Relative difference in laws of electrodynamics

"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.

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pervect
Staff Emeritus
"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.

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.

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"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.

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.

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?

"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.

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
2019 Award
"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.

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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.

Actually I see Dalespam already mentioned ROS in #26.

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
2019 Award
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.

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pervect
Staff Emeritus