Relative difference in laws of electrodynamics

[nouveau_riche]

Consider a train moving at speed 's' and there is a charge particle at rest relative to the observer at train. The second observer on a ground see the charge particle and observer moving relative to him, and infer the existence of a magnetic field strong enough that its field is significant at few centimeters from the charged particle. He decides to shoot an arrow with a circular loop hinged on it and there is a LED attach to the loop.
The observer at ground shoot the arrow as he see the train coming near him. The arrow passes near to the charge particle (say a few centimeteres away that he could feel the magnetic influence predicted by observer on ground).

according to the observer on ground the change in magnetic flux from the loop will induce an emf and current will flow, this will light up the LED, whereas from the point of view of observer on train there is no magnetic field so the LED should not glow

relativity says that the observer in each frame will conclude the same no matter their perception is different to explain the phenomena, but in scenario described above they both come at a different conclusion, where i got wrong(if i did)?

 

[Dale]

according to the observer on ground the change in magnetic flux from the loop will induce an emf and current will flow, this will light up the LED, whereas from the point of view of observer on train there is no magnetic field so the LED should not glow

relativity says that the observer in each frame will conclude the same no matter their perception is different to explain the phenomena, but in scenario described above they both come at a different conclusion, where i got wrong(if i did)?

In the trains frame the changing E field produces a current in the wire through electric induction.

 

[nouveau_riche]

In the trains frame the changing E field produces a current in the wire through electric induction.

but the observer at ground also knows that the electric effect would add to the magnetic one whereas the observer at train only account for electric induction

also if you presume that at the moment when arrow passes through the effective field region there is a component of velocity along the velocity of train, the change in electric field as expected by the observer at train will be less than that as expected by the observer at ground.

 

[Dale]

if you presume that at the moment when arrow passes through the effective field region there is a component of velocity along the velocity of train, the change in electric field as expected by the observer at train will be less than that as expected by the observer at ground.

This is correct. Different frames will have different E and B fields, but all experimental measurements will be agreed upon.

 

[nouveau_riche]

This is correct. Different frames will have different E and B fields, but all experimental measurements will be agreed upon.

what i want to say is that from the reference of observer on ground there must be a magnetic interaction and an electric but from the reference frame of observer on train the interaction should only be of electric kind.

the change of electric field as experienced by the observer at ground is more in magnitude than the change experienced by the observer on train had there been a component of velocity of arrow in the direction of train, therefore their prediction will be different

 

[Dale]

what i want to say is that from the reference of observer on ground there must be a magnetic interaction and an electric but from the reference frame of observer on train the interaction should only be of electric kind.

Yes, but a particle under an EM force only "knows" the net EM force on it, and it doesn't matter one bit to the particle if different frames disagree about how much of that net force is due to E and how much is due to B.

the change of electric field as experienced by the observer at ground is more in magnitude than the change experienced by the observer on train had there been a component of velocity of arrow in the direction of train, therefore their prediction will be different

No, it will not. You are free to work it out quantitatively by yourself if you like, but it can easily be seen simply from the Lorentz covariance of Maxwell's equations and the Lorentz force law, which govern all of classical EM. If you do work it out, do not forget that a moving sensor will be time dilated and length contracted when it makes any measurements.

 

[nouveau_riche]

Yes, but a particle under an EM force only "knows" the net EM force on it, and it doesn't matter one bit to the particle if different frames disagree about how much of that net force is due to E and how much is due to B.

No, it will not. You are free to work it out quantitatively by yourself if you like, but it can easily be seen simply from the Lorentz covariance of Maxwell's equations and the Lorentz force law, which govern all of classical EM. If you do work it out, do not forget that a moving sensor will be time dilated and length contracted when it makes any measurements.

the speed with which train and arrow moves is negligible in comparison to the speed of light, the effect of length contraction and time dilation would not alter the result much

 

[Dale]

the speed with which train and arrow moves is negligible in comparison to the speed of light, the effect of length contraction and time dilation would not alter the result much

You would be surprised. The EM interaction is so strong that even very small length contraction and time dilation effects become measurable. In fact, that is the whole basis of the relativistic explanation of magnetism, which is easily measurable even with drift velocities on the order of tenths of a mm per second.

 

[nouveau_riche]

You would be surprised. The EM interaction is so strong that even very small length contraction and time dilation effects become measurable. In fact, that is the whole basis of the relativistic explanation of magnetism, which is easily measurable even with drift velocities on the order of tenths of a mm per second.

could you suggest me a link where i can find an experiment that measures those small changes

 

[Nugatory]

what i want to say is that from the reference of observer on ground there must be a magnetic interaction and an electric but from the reference frame of observer on train the interaction should only be of electric kind.

An interesting historical note: This particular asymmetry is the one that Einstein chose to motivate his discussion of SR. The first paragraph of his 1905 paper provides the problem statement:

It is known that Maxwell's electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise—assuming equality of relative motion in the two cases discussed—to electric currents of the same path and intensity as those produced by the electric forces in the former case.

 

[Dale]

could you suggest me a link where i can find an experiment that measures those small changes

Any experiment involving the magnetic field of a straight wire is a good example. See the explanation in the link for the theory, particularly the section "Magnetism as a consequence of length contraction".

http://physics.weber.edu/schroeder/mrr/MRRtalk.html

Other good experiments would be to measure the EMF induced from moving a loop past a magnet or moving the magnet past the loop at the same speed. EDIT: Nugatory already mentioned this kind of experiment.

 

[ApplePion]

Consider very carefully the configuration of the magnetic field in the frame where the charge is moving. It does not seem that such a configuration will have a net flux thru the coil. Do it carefully.

 

[ApplePion]

DaleSpam: <<In the trains frame the changing E field produces a current in the wire through electric induction.>>

In the train's frame, the charge is perfectly at rest and is just producing a non-changing Coulomb field.

You seem to want to think there is a B field in the frame where the train is moving, and somehow then convert back to the frame where the train is at rest. But one can evaluate the situation in the rest frame of the train very directly--in that frame the charge is simply at rest, minding its own business, producing an unchanging Coulomb field.

 

[Dale]

In the train's frame, the charge is perfectly at rest and is just producing a non-changing Coulomb field.

You seem to want to think there is a B field in the frame where the train is moving, and somehow then convert back to the frame where the train is at rest. But one can evaluate the situation in the rest frame of the train very directly--in that frame the charge is simply at rest, minding its own business, producing an unchanging Coulomb field.

Sorry, looking back I can see that I worded it poorly.

I am aware that there is no B field, and that the E field is the standard static field given by Coulomb's law. I meant that the E field in the wire is changing over time as the wire moves over time through the spatially varying E field. This causes a current through electric (not magnetic) induction.

 

[ApplePion]

I've changed my mind. In my example I only considered the situation where the normal to the loop points in the direction of the charge, not the general case.

I now think this is a very interesting and non-trivial problem.

 

[ApplePion]

DaleSpam: "I meant that the E field in the wire is changing over time as the wire moves over time through the spatially varying E field. This causes a current through electric (not magnetic) induction."

I don't see why the thing you refer to as "electric induction" would cause a current.

For a situation where there is a non-vanishing time derivative of the electric field thru the coil, we need to look at it in a frame where the charge is moving towards the coil. The "curl of B" Maxwell's Equation refers to what is happening at a fixed point. So what happens is the time-varying electric field induces a tangential *magnetic* field around the stationary loop. But this does not produce any force on the charges in the loop! A time varying magnetic field, on the other hand, would produce a tangential E field, and *that* would have produced a force.

Even if you think that the "line integral of the magnetic field is equal to the time derivative of the electric flux" rule applies to flux caused by the loop moving, the magnetic field thus produced interacting with the velocity of the loop will not produce forces in the appropriate direction to cause current to flow thru the loop.

I probably should clarify what I mean by saying that the"line integral of the magnetic field is equal to the time derivative of the electric flux" rule applies to a stationary loop--lots of people, including myself, have been sloppy with this. Maxwell's Equations expplicitly apply to fixed points. However, everything obeys the Principle of Relativity, so there is complementary process. Indeed, what I am going to tell you comes from Einstein's original paper, and appears to have been much of his motivation. Consider a coil with current moving towards a second coil. Both coils are normal to the x axis, and their relative motion will be along that axis. By the "curl of E equation", the increasing magneticc flux thru the second coil will generate an E field which will cause charge to flow. Now consider this in the frame where the second coil is moving and the first coil is at rest. It might seem that in that frame the increased mnanetic flux causes an induced E field, causing current to flow. But *that* is not what is happening. What is happening is that the second coil is moving thru a magnetic field which happens to have components in the y and z directions, and in this frame it is the V x B force that is causing the current to flow, not changing of flux. I suspect some here will think I am wrong, and I would urge those people to read Einstein's original paper--this discussion featured prominently in his argument for the Principle of Relativity being operative for electromagnetism.

 

[Dale]

I don't see why the thing you refer to as "electric induction" would cause a current.

Consider the case where a conductor abruptly moves from a region of 0 field to a region of uniform field. Before the transition the conductor is uncharged, some time after the transition the conductor has a dipole charge. In order to go from uncharged to dipole there must be a current.

 

[ApplePion]

"Consider the case where a conductor abruptly moves from a region of 0 field to a region of uniform field. Before the transition the conductor is uncharged, some time after the transition the conductor has a dipole charge. In order to go from uncharged to dipole there must be a current."

But the geometery of the situation we are discussing will not lead to any net circulation of charge in the loop.

Consider a situation where everything is in the x y plane and there is a positive charge at the origin, and a rectangular loop whose verices are (x = 10, y = 3) (x= 10, y = -3) (x= 20, y =3) (x= 20, y = -3). The loop is moving in the x direction towards the origin. The point charge is moving in the positive y direction.

First ignore the motion of the point charge. As the loop gets closer to the point charge at the origin, the field from that charge will cause more negative charge to be at (x= 10, y = 3) and it will cause more more negative charge at (x= 10 y = -3) --just follow the field lines from the Cololoumb field. But this is not causing a dipole moment--the shifts would need to be opposite at those two points. Same analysis for the other two points. So there is no net circulation of charge around the loop.

If you want to take into account that the point charge is moving, you still won't get net circulation in the loop. Indeed, the actual criterion for net circulation on the loop is that the E field has a curl that curls around the loop. This will not occur for a uniformly moving charge.

Now contrast it with the situation in the frame where the point charge is moving in the x direction and the loop is not. The point charge produces a magnetic field due to its y direction motion; and the flux of this magnetic field increases as the point charges x direction velocity brings it it closer to the loop. So in this frame there *is* net circulation of charge in the loop.

So the electric induction you refer to does not resolve things.

 

[Dale]

But the geometery of the situation we are discussing will not lead to any net circulation of charge in the loop.

It doesn't have to be a net circulation in order to be a current. In both frames the current is transient and any illumination of the LED very brief.

But this is not causing a dipole moment--the shifts would need to be opposite at those two points.

That is because your loop is passing around the charge rather than next to the charge. The scenario in the OP, to my understanding, was a loop passing near a charge, which would induce a dipole charge distribution. That is what I was describing above.

The point charge produces a magnetic field due to its y direction motion; and the flux of this magnetic field increases as the point charges x direction velocity brings it it closer to the loop.

I don't think this is correct. With the geometry you have suggested I think that the magnetic flux will not change as the charge goes through the loop. [EDIT: actually I completely misunderstood your geometry, see below.]

Regardless of what scenario you choose to analyze, it is impossible to correctly use Maxwell's equations to predict some experimental outcome in one frame and to correctly use Maxwell's equations to predict a different outcome for the same experiment in another frame. Any configuration which results in any measurement, like a diode lighting, will give that same measurement in any frame. This follows immediately from the invariance of Maxwell's equations under the Lorentz transform.

 

[pervect]

There does , unfortunately, seem to be some misinformation out there, but a short proof of the invariance of Maxwell equation under the Lorentz transform can be found at http://hepth.hanyang.ac.kr/~kst/lect/relativity/x850.htm

Key parts of this can also be found in the wikipedia as well, http://en.wikipedia.org/w/index.php?title=Lorenz_gauge_condition&oldid=505739422, though the wiki article discusses the solution in the Lorentz gauge without specifically demonstrating that it's Lorentz invariant.

Any good E&M text should also have this information.

The fundamental idea behind the proof is to first introduce the electric potential V and the magnetic vector potential A, and then impose the Lorentz gauge condition. It can then be demonstrated that in this gauge, A transforms as a 4-vector, which means that it's Lorentz invariant.

I'm not sure if the OP is familiar with 4-vectors or not. 4-vectors are any sort of vector that transforms via the Lorentz transform. See for instance http://en.wikipedia.org/w/index.php?title=Four-vector&oldid=505607435. Griffiths EM textbook and Taylor and Wheeler's "Space time physics" should both mention 4 vectors for the unfamiliar.

So we are left with A being a 4-vector. A close inspection shows that E and B, by themselves are NOT 4-vectors, though they are closely related. They can be thought of as pieces of a bigger tensor, the so-called Faraday tensor.

See for instance http://en.wikipedia.org/w/index.php?title=Electromagnetic_tensor&oldid=505147584, Where the Faraday tensor defined by F_{ab} = \partial_a A_b - \partial_b A_ais a rank 2 Lorentz invariant tensor.&lt;br /&gt; &lt;br /&gt; To attack these results, one would either have to claim that:&lt;br /&gt; &lt;br /&gt; &amp;lt;br /&amp;gt; \frac{1}{c^2} \frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2}- frac{\partial^2}{\partial y^2}- \frac{\partial^2}{\partial z^2}&amp;lt;br /&amp;gt; &lt;br /&gt; &lt;br /&gt; was not invariant under the Lorentz transform, or that the charge current vector&lt;br /&gt; &lt;br /&gt; (rho, Jx, Jy, Jz), the charge-current density was not a 4-vector. If I had to guess where the fundamental confusion arose, I would guess it arose on this point.

 

[Dale]

Consider a situation where everything is in the x y plane and there is a positive charge at the origin, and a rectangular loop whose verices are (x = 10, y = 3) (x= 10, y = -3) (x= 20, y =3) (x= 20, y = -3). The loop is moving in the x direction towards the origin. The point charge is moving in the positive y direction.

First ignore the motion of the point charge. As the loop gets closer to the point charge at the origin, the field from that charge will cause more negative charge to be at (x= 10, y = 3) and it will cause more more negative charge at (x= 10 y = -3) --just follow the field lines from the Cololoumb field. But this is not causing a dipole moment--the shifts would need to be opposite at those two points.

Sorry, I completely misunderstood your geometry, which is inexcusable since you described it so carefully. I actually had to sketch it out on a piece of paper to get it right.

Your analysis is incorrect. There is a dipole moment in the wire due to the charge at the origin. There is more negative charge on the x=10 side and less negative charge on the x=20 side. The dipole moment is in the x direction.

You are correct that due to the symmetry in y there is no dipole moment in the y direction. But as the loop moves towards, over, and past the charge, the dipole moment in the x direction changes both direction and strength, which leads to a current in the wire.

 

[nouveau_riche]

Sorry, I completely misunderstood your geometry, which is inexcusable since you described it so carefully. I actually had to sketch it out on a piece of paper to get it right.

Your analysis is incorrect. There is a dipole moment in the wire due to the charge at the origin. There is more negative charge on the x=10 side and less negative charge on the x=20 side. The dipole moment is in the x direction.

You are correct that due to the symmetry in y there is no dipole moment in the y direction. But as the loop moves towards, over, and past the charge, the dipole moment in the x direction changes both direction and strength, which leads to a current in the wire.

suppose the observer on ground realizes that the wind speed available near the train can help him to alter the predictions of observer at train.

the observer at grounds threw the loop such that the air drifts the loop along with it in such a way so that the observer at train finds the there is no component of electric field perpendicular to the area and predict to see no change/current in loop. the prediction differ significantly?

 

[Dale]

What are you talking about? How does air drift affect current?

 

[ApplePion]

"It doesn't have to be a net circulation in order to be a current."

In one frame it has a net circulation, and in the other frame (according via your proposed mechanism) it would not. That is not acceptable.

 

[ApplePion]

"Your analysis is incorrect. There is a dipole moment in the wire due to the charge at the origin. There is more negative charge on the x=10 side and less negative charge on the x=20 side. The dipole moment is in the x direction. "

OK, you are correct on that.

But nevertheless under your explanation there is no circulation of charge around the loop in one frame, while there is in another frame. That is not acceptable within the context of the Principle of Relativity.

 

[Dale]

"It doesn't have to be a net circulation in order to be a current."

In one frame it has a net circulation, and in the other frame (according via your proposed mechanism) it would not. That is not acceptable.

Actually, it can be. Remember that simultaneity is relative. So what is purely current sloshing back and forth in one frame can have a component of circulation in another frame.

Bottom line is that Maxwells equations are Lorentz invariant, so it is simply impossible to use them in different frames and get a true contradiction. If you ever think that you have found one then you know you made a mistake.

 

[nouveau_riche]

What are you talking about? How does air drift affect current?

i cannot give you imagination

just imagine the situation if the observer on train always find that there is no component of electric field of charge which is perpendicular to the area vector.

 

[Dale]

just imagine the situation if the observer on train always find that there is no component of electric field of charge which is perpendicular to the area vector.

That isn't possible with a point charge. You could do it with a sheet of charge or two.

 

[nouveau_riche]

That isn't possible with a point charge. You could do it with a sheet of charge or two.

i think it is possible but the situation has to be ideal to make that happen but if you like to choose sheet over the point charge the prediction have altered as i said

 

[ApplePion]

"Actually, it can be. Remember that simultaneity is relative. So what is purely current sloshing back and forth in one frame can have a component of circulation in another frame. "

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.

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.

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.

"Bottom line is that Maxwells equations are Lorentz invariant, so it is simply impossible to use them in different frames and get a true contradiction. If you ever think that you have found one then you know you made a mistake."

All I'm arguing is that your specific proposal to resolve things will not work. I did make a comment saying the situation is not resolvable in some other way.

 

[Dale]

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.

It seems like a lot of unnecessary effort in order to reach a foregone conclusion. But if you do pursue it in more detail, I think that a point charge would be easier. The full relativistic field of a point charge moving arbitrarily is given by the Lienard Weichert potential: http://en.wikipedia.org/wiki/Liénard–Wiechert_potential

I don't know a similar solution for a line charge.

All I'm arguing is that your specific proposal to resolve things will not work.

That is certainly possible.

 

[ApplePion]

"It seems like a lot of unnecessary effort in order to reach a foregone conclusion. But if you do pursue it in more detail, I think that a point charge would be easier. The full relativistic field of a point charge moving arbitrarily is given by the Lienard Weichert potential"

A point charge is more difficult because as it moves up the y-axis the fields from it at some fixed point change. That does not happen with an infinite line of charge. (Actually there is a complication dealing with infinity, but I looked into that, and it is not a problem here.)

"I don't know a similar solution for a line charge."

The magnetic field from a line of charge is actually very easy to calculate. You use the Maxwell's Equation result that the line integral of the magnetic field in a circle around the line is 4 pi times the flux of current density, and choosing the proper symmetry situation you get 2 pi r B = 4 pi I, where I is the current. So B = 2 I/r. It's just the simple sort of thing done in Purcell. OK, so now that you know that, please go back and answer the two questions I asked in my earlier post--you should get a puzzling result.

 

[pervect]

If I'm understanding the problem correctly, I don't see the problem.

Because the line charge and the square loop are in the same plane, there will be no displacement flux \partial E / \partial t through the loop.

Regardless of whether the loop moves or the line charge moves, there WILL be a change in the total magnetic flux in the loop, and hence a circulation current. The edge of the loop closest to the wire will break more field lines than the edge of the loop furthest away.

 

[ApplePion]

"Regardless of whether the loop moves or the line charge moves, there WILL be a change in the total magnetic flux in the loop"

No, if the line of charge is not moving then it is not producing *any* magnetic field, and therefore there can be no magnetic flux or change in magnetic flux.

In one of the frames the line of charge is moving, and in the other it is not.

 

[pervect]

"Regardless of whether the loop moves or the line charge moves, there WILL be a change in the total magnetic flux in the loop"

No, if the line of charge is not moving then it is not producing *any* magnetic field, and therefore there can be no magnetic flux or change in magnetic flux.

In one of the frames the line of charge is moving, and in the other it is not.

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
-\ln \sqrt{x^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 = \partial_a A_b - \partial_b A_a.

 

[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.