what is misleading.

I'd add an arrow next to the magnet, which would point right and say "Moving in frame of reference 2".

 

[Hamish]

I heard this problem when someone was discussing relativity.
And how the laws of physics must be the same for all reference frames.
And in this problem the laws of physics seems to only work for the frame of the magnet.
I don't remember much more of the discussion, but I guess that the solution envolves some relativistic concept.

Yes, I vaguely remember this one from my undergradute EM courses.

The laws of physics don't change between inertial frames. A current carring wire in the same inertial frame as the magnet will always produce a torque. Bu observed quantities such as length, time and magnetic and electric fields will not be the same in two inertial frames and are transformed. It is possible to show (only please don't ask me how) that magnetism is actually the relativistic component of a moving electric field.

I could probably go and look up the mathematics. You can actually do relativistic EM quite easily if you have a grasp of tensors.

 

[RCNelson]

In the first frame of reference, the protons in the wire are standing still and the electrons in the wire are moving to the left.

In the second frame of reference, the electrons in the wire are standing still and the protons in the wire are moving to the right.

In both frames of reference the flow of charge is the same.

 

[Ziggurat]

In both frames of reference the flow of charge is the same.

No, the flow of charge in the two frames isn't the same. In fact, if you pick your reference frames right, in the moving frame there's no current at all, and hence no magnetic field at all. So the question is how do you restore the torque in the moving reference frame, where there is no magnetic field, since torque SHOULD be conserved between reference frames.

Half of the answer comes from the fact that in the moving reference frame, although there may be no NET current, the two charge components ARE moving with respect to each other. In the frame where the wire is stationary, these charge lines are balances, and we have net current but no net charge. Length contraction when changing reference frames means that in the moving reference frame, the charge lines no longer cancel. There is no net current, but there IS a net charge in the line.

The second half is somewhat related, and comes from the magnet itself. You can think of the magnet as being caused by circulating surface currents. In the moving reference frame, the magnet is no longer stationary. The surface currents experience length contraction as well, but now on one side of the magnet, they're moving faster compared to the reference frame, and on the other side they're moving slower, so the length contraction isn't the same for the two sides, and the charge density is therefore also different. In the moving reference frame, then, there's actually a charge imbalance between the two sides of the magnet (top and bottom in the picture). In the electric field caused by the line charge of the wire, this charge imbalance creates a torque on the magnet.

So basically, the torque IS conserved when changing reference frames, but the "cause" of the torque changes from the electric to the magnetic field. You can actually right the electro-magnetic field as a single antisymmetric 4x4 matrix (which has 6 free parameters, corresponding to the 3 components of the electric and magnetic fields), and the charge/current density as a single 4-component vector (3 for current, 1 for charge, analogous to 4-component relativistic momentum).
You can transform these directly in relativity the same way you transform your space-time coordinates to find out exactly how the electric and magnetic fields transform under changes of reference frame, and all invariant quantities (such as force) are guaranteed to be consistent between reference frames.

 

[epepke]

It is possible to show (only please don't ask me how) that magnetism is actually the relativistic component of a moving electric field.

This is actually pretty easy. Consider two parallel wires A and B with the same amount of current going in the same direction. Make them circular if you don't like infinite lengths. The protons in A will "see" about the same number of protons in B as if there were no current. However, since the electrons in B are going by quickly, due to Lorenz contraction, the distances between the electrons will appear shorter, so the protons in A will "see" more nearby electrons. Thus, the wires will be attracted magnetically to each other. It's the same for the protons in B and the electrons in both wires.

 

[Ziggurat]

This problem reminds me a bit of another class of E&M puzzles which look like they violate conservation of momentum because of "hidden" momentum. Here's a simple variant:

So everyone knows that photons can carry momentum. But static electric and magnetic fields can also carry momentum (but you need both, and only the components perpendicular to each other carry momentum). So imagine a coaxial cable (a conducting wire surrounded by a conducting cylinder). If we send current down the central wire and back through the cylinder, we create a magnetic field between the two. If we now apply a voltage difference between the two, there's an electric field there as well. The electromagnetic field between the wire and the cylinder now has momentum pointing in the direction of the cable. But of course, we can make this a static situation, so there should be no net momentum. Where's the missing momentum?

Solution: the momentum is "hidden" in the fact that the momentum of electrons carrying the current isn't linear with velocity in special relativity. Classically, the electrons flowing down the wire and back up the cylinder should have the same momentum: since momentum is linear with velocity, it's given directly by the current (for a fixed total current, we can flow twice as many electrons at half the speed, or vice versa, without changing momentum), which is the same for both the wire and the cylinder. But in special relativity, momentum isn't linear with speed, so if you flow half as many electrons at twice the speed, you have the same current but a different momentum. Now the tricky part is that since there's a voltage difference between the wire and the cylinder, there's an energy difference between electrons on the inside and the outside, and the electrons are actually traveling faster in one than in the other. So even though the currents balance out, because of the potential difference the momentum of electrons in the two wires don't match. And that mismatch cancels the momentum of the electromagnetic field inside the cable.

 

[RCNelson]

Ziggurat:
No, the flow of charge in the two frames isn't the same. In fact, if you pick your reference frames right, in the moving frame there's no current at all, and hence no magnetic field at all.

I will attempt to explain it more clearly:

This figure shows a section of wire as viewed from three frames of reference.
The circles with a minus sign inside are electrons.
The circles with a plus sign inside are protons.

In all cases there is the same amount of current. Since current is defined as "the amount of electric charge flowing past a specified circuit point per unit time", it doesn't matter if the current takes the form of electrons moving left or protons moving right.
Negatively charged particles moving in a negative direction make the same current as positively charged particles moving in a positive direction.

Frame of reference A is stationary relative to the protons.
Frame of reference B is moving at half the velocity of the electrons in the same direction as the electrons.
Frame of reference C is moving at the full velocity of the electrons in the same direction as the electrons.

In reference frame C we see the positively charged particles moving to the right creating the magnetic flux as indicated by the arrow.

In reference frame A we see the negatively charged particles moving to the left creating the same magnetic flux as indicated by the arrow. Notice that the flux lines are in the same direction in both reference frames A and C. This is because, though the particles are moving on the opposite direction, they are also negatively charged.

In reference frame B we see positively charged particles moving right and negatively charged particles moving left. The speed of these particles is half what they were in reference frames A and C, so the flux lines due to the protons are half what they were in reference frame C, and the flux lines due to the electrons are half what they were in reference frame A. The cumulative effect of these moving electrons and protons add together to give the same net flux as in reference frames A and C.

 

[Ziggurat]

I will attempt to explain it more clearly:

Yes, I did manage to confuse myself about the existence of the magnetic field. However, the basic problem still exists that the magnetic field is not the same for the three reference frames. Because of length contraction, when you shift reference frames the charge density changes differently for the positive and negative charges, so you are incorrect that the current is equal in all cases, and what I described (the charge density on both the wire and the magnet changing with reference frame) is still a critical part of the consistency between the two pictures.

I think what I was thinking about is the related and slightly more clear example of a single charge line (say, only electrons, no protons) moving past a magnet, so that you get a current, magnetic field, and torque as described, but if you change to the reference frame of the moving charge line there really is no magnetic field.

 

[RCNelson]

Ziggurat:
However, the basic problem still exists that the magnetic field is not the same for the three reference frames. Because of length contraction, when you shift reference frames the charge density changes differently for the positive and negative charges, so you are incorrect that the current is equal in all cases, and what I described (the charge density on both the wire and the magnet changing with reference frame) is still a critical part of the consistency between the two pictures.

Given the velocity of the electrons due to current flow is typically less than 1 cm per second, would there be enough length contraction to matter?

 

[Ziggurat]

Given the velocity of the electrons due to current flow is typically less than 1 cm per second, would there be enough length contraction to matter?

I guess it depends what you mean by "matter". The resulting electric field would probably be negligible compared to other factors involved in a real situation, but in terms of providing consistency (physics should be exactly the same, not almost the same, in every inertial reference frame), it's still absolutely critical.

 

[69dodge]

Originally posted by RCNelson
Given the velocity of the electrons due to current flow is typically less than 1 cm per second, would there be enough length contraction to matter?

I'm not sure exactly what situation you're talking about, and I probably couldn't do the calculations even if I were, but I'll still venture a guess and say, yes, I think it matters. The electric field is amazingly strong; it's easy to lose sight of this fact because everyday objects almost always have nearly exactly the same amount of positive and negative charge, which cancel each other out almost perfectly. Even a small fractional difference results in a large force.