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 [itex]F_{ab} = \partial_a A_b - \partial_b A_a[itex] is a rank 2 Lorentz invariant tensor.

To attack these results, one would either have to claim that:

$$\frac{1}{c^2} \frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2}- frac{\partial^2}{\partial y^2}- \frac{\partial^2}{\partial z^2}$$

was not invariant under the Lorentz transform, or that the charge current vector

(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 $F = \partial_a A_b - \partial_b A_a$.