Relative difference in laws of electrodynamics

[pervect]

I think I have a resolution, more or less, though it's not terribly well documented at this point.

We can formally use the above vector potential to show that the integral of the E-field around any closed curve in which the line charge is at rest is zero. This is in accord with the induction law, that the integral of E around any curve is equal to the rate of change of magnetic flux.

I've been lazy and haven't gone through the math in detail on the case where there is a line current (rather than a line charge) but I'm pretty confident that one will find that around a closed loop, there is a magnetic flux, and if you move the loop correctly you can get it to change. I'm confident that Maxwell's equations will work and give that the integral around the curve of E is nonzero and is equal to the rate of change of the magnetic flux, which is also nonzero.

[add]I've also been lazy and assumed that the "loop of wire" has a high enough resistance that it doesn't affect the field configuration, rather than a zero resistance, which would mean placing the charges on the wire correctly to make the electric potential constant throughout the wire. Mainly because it's so much easier to do, and we can tell which way the current "wants to flow" if there is a high resistance, we don't need to get into that level of detail.Where does this leave us on a practical level? Formally, we are saying that due to the relativity of simultaneity, we are integrating around a different curve. But what about the questio about whether or not the light lights up? I submit that a good part of the answer is that while Maxwell's equations are frame independent, the constituitive equations, relating the material properties, ARE frame dependent.

I found one reference on this point which is, alas, behind a paywall. There's probablly more.

http://www.springerlink.com/content/n726472450j38731/

"On the frame dependence of electric current and heat flux in a metal
Zur Bezugssystem-Abhängigkeit des elektrischen und des Wärmestroms in einem Metall"

The other part of the issue I think is the issue of proving what's called the "lumped circut approximation". One of my old E&M books had a formal proof, but I don't recall where it wandered off to. At any rate, the point is that in the approximation, the current anywhere around a loop will be constant, if and only if you can ignore the capacitive effects that allow charge to accumulate at "nodes".

This follows from the conservation of charge.

IF we had a wire with a current rather than a line charge, I'd feel comfortable in ignoring the capacitive effects. With the really high electric fields that an un-neutralized line charge will have, I'm not sure if the capacitive effects can be ignored. I'm not terribly interested in modelling them (the capacitive effects) in detail, but it doesn't see like a "big mystery" to me, just a lot of work.

So my basic resolution is that while Maxwell's equations are frame independent, the behavior of the wire isn't. To use familiar analysis techniques, its important that we analyze the problem in the frame of the wire - or at least a frame where the wire is moving very slowly compared to the speed of light.

 

[PAllen]

A few obvious conceptual points:

- Current must be frame dependent. A line of charge in one frame will be a current of some type in frame moving relative to the line of charge.

- The LED will go on, if, and only if, it experiences current in its rest frame. This implies that a complete description of a moving LED and currents will incorporate the possibility of a moving LED not responding to current (both as determined in some frame where the LED is moving).

 

[ApplePion]

##################$# Ah, I didn't understand the problem. I don't see an obvious resolution yet. For what it's worth, the potential of a stationary line charge should just be something like phi = -ln(r). In coordinates more adapted to the problem, it'd be something more like −lnx 2 +z 2 − − − − − − √ . The 4-potential is just (phi,0,0,0) Boost as needed to get the 4- potential for the line charge, and differentiate as needed to get the Faraday tensor F=∂ a A b −∂ b A a .$######################

Everything you wrote is correct, but there is a much simpler way to get there.

Draw an imaginary circle whose plane the line charge is normal to, and which intersects the circle in the very center point. Use the Curl of B Maxwell's Equation in integral form-- the line integral of B is equal to 4 pi times the flux of the current. That gives 2 pi r B = 4 pi I. You get B immediately.

Furthermore, you derivation of the electric field from the potential is a circular derivation. The most reasonable way to derive the potential you used is to find the electric field via Gauss' Law--it is a very easy Gauss' Law application, and then to integrate to get the potential. So essentially, to get the electric field you are finding the electric field via Gauss' Law, integrating it to get the potential, and the taking a derivative to get back the electric field which you had all along.

 

[ApplePion]

"We can formally use the above vector potential to show that the integral of the E-field around any closed curve in which the line charge is at rest is zero. "

Right, so in the frame where the charge is at rest there is no integral of the E field asround the loop. But what about in the frame where the charge is not at rest?"So my basic resolution is that while Maxwell's equations are frame independent, the behavior of the wire isn't"

The Principle of Relativity would require that if the light bulb lights up in one frame then in any other frame it would. This can be made more clear it instead of there being a light bulb in the circuit there was an ignition device for dynamite which would blow up the building. Clearly you cannot have the bulding blown up in one frame, but then make a coordinate transformation such that the building was fine.

 

[ApplePion]

"The LED will go on, if, and only if, it experiences current in its rest frame. This implies that a complete description of a moving LED and currents will incorporate the possibility of a moving LED not responding to current (both as determined in some frame where the LED is moving)."

Whether the current will light up the bulb depends on (put crudely) whether the electrons are crashing into the bulb. So it depends on whether the bulb and the electrons have a *relative* velocity. If the light bulb has a *relative* velocity in one frame then they will have a relative velocity in another frame. So your attempted resolution is not physically correct.

And if it was not contradicted by the argument I made in the above paragraph, we would have a very strange situation. Your argument would imply that a light bulb hooked up to a battery would not light up the battery, being that in some frame there was no current. Your argument, attempted to resolve a difficult situation, would plunge acceptable situations into trouble.

I also note that with a normal wire you really can't make the current disappear by a coordinate transformation, anyway. Suppose that electrons in a wire in the original coordinate system was traveling North. You then make a coordinate transformationn so that in the new frame the electrons are at rest. Is the current in the wire in the new frame now zero? Well, not quite. You, in the new frame, have protons moving South! So you still have current.

 

[PAllen]

"The LED will go on, if, and only if, it experiences current in its rest frame. This implies that a complete description of a moving LED and currents will incorporate the possibility of a moving LED not responding to current (both as determined in some frame where the LED is moving)."

Whether the current will light up the bulb depends on (put crudely) whether the electrons are crashing into the bulb. So it depends on whether the bulb and the electrons have a *relative* velocity. If the light bulb has a *relative* velocity in one frame then they will have a relative velocity in another frame. So your attempted resolution is not physically correct.

Everything you write before the last line in above paragraph is consistent with what I said, and I would take as a restatement of the same thing in different terms. Therefore, your last line is irrelevant.

And if it was not contradicted by the argument I made in the above paragraph, we would have a very strange situation. Your argument would imply that a light bulb hooked up to a battery would not light up the battery, being that in some frame there was no current. Your argument, attempted to resolve a difficult situation, would plunge acceptable situations into trouble.

Sorry, this is the opposite of what I said: If the LED experiences current in its rest frame, then it will light up in all frames, but the description may be more complex.

I also note that with a normal wire you really can't make the current disappear by a coordinate transformation, anyway. Suppose that electrons in a wire in the original coordinate system was traveling North. You then make a coordinate transformationn so that in the new frame the electrons are at rest. Is the current in the wire in the new frame now zero? Well, not quite. You, in the new frame, have protons moving South! So you still have current.

This is generally true. However, for a case like the point charge in the OP, you have a transient sloshing of charge distribution. My observation is that if you determine this [looks/doesn't look] like a current in the rest frame of the LED, then that is all you need to predict the actual result.

 

[ApplePion]

"Sorry, this is the opposite of what I said: If the LED experiences current in its rest frame, then it will light up in all frames"

In the thought experiment it does not light up in the rest frame but *does* light up in the moving frame--that is what makes it a paradox that needs to be resolved.

" but the description may be more complex."

OK, then please explain.

"This is generally true. However, for a case like the point charge in the OP, you have a transient sloshing of charge distribution."

I don't know what "OP" means, but the way the problem is set up, in the frame where the loop is moving the bulb is "shown" to light up, and in the frame where it is not moving it is "shown" not to light up. That is why we have a paradox that needs to be resolved. (In no frame is there sloshing.)

" My observation is that if you determine this [looks/doesn't look] like a current in the rest frame of the LED, then that is all you need to predict the actual result"

Yes, it *should* be that way. We both agree that nature has to work that way. But when one goes through the specific details it appears that we get something neither of us find acceptable.

You are being presented with something that gives a result neither of us believe can really be true, and rather than you resolving the paradox directly you are saying that we have a paradox and therefore something must be wrong, and so the paradox is resolved.

 

[PAllen]

I don't know what "OP" means, but the way the problem is set up, in the frame where the loop is moving the bulb is "shown" to light up, and in the frame where it is not moving it is "shown" not to light up. That is why we have a paradox that needs to be resolved. (In no frame is there sloshing.)

...
You are being presented with something that gives a result neither of us believe can really be true, and rather than you resolving the paradox directly you are saying that we have a paradox and therefore something must be wrong, and so the paradox is resolved.

OP means original post. The original post had a point charge scenario.

I wasn't proposing to resolve the paradox specifics. I don't have the time to try it. My post to which you take offence was described as " a few obvious conceptual points". My only reason for posting it was to possibly provide a hint of where to look for the problem. For example, if you were more convinced of the solidity (e.g. absence of simplifying assumptions) in the analysis in the LED rest frame, then look at issues like motion of the LED as well as the motion of the wire loop.

 

[PAllen]

Let's work it out carefully. Also, it will be easier to deal with if instead of a point charge at the origin we have a line of charge in along the y axis.

1) In the frame where the line of charge is at rest and the loop has a velocity component in the y direction and is also moving in the x direction towards the line of charge, do you not agree that despite the increasing dipole moment at places, a light bulb at x=10 y= 0 will not light up? Just work in that frame to directly see what happens in that frame.

2) How about in the frame where the line of charge is moving along the y-axis and the loop is only moving in the x direction? The v x B force will be greater at x = 10 than at x = 20 because B is greater at x = 10 than at x = 20, so the light bulb will light up then. Do you not agree? So we have an apparent paradox.

I can propose a plausibility argument for the resolution, but do not have time to support it mathematically. I have numbered the key paragraphs in Applepion's initial introduction of this scenario. I believe no one disputes (1). Further, I introduced:

3) How about the frame where the loop is at rest. The line of charge will be approaching with both x and y velocity components. There will obviously be a both electric and magnetic fields. However, if we take the simplifying assumption that charges in the stationary loop are effectively stationary in the absence of EM fields, and also effectively stationary if there is only a dipole charge distributions due to E field, then the B field should have no effect. Thus simplified analysis of (3) agrees with (1).

The key to resolution is then (2). I agree that for a suitably oriented loop, there is a stronger tangential B force on the part of the loop closest to the moving charge line, and a weaker tangential force on the opposite side (again, assuming a simple, classical model of balanced charges in the wire). However, a key point is that both tangential forces point in the same direction. Suppose they were the same strength. Then the effect would just be charge clumping at midway points on the loop. The amount of clumping would be when repulsion within the clump balanced the magnetic forces. So what if the forces are not equally strong? I claim that simply shifts the location of peak clumping, so that all forces balance, and does not produce a circulating current. Then, the electric field will simply further adjust the location of the charge asymmetry. Thus (2) will look just like the other cases - clumping of negative charges at one point in the loop, and positive at an opposite point.

 

[ApplePion]

"However, a key point is that both tangential forces point in the same direction. Suppose they were the same strength. Then the effect would just be charge clumping at midway points on the loop. The amount of clumping would be when repulsion within the clump balanced the magnetic forces. So what if the forces are not equally strong? I claim that simply shifts the location of peak clumping, so that all forces balance, and does not produce a circulating current. "

You are arguing that if there is a net EMF around a loop, it will *not* cause current to flow. I think if such a claim was made by another person under different circumstances you would quickly realize that such a belief is incorrect.

 

[pervect]

Everything you wrote is correct, but there is a much simpler way to get there.

Draw an imaginary circle whose plane the line charge is normal to, and which intersects the circle in the very center point. Use the Curl of B Maxwell's Equation in integral form-- the line integral of B is equal to 4 pi times the flux of the current. That gives 2 pi r B = 4 pi I. You get B immediately.

Furthermore, you derivation of the electric field from the potential is a circular derivation. The most reasonable way to derive the potential you used is to find the electric field via Gauss' Law--it is a very easy Gauss' Law application, and then to integrate to get the potential. So essentially, to get the electric field you are finding the electric field via Gauss' Law, integrating it to get the potential, and the taking a derivative to get back the electric field which you had all along.

The approach you mentioned, using Gauss law, is how I decided that I'd recalled the 4-potential correctly. I agree it's equivalent , i.e. using the 4-potential isn't any different than the integral approach.

In order to finish off the problem, one wants to find the fields in the frame where the line charge is moving "diagoanally", and the wire loop is stationary. This is where the 4-potential approach comes in handy. I don't see an easier way to do it than boosting the 4-potential.

 

[pervect]

"The LED will go on, if, and only if, it experiences current in its rest frame. This implies that a complete description of a moving LED and currents will incorporate the possibility of a moving LED not responding to current (both as determined in some frame where the LED is moving)."

Whether the current will light up the bulb depends on (put crudely) whether the electrons are crashing into the bulb. So it depends on whether the bulb and the electrons have a *relative* velocity. If the light bulb has a *relative* velocity in one frame then they will have a relative velocity in another frame. So your attempted resolution is not physically correct.

I think Pallen is basically on the same page as I am. The frame dependence here is not in Maxwell's equations. I think the frame dependence appears when we introduce the light bulb. The E and B fields all obey the appropriate mathematical transformation laws, i.e. A transforms as a 4-vector.

 

[PAllen]

"However, a key point is that both tangential forces point in the same direction. Suppose they were the same strength. Then the effect would just be charge clumping at midway points on the loop. The amount of clumping would be when repulsion within the clump balanced the magnetic forces. So what if the forces are not equally strong? I claim that simply shifts the location of peak clumping, so that all forces balance, and does not produce a circulating current. "

You are arguing that if there is a net EMF around a loop, it will *not* cause current to flow. I think if such a claim was made by another person under different circumstances you would quickly realize that such a belief is incorrect.

You are right. I retract this argument.

 

[pervect]

OK, here's my take.

The crude model of a light bulb I'd suggest is just a cloud of - and + charges. If there's an electric field, there will be relative motion between the charges, the positive and negative charges will move in different directions. The detailed physics of a wire is more complex, but if there's an E field pointing in the direction of the wire, we expect a current flowing through it. Well, at least I would, I'm hoping there will be some agreement on this point.

Relative motion can determine if our light bulb lights up or not. If we have only a B-field, and our cloud of + and - charges is stationary, then we would say that the "light bulb" does not light up. If we make our collection of + and - charges move, the + and - charges will separate from each other, the B field will make them curve in opposite directions. In fact with the tensor approach we can say that an E field exists in the moving frame, and not in the stationary frame.

The notion that the faraday tensor transforms as a tensor means that if we know E and B at a point in one frame, we can determine E and B at that point in ANY frame, without recomputing them from the bondary conditions, simply by transforming E and B at that point using the appopriate tensor transformation laws to the new frame.

As far as the idea of integrating the voltage around the loop - that idea comes from lumped circuit theory. Lumped circuit theory is a good approximation in some circumstances, but not in this one.

It seems likely that lumped circuit theory is not Lorentz invariant from this example. It's a bit surprisng, maybe, but then we usually do lumped circuit theory in the frame of the apparatus.

It's also clear that lumped circuit theory won't work for other reasons other than the voltage integration issue. Specifically, lumped circuit theory says that the current through a loop is constant everywhere, because charge that flows into a node must flow out of the node. If the node has a "capacitance", this is no longer true. So lumped cirucuit theory exists only when we can ignore the capacitances. In this example, the capacitance to the line charge is an important part of the problem, significant currents flow through it, so we can't satisfy the condition that "nodes" do not acquire charge needed to make lumped cirucuit theory work, unless we improve the model by having more than just a wire, by adding into the circuit theory model the necessary capacitances (to infinity and to the line charge). We'd also add in resistances and inductances for the wire. As we add in a large number of these elements, we'd recover the field theory result. It seems like we might have to do this all in the frame of the loop, however.

 

[Dale]

It seems likely that lumped circuit theory is not Lorentz invariant from this example. It's a bit surprisng, maybe, but then we usually do lumped circuit theory in the frame of the apparatus.

Sorry I have lost track of this thread while traveling this week. However, I was surprised by this comment and then after thinking about it I can prove it.

There are three fundamental assumptions to lumped circuit theory:
1) No circuit element has a net charge
2) There is no magnetic coupling between circuit elements
3) The speed of light is large compared to the frequencies and size in the circuit

Obviously, 3 is violated under a general Lorentz transform, but perhaps more importantly 1 is also violated. If a circuit element is uncharged but carries a current in one frame then it will carry a current and be charged in all other frames (or rather all other frames boosted along the direction of the current). Therefore, a circuit which satisfies those three assumptions in one frame will not necessarily satisfy them in other frames, making circuit theory a frame variant approximation to Maxwell's equations.

If you are doing a problem involving relativistic effects or Lorentz transforms, then you cannot use circuit analysis techniques, you need to use the full Maxwell's equations. Once you do that, of course, it is clear that everything is consistent.

 

[ApplePion]

"In order to finish off the problem, one wants to find the fields in the frame where the line charge is moving "diagoanally", and the wire loop is stationary"There is no such frame the way the problem is set up.. Here are the two frames:

First frame: Line of charge moving in the positive y direction, and the loop moving in the negative x direction.

Second frame: Line of charge not moving, and the loop of charge having velocity components in the negative y direction and in the negative x direction.

 

[pervect]

"In order to finish off the problem, one wants to find the fields in the frame where the line charge is moving "diagoanally", and the wire loop is stationary"


There is no such frame the way the problem is set up.. Here are the two frames:

First frame: Line of charge moving in the positive y direction, and the loop moving in the negative x direction.

Second frame: Line of charge not moving, and the loop of charge having velocity components in the negative y direction and in the negative x direction.

The frame in which the wire loop is stationary (and the line charge is moving diagonally) is a very important frame, as that's the frame in which one can apply lumped circuit theory (though if the wire velocity is small enough the errors using lumped circuit theory should be relatively small), and the frame in which one can tell whether or not the light bulb at a specific location lights by looking at the electric field.

I don't see why you'd want to omit it from the problem - it's of great physical significance, if you're trying to find out whether or not the light bulb lights up.

 

[Q-reeus]

nouveau_riche, while you seem to have dropped out of this, and #18 did tighten things up with a specific scenario, none of the deliberations so far may be touching just exactly what you had in mind. Please clarify then the following, which no-one bothered to ask of you: From your #1:
* 'The observer at ground shoot the arrow as he see the train coming near him.' That leaves a lot of leeway for relative angular orientation - which did you have in mind (presumably at the point of nearest approach between charge at/on train, and loop-on-arrow)?
* 'an arrow with a circular loop hinged on it' - tells me nothing about relative orientation between arrow shaft axis and loop axis (i.e. the normal to plane of loop). please clarify.

As far as the scenario in #18 or later versions go, the resolution is quite simple imo. If in any given frame S having a stationary loop threaded by a time-changing B owing to some moving charge q, there is an emf in the loop, which must be experienced in any other frame. In the charge's rest frame S', there is no B field and a Coulombic (thus irrotational) E field. So where's the emf coming from as determined in S'? Big mystery it seems from the posts so far. Short answer is non-simultaneity. The E field is irrotational as determined in S', but not as determined by the loop which is an extended entity. It responds to the line-integral of the E field around the loop - *as evaluated at a given instant in it's own rest frame*.

That means a series of 'E field evaluation clocks' placed around the periphery must be synchronized in the loop rest frame. They will not be synchronized in the charge's rest frame S' - according to the well known expression t' = γ(t-ux/c²), with u the velocity of loop in S', and x the longitudinal component of displacement of a given point on the loop from the charge q. Thus various parts of the loop samples E field of q differently to what one naively supposes it will in S'. In particular one gets that assuming the loop is moving towards the charge in S', in order for 'evaluation clocks' to synchronously sample, parts of the loop further back from q must be evaluated at [STRIKE]a later[/STRIKE] an earlier instant [made a sign error!] as determined in S', and it's not too hard to figure this will bias things to give a net circulation of E as required. In short, simultaneous evaluation of E field by the loop in S, implies non-simultaneous evaluation of E' in S'. Hence the loop must move a certain distance in S' for this to be satisfied. Too lazy (or exhausted) to give a laborious calculation proving it, but I believe above gives the qualitative answer. Nothing whatsoever to do with induced dipole moments or the limitations of lumped element equivalent circuits.

 

[ApplePion]

"The frame in which the wire loop is stationary (and the line charge is moving diagonally) is a very important frame, as that's the frame in which one can apply lumped circuit theory (though if the wire velocity is small enough the errors using lumped circuit theory should be relatively small), and the frame in which one can tell whether or not the light bulb at a specific location lights by looking at the electric field."

Let's refer to this frame you describe as "Frame 3". As before, Frame 1 will be the frame where the line of charge is moving in the positive y direction, and the loop moving in the negative x direction; and Frame 2 will be the frame where the line of charge not moving, and the loop of charge having velocity components in the negative y direction and in the negative x direction.

You want to do the calculation in Frame 3. OK, suppose you find that your calculation in Frame 3 leads you to conclude that the LED does not light up. I would then say to you "But look at the calculation in Frame 1. It lights up in frame 1. So we have a conflict. We have a problem." Suppose your calculation in Frame 3 leads you to conclude that the LED does not light up. I would then say to you "But look at the calculation in Frame 2. It does not light up in Frame 2. So we have a conflict. We have a problem."

So doing your calculation in Frame 3 cannot get us out of the problem.

 

[ApplePion]

Q-reeus: " In the charge's rest frame S', there is no B field and a Coulombic (thus irrotational) E field. So where's the emf coming from as determined in S'? Big mystery it seems from the posts so far. Short answer is non-simultaneity. The E field is irrotational as determined in S', but not as determined by the loop which is an extended entity."

The loop does not get to determine anything in the charge's rest frame. You have to use the specific laws of physics. In the frame where the charge is at rest you have a charge at rest producing a Coloumb field, and thus no net EMF around the loop.

Your appeal to how the loop feels things is really you switching to the other frame, the frame where the loop is at rest. So what you have done is avoid explaining why things are different in the two frames, and rather compare the same frame (the frame where the loop is at rest) to itself.

 

[Q-reeus]

The loop does not get to determine anything in the charge's rest frame. You have to use the specific laws of physics. In the frame where the charge is at rest you have a charge at rest producing a Coloumb field, and thus no net EMF around the loop.

Your appeal to how the loop feels things is really you switching to the other frame, the frame where the loop is at rest. So what you have done is avoid explaining why things are different in the two frames, and rather compare the same frame (the frame where the loop is at rest) to itself.

I disagree ApplePion. Suppose one moves a magnet past a coil one way, then a moment later moves it back in exact reverse motion. The summed emf - without regard to temporal separation - is zero. Nevertheless there will be currents generated, first one way, then the other. It matters much that we respect that emf in that coil is defined as the line integral sum of tangent E component around the coil - at a given moment in the coil's rest frame. Consequently the very much SR physics phenomenon of non-simultaneity forces us to accept the coil does not evaluate simultaneously in all it's parts in charge's frame S'. I very much doubt there is any other way to resolve this problem. You do agree if an emf exists in one frame it must also exist in any other?

 

[ApplePion]

"You do agree if an emf exists in one frame it must also exist in any other?"

Yes, I do agree that is what must happen. That is why it appearing not to happen is unacceptable.

 

[Q-reeus]

Yes, I do agree that is what must happen. That is why it appearing not to happen is unacceptable.

Good. We are agreeing on something here! :tongue2: Think about my proposal a bit more please.

 

[PAllen]

"The frame in which the wire loop is stationary (and the line charge is moving diagonally) is a very important frame, as that's the frame in which one can apply lumped circuit theory (though if the wire velocity is small enough the errors using lumped circuit theory should be relatively small), and the frame in which one can tell whether or not the light bulb at a specific location lights by looking at the electric field."

Let's refer to this frame you describe as "Frame 3". As before, Frame 1 will be the frame where the line of charge is moving in the positive y direction, and the loop moving in the negative x direction; and Frame 2 will be the frame where the line of charge not moving, and the loop of charge having velocity components in the negative y direction and in the negative x direction.

You want to do the calculation in Frame 3. OK, suppose you find that your calculation in Frame 3 leads you to conclude that the LED does not light up. I would then say to you "But look at the calculation in Frame 1. It lights up in frame 1. So we have a conflict. We have a problem." Suppose your calculation in Frame 3 leads you to conclude that the LED does not light up. I would then say to you "But look at the calculation in Frame 2. It does not light up in Frame 2. So we have a conflict. We have a problem."

So doing your calculation in Frame 3 cannot get us out of the problem.

Obviously best would be a coordinate independent computation of involving the Faraday tensor and the conductor world tube, with some general model of a conductor. The inconsistencies are necessarily a result of applying methods outside of the bounds of the simplifications under which they are derived.

My best guess at present is that there is a current in all frames. The case I screwed up before, (frame 1 as you've called it above), except for having a non-uniform (but static) E field, is otherwise identical to a classic demonstration in elementary EM. I really don't see how the E field could make the current go away, compared to the elementary exercise. So, this seems like the most robust case.

In the other frames, there are issues not normally dealt with for circuits:

Frame 2: a conducting loop moving in a non-uniform (static) E-field
Frame 3: a stationary conducting loop in non-uniform, time varying, E-field (also non-uniform, time varying B field, but by normal rules, this should be irrelevant; but maybe normal rules should be questioned - in a real conductor, electrons are really (approximately) a thermal gas with high individual velocities. In many cases, that can be ignored, but not all. For example, such a model is crucial for deriving the criteria for breakdown of Ohm's law.)

Note, I do not think simultaneity is relevant. All it does is require that the exact shape and size of the loop is different in each frame. I sincerely doubt the answer to these questions is at all dependent on the shape of the loop. I should add that the orientation of loop I am assuming is that it is in the same plane as the line of current and motion of the loop toward the line. Thus the whole setup is coplanar. I assume this because that gives the maximum current per frame 1 analysis.

 

[TrickyDicky]

It is IMHO as Q-reeus says, a paradox that is fully solved by the relativity of simultaneity set in Minkowski spacetime.
Just like two spacelike separated observers may not agree about the causality of certain events, if we consider the loop as spatially extended, two observers may disagree about what happens, that is because in relativity only spacetime is well defined, not the space slice.

 

[TrickyDicky]

Actually I see Dalespam already mentioned ROS in #26.

 

[Q-reeus]

It is IMHO as Q-reeus says, a paradox that is fully solved by the relativity of simultaneity set in Minkowski spacetime.
Just like two spacelike separated observers may not agree about the causality of certain events, if we consider the loop as spatially extended, two observers may disagree about what happens, that is because in relativity only spacetime is well defined, not the space slice.

Quite agree TrickyDicky and more succinctly expressed than myself, but (and it won't be real soon from me) looks like some convincing calculations will be in order, to prove it one way or the other. One thing I am almost 100% certain about is that length contraction of the moving loop in frame S' does not capture the non-simultaneity relevant here - those clocks on the periphery will be quite out of sync in S' notwithstanding the squashed shape of loop they ride on. Also, the circulation of E must be an intrinsic, intensive feature, not some effective internal field arising from any special material interactions in the loop conductor. Hence a Feynman disk or similar will rotate in response to an intrinsic curl E no differently than a current will circulate in a conducting loop - with the conceivable caveat that induced surface charges are negligible; well satisfied for a very thin loop. Must go.

 

[PAllen]

Quite agree TrickyDicky and more succinctly expressed than myself, but (and it won't be real soon from me) looks like some convincing calculations will be in order, to prove it one way or the other. One thing I am almost 100% certain about is that length contraction of the moving loop in frame S' does not capture the non-simultaneity relevant here - those clocks on the periphery will be quite out of sync in S' notwithstanding the squashed shape of loop they ride on. Also, the circulation of E must be an intrinsic, intensive feature, not some effective internal field arising from any special material interactions in the loop conductor. Hence a Feynman disk or similar will rotate in response to an intrinsic curl E no differently than a current will circulate in a conducting loop - with the conceivable caveat that induced surface charges are negligible; well satisfied for a very thin loop. Must go.

The problem is that in the frame of the wire loop at rest, you have some shape at rest. There can be nothing influencing the current except the precise E and B fields. These are the most complex in this frame (for this problem), but they are still nothing but a Poincare transform of an axially symmetric Coulomb field. This does lead to a field with mixed E and B, that does not have axial symmetry, and is time dependent. But the complete explanation must, then, boil down to how this field interacts with a stationary conducting loop of some general shape.

 

[pervect]

I spun off a new (but related) thread to explore what happens when you "boost" a neutral loop of wire carrying a current I. My initial results (I don't thin I've made an error, but it's possible) are that Kirchoff's current law is not satisfied in the usual manner after the boost.