Magnetism seems absolute despite being relativistic effect of electrostatics

[Dale]

In case I was misunderstanding universal's previous question, let me clarify the situation:

Frame 1 (lab frame):
Wire is neutral and carries a current. Test charge is moving. Electrostatic force on test charge is 0 because wire is neutral. Magnetic force on test charge is non-zero since charge is moving.

Frame 2 (test-charge frame):
Wire is charged and carries a current. Test charge is at rest. Electrostatic force on test charge is non-zero because the wire is charged. Magnetic force on test charge is 0 since charge is not moving.

 

[Per Oni]

Yes, are you familiar with four-vectors or tensors?

No, one day I’ll learn that concept, it looks like it’s really useful.

I think I have the same (perhaps faulty) thought process as universal, when he says:

Whereas, my question is, why don't we see this magnetic force to come in play(or act) when the test charge is STATIONARY w.r.t the wire. Since, even without motion of the test charge there should be the length contraction of the moving charges in the wire, when there is current, and therefore there should be a force even on the stationary test charge.

This situation refers to your frame 1 but without the test charge moving. “If” there’s a Lorentz boost as seen from the test charge, it should feel an extra electrostatic force.

My take on it is (and the reality is) that no such extra electrostatic force is present therefore this whole idea of a magnetic field being a Lorentz boosted electrostatic field is for a lot of us hard to believe.

 

[harrylin]

You refer perhaps to explanations (often accompanied by nice looking calculations) according to which magnetism is claimed to be a kind of illusion due to length contraction.

The most basic and simple case (although very high tech) that I can imagine, as it completely avoids issues with electron source and drain, is that of a closed loop superconductor in which a current is induced.

We thus start with, I think, an insulated wire containing a number of electrons N and an equal number of protons N.

I think that the following situation sketch is correct:

In the wire's rest frame:
- length contraction can play no role at all
- a magnetic field is observed

In any inertial moving frame:
- length contraction plays a role in predicting non-zero electric fields
- a magnetic field is observed that can't be transformed away

Is that correct?
Such a magnetic field looks reasonably "absolute" to me.

Harald

 

[Dale]

This situation refers to your frame 1 but without the test charge moving. “If” there’s a Lorentz boost as seen from the test charge, it should feel an extra electrostatic force.

Why would there be a Lorentz boost for a stationary test charge?

In this physically different situation there is no force since the wire is uncharged (no electrostatic force) and the test charge is not moving (no magnetic force). In other frames there will be both an electrostatic and a magnetic force, but they will cancel each other for 0 total EM force.

My take on it is (and the reality is) that no such extra electrostatic force is present

This is completely incorrect. If there is an EM force on a charged particle then it is always possible to transform to a frame where the particle is at rest. In this frame the magnetic force is 0 so the force on the particle is entirely due to the electrostatic force.

therefore this whole idea of a magnetic field being a Lorentz boosted electrostatic field is for a lot of us hard to believe.

Many things are hard for people to believe, that does not make them wrong.

However, I would also disagree with the idea of a magnetic FIELD being a boosted electrostatic FIELD, the math doesn't support that. The correct statement would be that a magnetic FORCE is a "boosted" electrostatic FORCE, which the math supports.

If you just have the field then you don't have a rest frame. You need a test charge on which there is a force so that you can boost to the test charge's rest frame where there is no magnetic force.

 

[Per Oni]

Why would there be a Lorentz boost for a stationary test charge?

Because of the fact that the test charge sees a current flowing in the conductor.

This is completely incorrect. If there is an EM force on a charged particle then it is always possible to transform to a frame where the particle is at rest. In this frame the magnetic force is 0 so the force on the particle is entirely due to the electrostatic force.

Let’s look at the situation I thought I was referring to, which is a test charge stationary wrt a wire. What I ment to say is that in reality there are no different (although I said extra) electrostatic forces on the test charge whether a current flows or not. Can we agree on this?

 

[universal_101]

In case I was misunderstanding universal's previous question, let me clarify the situation:

Frame 1 (lab frame):
Wire is neutral and carries a current. Test charge is moving. Electrostatic force on test charge is 0 because wire is neutral. Magnetic force on test charge is non-zero since charge is moving.

Frame 2 (test-charge frame):
Wire is charged and carries a current. Test charge is at rest. Electrostatic force on test charge is non-zero because the wire is charged. Magnetic force on test charge is 0 since charge is not moving.

First, Yes, you did misunderstand my question,

Second, I believe that whole point of this debate/discussion is that we want to understand magnetic force as a relativistic effect of electrostatics. But, you are explicitly using the two types of forces to explain the experimental observations. Which puts the magnetic force in the categories of absolute forces. Because, if we are unable to explain the motion of charged particles without introducing the magnetic force then it compels us to think of it as an absolute property.

 

[kmarinas86]

You refer perhaps to explanations (often accompanied by nice looking calculations) according to which magnetism is claimed to be a kind of illusion due to length contraction.

The most basic and simple case (although very high tech) that I can imagine, as it completely avoids issues with electron source and drain, is that of a closed loop superconductor in which a current is induced.

We thus start with, I think, an insulated wire containing a number of electrons N and an equal number of protons N.

I think that the following situation sketch is correct:

In the wire's rest frame:
- length contraction can play no role at all
- a magnetic field is observed

In any inertial moving frame:
- length contraction plays a role in predicting non-zero electric fields
- a magnetic field is observed that can't be transformed away

Is that correct?
Such a magnetic field looks reasonably "absolute" to me.

Harald

Why would the length contraction play a role in predicting non-zero fields when the number of positive charge and negative charges are equal?

In any case, does not the relative velocity between the charges pay a role here?

It is as if both relative velocity between charges and relative velocity between the the charges and the observer both collude in determining whether or not non-zero fields are observed.

I cannot simply call that "length contraction". Like another had stated, it is "length contraction plus something else".

Also, what if we inverted things and called the protons the "current" and the electrons the "wire"? In this case, the magnetic field would be seen as being produced by the positive charges. In contrast, the negative charges would not be seen as responsible for the magnetic field. The electron frame would be the "rest" frame of the wire. Would then we say that there exists zero electric field outside the wire from the "rest" frame of the electrons? And if we would move to the proton frame, now regarded as the "rest" frame of the current, would we then say that there is a non-zero electric field outside the wire?

This begs the question, "Could the problem be interpreted such that there is ALWAYS a non-zero electric field outside the wire, depending on what one regards as being 'current'?" Obviously there is a gap in reasoning going on here. So what's up with that?

Alternatively, if you consider the fact that "ionic current" or "positive charge" current can be just as guilty in producing magnetic fields as the electron current, one would realize that for the case of a neutral wire, different Lorentz transformations do not lead to differences in the magnetic flux. The magnetic field produced by a + charge is equal and opposite of that produced by a - charge if their movements are the same. So the magnetic flux produced by the neutral wire should be frame invariant. What changes is the magnetic flux intensity (a.k.a. magnetic flux density) and corresponding area of integration (an area which is itself subject to Lorentz transformations). This is same as with the electric flux; the Lorentz transformation leaves it unaltered (with the electric field intensity (a.k.a. electric flux density) and corresponding integration being subject to exact same transformation as that of their magnetic counterparts).

Now, if we the consider the case where have only an electron beam (no positive charges), we must realize that a charge in a co-moving frame same as that of another charge, is not going to experience a magnetic force from that other charge, but only an electrostatic force. This is the same electrostatic force that one would expect if you simply had the two charges at so-called rest, separated by the same distance (correcting for the Lorentz transformation of course). However, if you have the charges moving at different speeds, there is a relative velocity between them. Only then can you say that they interact magnetically.

It appears it is the relative velocity (or lack thereof in other cases) that determines whether or not there is magnetic interaction between particles in a system.

Of course, you can predict that two electrons co-moving relative to an external observer will have a magnetic field around them, but that magnetic field is not something that the charges interact with, because in their frame, that field simply does NOT exist. So as long as nothing in the frame of the external observer interacts with those particles, the magnetic field as seen from the point of view of the external observer may as well not exist, for lack of physical significance.

 

[Dale]

Let’s look at the situation I thought I was referring to, which is a test charge stationary wrt a wire. ... whether a current flows or not.

This is kind of a strange thing to try to analyze in a discussion about relativity. The current flowing or not flowing are two physically different situations in the same reference frame. Usually relativity is used to analyze the same physical situation from two different reference frames.

Certainly your approach could be useful for learning EM, but not for learning relativity nor for learning about the relativistic connection between electric and magnetic forces and fields. I would suggest starting a new thread in the Classical Physics forum.

What I ment to say is that in reality there are no different (although I said extra) electrostatic forces on the test charge whether a current flows or not. Can we agree on this?

In the wire frame, yes, I agree. In other frames it is not correct. The electrostatic and magnetic forces are frame-variant, so you have to specify the frame whenever you talk about them.

 

[Dale]

But, you are explicitly using the two types of forces to explain the experimental observations. Which puts the magnetic force in the categories of absolute forces. Because, if we are unable to explain the motion of charged particles without introducing the magnetic force then it compels us to think of it as an absolute property.

The magnetic force was 0 in one frame and non-zero in another frame. Physics quantities don't get any more relative than that. I don't know how you came to your conclusion, but it is wrong. If a quantity depends on the reference frame then by definition it is relative, not absolute.

 

[kmarinas86]

The magnetic force was 0 in one frame and non-zero in another frame. Physics quantities don't get any more relative than that. I don't know how you came to your conclusion, but it is wrong. If a quantity depends on the reference frame then by definition it is relative, not absolute.

You have positive charges. Positive charges can produce a magnetic field.

Calling electrons the current, while yet ignoring the contribution of the magnetic field by the positive charges, is completely wrong and arbitrary.

For the magnetic field of a neutral entity, a Lorentz transformation should result in equal and opposite changes in the magnetic flux contributions by the negative and positive charges. (Net field intensity should depend on length contraction, yes, but not the flux).

 

[Dale]

You have positive charges. Positive charges can produce a magnetic field.

Calling electrons the current, while yet ignoring the contribution of the magnetic field by the positive charges, is completely wrong and arbitrary.

I agree. I don't know if others are ignoring some of the charge carriers, but I am not.

For the magnetic field of a neutral entity, a Lorentz transformation should result in equal and opposite changes in the magnetic flux contributions by the negative and positive charges.

I doubt that this is correct, but I almost never work with the integral form of Maxwell's equations so I am not certain. I would want to see a proper derivation before making a strong statement either way, or at least work a couple of examples.

 

[universal_101]

The magnetic force was 0 in one frame and non-zero in another frame. Physics quantities don't get any more relative than that. I don't know how you came to your conclusion, but it is wrong. If a quantity depends on the reference frame then by definition it is relative, not absolute.

This is very well put, but I'm not questioning the relative nature of magnetic force in different frames, actually, nobody is. The problem is why do we have to invoke the magnetic force if the magnetic force itself is supposedly can be explained as the relativistic effect of the electrostatics.

 

[jartsa]

I know that magnetic force due to a current carrying wire on a test charge moving w.r.t the wire(along the wire), can be interpreted as the electrostatic force if we use the first order relativistic corrections for Time Dilation or Length contraction of the charges of the wire, in the frame of the the test charge.

But what I don't seem to understand is rather very simple situation.

Let's consider a simple model of a conducting wire,

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Now, let's suppose there is some current in the wire and the electrons are moving at speed 'v' w.r.t the the wire,
secondly, a stationary test charge w.r.t the wire lying around.

Naming the above scenario as (1)

Now, the test charge starts moving in the direction of electrons with the same speed 'v'.
This time in the reference frame of the test charge, electrons are stationary and nucleus(positive charge) is moving at speed 'v'.

Naming this scenario as (2)

And so the question arise, the two scenario are identical w.r.t principle of relativity. That is, in the first case only negative charges are moving, but there is no force on the charge. But in the second case when positive charges are moving there is a force on the test charge(magnetic force towards wire). Whereas, the two cases are essentially identical w.r.t principle of relativity.

When line of objects starts moving, length contraction may be observed, or length contraction may not be observed. Rigid bodies contract, other things may either contract or not.

When observer observing line of objects, starts moving, he will observe length contraction.

So I'm saying scenario1 and scenario2 differ this way. (motion without length contraction vs. motion with length contraction)

 

[Dale]

why do we have to invoke the magnetic force if the magnetic force itself is supposedly can be explained as the relativistic effect of the electrostatics.

You don't have to invoke the magnetic force in the rest frame of the test particle. Once you have the total force in the rest frame, you can then transform and find the total force in any other frame, without ever invoking the magnetic force.

The magnitude of the total force in the other frame is a relativistic effect of electrostatics. It differs from the electrostatic force in the other frame by a certain amount, that amount is called the magnetic force. Thus, the magnetic force is a relativistic effect of electrostatics.

 

[kmarinas86]

When line of objects starts moving, length contraction may be observed, or length contraction may not be observed. Rigid bodies contract, other things may either contract or not.

How do the formulas of SR recognize rigidity? Is it quantifiable? What happens if I have an egg container, and in each egg placeholder I have either a "rigid" body or a non-rigid body. How does that work? I thought that SR does not permit "fully" rigid bodies.

 

[jtbell]

why do we have to invoke the magnetic force if the magnetic force itself is supposedly can be explained as the relativistic effect of the electrostatics.

We don't have to. In principle, we can start in the particle's rest frame and Lorentz-transform the force (which is purely electrostatic in that frame) into the frame that we actually use for observing the particle.

However, for practical problem-solving, it's usually more convenient to use only one reference frame, namely the one in which we observe the particle. So we invoke both electric and magnetic forces in that frame, and the net force gives the same result as the first approach.

Doing the Lorentz transformation is fairly simple if the motion is uniform. If the motion isn't uniform, it becomes messy.

 

[jartsa]

How do the formulas of SR recognize rigidity? Is it quantifiable? What happens if I have an egg container, and in each egg placeholder I have either a "rigid" body or a non-rigid body. How does that work? I thought that SR does not permit "fully" rigid bodies.

Move one part of rigid body, other parts of rigid body will follow.

Well maybe I meant elastic body.

Let's consider a line of cars driving 50 mph, making a 90 degrees turn at street corner. We newer see any car moving at any other speed than 50 mph, so the line does not contract. Cars do contract. When a car contracts its rear end moves faster than its front.

Electron flow in rectangular circuit behaves the same way.

 

[universal_101]

We don't have to. In principle, we can start in the particle's rest frame and Lorentz-transform the force (which is purely electrostatic in that frame) into the frame that we actually use for observing the particle.

However, for practical problem-solving, it's usually more convenient to use only one reference frame, namely the one in which we observe the particle. So we invoke both electric and magnetic forces in that frame, and the net force gives the same result as the first approach.

Doing the Lorentz transformation is fairly simple if the motion is uniform. If the motion isn't uniform, it becomes messy.

Well, transformation of one force into another is easy to understand and follow, but as long as the transformation is consistent all the way.

Please, try to analyse the situation,

when there is a current, the charges in the wire start moving in a particular direction, but when there is NO current there is NO motion. Therefore, according to the transformation of one force into other, there should be a force on a stationary charge standing near by, towards the current carrying wire, when there is current.

Remembering, that my original post/question is exactly same situation, to which the answer was the transformation of one force into another, to explain the magnetic force.

 

[harrylin]

Why would the length contraction play a role in predicting non-zero fields when the number of positive charge and negative charges are equal?
In any case, does not the relative velocity between the charges pay a role here? [..]

In a frame in which the charges are moving at different velocities, the charge distribution may be inhomogeneous. However I suppose that that is not the point of this discussion: I merely noticed that for such a case length contraction may play a role in what is not under discussion here. My point is that according to my analysis, magnetic fields can in general not be transformed away.

Also, what if we inverted things and called the protons the "current" and the electrons the "wire"? In this case, the magnetic field would be seen as being produced by the positive charges. [...]

Sorry I can't follow that; a magnetic field appears as the result of whatever moving charges - no matter if those are the protons or electrons or both.

This begs the question, "Could the problem be interpreted such that there is ALWAYS a non-zero electric field outside the wire, depending on what one regards as being 'current'?" Obviously there is a gap in reasoning going on here. So what's up with that?

A non-zero (at least, non negligible) electric field can be measured.

Alternatively, if you consider the fact that "ionic current" or "positive charge" current can be just as guilty in producing magnetic fields as the electron current, one would realize that for the case of a neutral wire, different Lorentz transformations do not lead to differences in the magnetic flux. The magnetic field produced by a + charge is equal and opposite of that produced by a - charge if their movements are the same. So the magnetic flux produced by the neutral wire should be frame invariant.

For a current-free wire, indeed. That isn't an issue.

What changes is the magnetic flux intensity (a.k.a. magnetic flux density) and corresponding area of integration (an area which is itself subject to Lorentz transformations). This is same as with the electric flux; the Lorentz transformation leaves it unaltered (with the electric field intensity (a.k.a. electric flux density) and corresponding integration being subject to exact same transformation as that of their magnetic counterparts).

Now, if we the consider the case where have only an electron beam (no positive charges), we must realize that a charge in a co-moving frame same as that of another charge, is not going to experience a magnetic force from that other charge, but only an electrostatic force. This is the same electrostatic force that one would expect if you simply had the two charges at so-called rest, separated by the same distance (correcting for the Lorentz transformation of course). However, if you have the charges moving at different speeds, there is a relative velocity between them. Only then can you say that they interact magnetically. [..]

The Lorentz transformation does the same as the introduction of a magnetic force; that realisation is the basis for this kind of discussions. However, as I illustrated, more insight can be obtained if one doesn't limit one's thinking to linear cases and instead considers a more basic magnet such a made with a single wire loop.

Of course, you can predict that two electrons co-moving relative to an external observer will have a magnetic field around them, but that magnetic field is not something that the charges interact with, because in their frame, that field simply does NOT exist. So as long as nothing in the frame of the external observer interacts with those particles, the magnetic field as seen from the point of view of the external observer may as well not exist, for lack of physical significance.

Your argument doesn't apply to more general cases as I illustrated. Magnetic and electric fields are relative in the way that length contraction and simultaneity are relative; it doesn't mean that one of the concepts should be discarded.

 

[kmarinas86]

When line of objects starts moving, length contraction may be observed, or length contraction may not be observed. Rigid bodies contract, other things may either contract or not.

How do the formulas of SR recognize rigidity? Is it quantifiable? What happens if I have an egg container, and in each egg placeholder I have either a "rigid" body or a non-rigid body. How does that work? I thought that SR does not permit "fully" rigid bodies.

Move one part of rigid body, other parts of rigid body will follow.

Well maybe I meant elastic body.

Let's consider a line of cars driving 50 mph, making a 90 degrees turn at street corner. We newer see any car moving at any other speed than 50 mph, so the line does not contract. Cars do contract. When a car contracts its rear end moves faster than its front.

Electron flow in rectangular circuit behaves the same way.

What is the quantity and its symbol? How is it used in the equations of SR? Can it be plugged into Lorentz transforms?

 

[jartsa]

What is the quantity and its symbol? How is it used in the equations of SR? Can it be plugged into Lorentz transforms?

Are you trying to bully me or what? Collection of free particles may retain its shape when accelerated, without any stresses. When free particles are glued together, we have a rigid body, which must length contract, when accelerated, or else stresses are generated in the body.

 

[kmarinas86]

Your argument doesn't apply to more general cases as I illustrated. Magnetic and electric fields are relative in the way that length contraction and simultaneity are relative; it doesn't mean that one of the concepts should be discarded.

Right, length contraction is relative.

The problem is:
* I can have the positive charge have a greater length contraction in the frame of the negative charge.
* I can have the negative charge have a greater length contraction in the frame of the positive charge.

Following the claims of DaleSpam's comments, this would mean that the wire can appear to have net positive charge or a net negative charge, depending on the frame of reference. There is also a frame in which the length contractions of the positive and negative charges match. I suppose that is when the electric-field outside the wire disappears.

Now that I think about it terms of length contraction, the changes of the electric field with respect to the frame given is NOT linear because the equations for length contraction do not have constant derivative with respect to relative velocity with the observer. Therefore, the LT would result in different change "factors" for the electric field of the electrons and the electric field of the protons in the case when there is current.

 

[kmarinas86]

Are you trying to bully me or what? Collection of free particles may retain its shape when accelerated, without any stresses. When free particles are glued together, we have a rigid body, which must length contract, when accelerated, or else stresses are generated in the body.

I though that force dynamics weren't a part of the Lorentz transformation.

This above sounds to me more like the "deformable electron" concept of Lorentzian-Ether theory.

 

[kmarinas86]

The idea that the electric field intensity of each charge being variant with respect to the observer isn't strange to me.

What's strange is the idea that steady-state (read: DC) current should somehow be uniform through out the wire when the protons and electrons clearly cannot be subject to the same length contraction.

Let's keep this REALLY simple. Assuming that the wire is neutral (no net charge) and that the wire is 1 meter long and that I have a length contraction of electrons, why should I get from that a uniform charge distribution when the electrons are drifting through wire (current)?

In the steady state the four-current (density) is uniform and constant in the lab frame, therefore it is uniform and constant in the test-charge frame also.

I would TOTALLY expect an un-uniform distribution, assuming length contraction applies to the bulk flow of electrons.

Why? Why do you expect a gap of any kind in the steady state?

I STILL don't have an answer to my question as to what do the electrons actually length contract towards.

Length contraction occurs, as always, in the direction of motion. The word "towards" doesn't make any sense in this context. The word "towards" implies something changing over time. Length contraction does not change over time in an inertial frame.

Ok, then let me ask it this way: From the lab frame, where is the center of contraction for the bulk of electron flow in a straight wire conductor? The contraction is only "linear", so I assume that this "center" of contraction must be a geometric plane. Where is that located in relation to the observer?

SR says that objects (read: multiple particles) will length contract. So, logically speaking, you can treat the + charges and - charges as two separate "objects" at different speeds. I assume this to mean not only the particles by themselves, but the entire bulks of the particles as a whole. For an object to contract, the distance in-between also has to contract. You don't have just the fundamental particles contracting. In the extreme case, going from 0 current to a very high current would cause the following to occur:

This

+       +       +       +       +
-       -       -       -       -

into this

+       +       +       +       +
              -----

or

+       +       +       +       +
-----

or

+       +       +       +       +
                            -----

et cetera

going from 0 current to a very high current would cause the following to occur:

No, I already covered the non-steady state situation in post 8. None of your suggestions are correct, neither in the transient nor in the steady-state conditions.

If the quantity of charge in a length measured by the observer were really to vary depending of the length contraction of each set of charges (the + set vs. the - set) whose length contraction values are different, we would see not only an frame-variant electric field intensity, but also, we would see an frame-variant electric FLUX as well in that length. In reality, if we Lorentz transform a system, we do NOT create positive charge and negative charges out of nowhere. Those additional electrons somehow fitting into the wire must be present with and without the Lorentz transformation. So if the wire was uniformly charged before the length transformation and after it, then some of the - charge that was OUTSIDE the wire without the LT is instead seen as being INSIDE the wire with that LT. The same would go with the + charge.

I guess that the difference of electric flux between different LT frames means that time-retardation effects apply to electric flux as well.

 

[kmarinas86]

Alternatively, if you consider the fact that "ionic current" or "positive charge" current can be just as guilty in producing magnetic fields as the electron current, one would realize that for the case of a neutral wire, different Lorentz transformations do not lead to differences in the magnetic flux. The magnetic field produced by a + charge is equal and opposite of that produced by a - charge if their movements are the same. So the magnetic flux produced by the neutral wire should be frame invariant.[ What changes is the magnetic flux intensity (a.k.a. magnetic flux density) and corresponding area of integration (an area which is itself subject to Lorentz transformations). This is same as with the electric flux; the Lorentz transformation leaves it unaltered (with the electric field intensity (a.k.a. electric flux density) and corresponding integration being subject to exact same transformation as that of their magnetic counterparts).]

For a current-free wire, indeed. That isn't an issue.

I thought that the observed magnetic field was directly proportional to relative velocity v. The electric field's dependence on \gamma should contrast with the magnetic field's dependence v.

In that case, I cannot at all see how changes in the E-field can compensate precisely for changes in the B-field. They simply do not match. So it can undershoot or overshoot the requirement for compensating for the difference of the B between different LT frames.

Alternatively, if B varied with the rapidity \varphi (with respect to LT frames, not time or acceleration, mind you), it would not be an exact match either:

Column 1: v/c
Column 2: \varphi
Column 3: \gamma
Column 4: Change in Column 2
Column 5: Change in Column 3
Column 6: Column 2 / Column 3

0.00	0.00	1.00			
0.10	0.10	1.01	0.10	0.01	19.92
0.20	0.20	1.02	0.10	0.02	6.57
0.30	0.31	1.05	0.11	0.03	3.86
0.40	0.42	1.09	0.11	0.04	2.67
0.50	0.55	1.15	0.13	0.06	1.98
0.60	0.69	1.25	0.14	0.10	1.51
0.70	0.87	1.40	0.17	0.15	1.16
0.80	1.10	1.67	0.23	0.27	0.87
0.90	1.47	2.29	0.37	0.63	0.60

Only one other possibility: The B normal to the wire is proportional to \gamma_{v\ parallel\ to\ the\ wire}. The problem is that I never heard of it.

Meanwhile, in SR, the "relativistic energy" of a particle is relative to LT frames. So the idea that the magnetic field is simply the relativistic component of the electric field appears doomed. SR would have no problem having the change in the E field be more than and/or less than what would be needed to compensate for the magnetic field, for it appears to be required to have the "relativistic energy" of a particle to vary.

By the way, if some E fields and some B fields cannot transform away, then the claim that electric fields and magnetic fields are part of the same "electromagnetic field" seems dubious at best.

Maybe we should move away from the field concepts and stick with the vector potential instead.

 

[DrGreg]

Attached to the bottom of this post is a diagram to help explain things. As was mentioned earlier in this thread, one way to approach the problem is to consider it a variant of the ladder paradox, and consider the different definitions of simultaneity.

But my approach here considers length contraction only. And I am going to consider a complete circuit: not just a single wire with a left-to-right electron flow, but also a return wire with a right-to-left flow. Apart from the ends of the wires, we keep the two wires far apart so they have negligible influence on each other. The diagram is a highly idealised simplification, considering just 16 electrons in the circuit. The ends of the wires should be in contact with each other but I've drawn them as separated to keep the diagram simple.

The top left part of the diagram shows the wires with no current flowing, in the rest-frame of the wires. 16 electrons equally spread out along the wire.

The top right part of the diagram again shows the wires with no current flowing, but now in a frame moving at the velocity that electrons would flow in the bottom wire if the current were on. We see length contraction as indicated by the yellow arrows. I'm assuming a Lorentz factor γ=2. So far so good.

The two bottom diagrams now show what happens when the current is flowing.

In the bottom left diagram, as we are told the wires remain electrically neutral, there must still be 16 electrons in the wires. There's no reason for the electrons to bunch together anywhere, they will remain spread out around the whole circuit as shown.

Finally, let's look at the bottom right diagram, which I think some people are having difficulty to imagine. We already know what happens to the red positive ions, their separation contracts just as before. The electrons in the lower wire are now stationary, so their separation must be larger than the bottom left diagram as shown. On the other hand, the electrons in the upper wire are moving faster than in bottom left diagram, so their separation must be less than in bottom left diagram. No electrons have escaped so the total number of electrons in circuit must still be 16. But now there are fewer electrons in the lower wire and more in the upper wire. So the lower wire has a positive charge and the upper wire has a negative charge.

 

[DrGreg]

It doesn't seem to have been mentioned in this thread yet. The E and B fields are used to construct a 4×4 matrix<br /> F^{\mu\nu} = \begin{bmatrix}<br /> 0 &amp; -E_x/c &amp; -E_y/c &amp; -E_z/c \\<br /> E_x/c &amp; 0 &amp; -B_z &amp; B_y \\<br /> E_y/c &amp; B_z &amp; 0 &amp; -B_x \\<br /> E_z/c &amp; -B_y &amp; B_x &amp; 0<br /> \end{bmatrix}<br />This is a rank-2 tensor whose components transform as a tensor, i.e. there's a double Lorentz transformation involved.

 

[kmarinas86]

Attached to the bottom of this post is a diagram to help explain things. As was mentioned earlier in this thread, one way to approach the problem is to consider it a variant of the ladder paradox, and consider the different definitions of simultaneity.

But my approach here considers length contraction only. And I am going to consider a complete circuit: not just a single wire with a left-to-right electron flow, but also a return wire with a right-to-left flow. Apart from the ends of the wires, we keep the two wires far apart so they have negligible influence on each other. The diagram is a highly idealised simplification, considering just 16 electrons in the circuit. The ends of the wires should be in contact with each other but I've drawn them as separated to keep the diagram simple.

The top left part of the diagram shows the wires with no current flowing, in the rest-frame of the wires. 16 electrons equally spread out along the wire.

The top right part of the diagram again shows the wires with no current flowing, but now in a frame moving at the velocity that electrons would flow in the bottom wire if the current were on. We see length contraction as indicated by the yellow arrows. I'm assuming a Lorentz factor γ=2. So far so good.

The two bottom diagrams now show what happens when the current is flowing.

In the bottom left diagram, as we are told the wires remain electrically neutral, there must still be 16 electrons in the wires. There's no reason for the electrons to bunch together anywhere, they will remain spread out around the whole circuit as shown.

Finally, let's look at the bottom right diagram, which I think some people are having difficulty to imagine. We already know what happens to the red positive ions, their separation contracts just as before. The electrons in the lower wire are now stationary, so their separation must be larger than the bottom left diagram as shown. On the other hand, the electrons in the upper wire are moving faster than in bottom left diagram, so their separation must be less than in bottom left diagram. No electrons have escaped so the total number of electrons in circuit must still be 16. But now there are fewer electrons in the lower wire and more in the upper wire. So the lower wire has a positive charge and the upper wire has a negative charge.

I think that using the return wire to prove a point is cheating. It does nothing for the original scenario without a return wire.

The ladder paradox also has some asymmetries that seem to be missing in your example:

Figure 4: Scenario in the garage frame: a length contracted ladder entering and exiting the garage

Figure 5: Scenario in the ladder frame: a length contracted garage passing over the ladder

The two frames do not see the same number of rungs inside the garage in each case.

If we assumed that the protons were represented as tiles on the garage floor, the garage as the wire, and the ladder as the electron current in and out of the wire, then clearly the charge inside the boundary of the garage is not invariant.

However, considering that the electric field intensity increases by the same amount that the boundary of the garage in the LT frame is length contracted, this would keep the electric flux around that boundary of the garage a constant.

 

[DrGreg]

I think that using the return wire to prove a point is cheating. It does nothing for the original scenario without a return wire.

Well it has the advantage of charge conservation in a closed system, which doesn't apply to an open-ended wire.

The ladder paradox also has some asymmetries that seem to be missing in your example:

The two frames do not see the same number of rungs inside the garage in each case.

If we assumed that the protons were represented as tiles on the garage floor, the garage as the wire, and the ladder as the electron current in and out of the wire, then clearly the charge inside the boundary of the garage is not invariant.

However, considering that the electric field intensity increases by the same amount that the boundary of the garage in the LT frame is length contracted, this would keep the electric flux around that boundary of the garage a constant.

Sorry, somehow the image attachment to my post failed to upload correctly. I have now re-uploaded it and added it to that message.

If you ignore my return wire and concentrated on my lower wire only, it seems to me that my diagram agrees with your ladder diagram, so I haven't grasped what your problem is.

 

[kmarinas86]

Well it has the advantage of charge conservation in a closed system, which doesn't apply to an open-ended wire.

Sorry, somehow the image attachment to my post failed to upload correctly. I have now re-uploaded it and added it to that message.

If you ignore my return wire and concentrated on my lower wire only, it seems to me that my diagram agrees with your ladder diagram, so I haven't grasped what your problem is.

The problem is that we are talking about a single current and the fact that charge has to be conserved between frames for that single current.

There can be charge outside the wire ends (say at the ends of a capacitor or what not).