First, what is the electrostatic force, and what is the magnetic force?
The electrostatic force is what causes opposite charges to attract, and like charges to repel. Electrons, negatively charged, tend to stick to protons, positively charged. Two protons would repel each other, as would two electrons. On a macroscopic scale, the electrostatic force is what causes static electricity, in which an object accumulates an excess or shortage of electrons. It also causes lightning, which is basically static electricity on an even larger scale.
The magnetic force acts upon moving charges. If I've got an electric current, where the electrons are moving forward in a wire, the current creates a magnetic field. If I place two of these wires next to each other with the currents going in the same direction, they will attract. If the currents go in opposite directions, the wires repel. What we call magnets are materials with permanent circular currents on an atomic scale. The north pole of a magnet has currents going counter-clockwise, and the south pole has currents going clockwise. The north and south poles attract because when they are placed together, the currents go in the same direction.
The magnetic and electric forces interact and affect each other, but it is not clear why. Why should currents in the same direction attract? The wires, after all, have no net charge. There are just as many electrons as protons in each wire. So it can't be that the electric force is somehow sneaking in, disguised, right?
There is, in fact, a paradox associated with magnetism. Magnetic forces only act upon moving charges. But if we consider a moving particle's reference frame, the particle always has zero speed relative to itself. Therefore, from the particle's reference frame, it cannot be affected by magnetic forces. These forces shouldn't be disappearing just because our reference frame is different!
Let's consider a specific case: two wires with current going in the same direction. Wires, along with most everyday objects, consist of equal numbers of protons and electrons. If a wire has electric current going through it, that means that the protons are remaining still while the electrons are moving in one direction along the wire. The electrons, in fact, are moving at a large range of speeds, but for simplicity's sake I will assume that they are all moving at one constant speed.
Let's consider the wires with Relativity in mind. Of course, from the protons' motionless frame of reference, the wire is electrically neutral. But what happens if we consider the frame of reference of a moving electron? From the electron's point of view, the other wire contains a bunch of motionless electrons and a bunch of backwards-moving protons. Since the electrons and protons are moving relative to each other, we must take into account Lorentz contraction. If you don't recall, Lorentz contraction makes all distances in the direction of motion smaller. Lorentz contraction causes the protons to be closer together, more densely packed. As a result, the other wire has an overall positive charge, creating an electrostatic force. The electron will be attracted by this force.
So from my point of view, standing still, the wires attract because of magnetic forces. From the electron's point of view, they attract because electric forces. Both of us are correct, much in the same way that we would both be correct in thinking the other's clock ticks slower than our own. The resolution to the paradox is that electrostatic and magnetic forces transform into each other as we change reference frames. It turns out that magnetism is necessary for Special Relativity and electrostatics to make any sense together.
What's interesting about this is that it occurs at extremely low velocities. I did a bit of math, and I found that if we have 10 amps (a quantity of current) going through a copper wire of diameter 1mm, then the average velocity of electrons is
Usually, when you learn Special Relativity, teachers are quick to say that it is entirely ignorable at everyday speeds. But it turns out that even at microscopic speeds, Relativity does no less than power the modern age.
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45 comments:
I've been trying for the past few weeks to find out why do currents attract and repel each other and you wouldn't belive how hard it is to find explanations that show how this is in fact a relativistic effect. These seem to be reserved only to physicists, wich is a pity since a lot of insight can be obtained from this perspective. Your post is helpful without going into a sea of equations,but I do have a question(or several): what wouhld happen if two electrons are traveling in parallel in a vacum without any positive charges in the near vecinity and with no magnetic fields (other than their own) interfering? I guess that if they were going in the same direction they would repel each other but what if they went in oposite directions? or if one is static while the other moves?
If you have two electrons in a vacuum, they will repel in all reference frames, regardless of their motion. In order for the magnetic force to fully cancel out the electric force, the electrons must be moving together at the speed of light.
Of course, the total force may have a different magnitude in different reference frames. I think that this is accounted for by time dilation or Lorentz contraction, but I'm not sure. Unfortunately, as a student, I don't yet have knowledge of the specific equations (and I agree that it's hard to find them).
Here's a thought. In the electrons' frame of reference we observe two currents of protons flowing in the same direction. The first proton current is generating a magnetic field, which exerts an atractice force on the other proton current and vice versa. No need to involve electrostatic field. To sum up: I explain the forces in both frames without Lorentz contraction. Where am I wrong? (Hint: Hall effect)
Good example!
That seems to explain the forces in both frames, but if we look closer, it is inconsistent. In the proton's frame of reference, it's only the electrons which attract each other. In the electron's frame of reference, it's only the protons which attract each other. It cannot be both. And to definitively determine which it really is, we can simply measure the Hall effect. For instance, if it's the electrons which attract each other, we would expect the electrons in each wire to be more concentrated towards the second wire.
The inconsistency will remain until we consider Relativity.
If one can say that magnetism is a relativistic effect, can’t one also say that relativity is an electromagnetic effect? In other words, instead of using the obtuse fact that the speed of light is a constant irrespective of reference frame velocity, use Maxwell’s equations to derive relativity.
Yes, that's sort of how it was actually done. See, electromagnetic waves can be derived directly from Maxwell's equations. The equations showed that these waves move at the speed of light. Scientists thought this was a little odd, since the equations don't specify any particular reference frame. At first, the popular theory was that Maxwell's equations are only accurate when we are motionless with respect to the "ether". This theory predicted that light would move slower when going against the "ethereal wind". After it became apparent that light moves at the same speed in all directions, Einstein created Special Relativity as another solution to the problem.
In brief, Special Relativity was created with electromagnetism in mind. That's why Einstein's famous paper was called "On the Electrodynamics of Moving Bodies".
So what happens when there are only two electrons moving parallel to each other?
There are no protons, so there is no excess of positive charge due to contraction.
What makes electrons still attract each other? Where magnetic force comes from?
The electrons will not attract each other. They will repel because like charges repel each other. The attraction due to magnetic force will always be smaller than the repulsion due to electric force, no matter what your reference frame is.
However, the magnetic and electric forces will cancel out to some extent. I believe this "weakened" force is equivalent to a force that has been slowed down by time dilation (though I haven't checked this mathematically).
Very informative post. My question is this: I see that the
electrons in wire A see and increased positive charge density in wire B (with the backward-moving protons). But what about the backward-moving protons in wire A by the electrons moving in A?
In the moving reference frame, two forces are acting on the backwards-moving protons. The electric force causes protons in A to be repelled by the increased positive charge density in B. The magnetic force causes protons in A to be attracted by the current in B. These two forces should cancel out exactly.
Time dilation explanation is not correct. Don't know the real one, but this can't work. Suppose you had two uncharged masses moving parallel to each other. They're attracted by gravity, just as two charged particles are repelled by charges. They have the same time dilation effect but do not require a new force to explain it. If you're going to take issue with using gravity and say it's in the general relativity spectrum, instead consider two masses connected by a spring.
...
I think you're absolutely correct. Time dilation cannot explain the magnetic force between two particles which are moving together. I will have think on this some more, because I don't know what the explanation is.
After some thought, I realized that if we have a single moving charged particle, that means that the electric and magnetic fields are changing with time. According to Maxwell's equations, changing electromagnetic fields induce more electromagnetic fields. It gets complicated really fast. Therefore, if we have two charged particles moving, the interaction is much more than a simple Biot-Savart law. I think the explanation lies somewhere in here, but I'll get back to you when I've gone through the math.
Okay, I think I've got it this time! If we've got a moving charged particle, its electric field has been lorentz contracted. Therefore, the electric field lines which go perpendicular to its motion are more dense, and therefore stronger. The boosted strength of the electric force exactly counteracts the magnetic force when both particles are moving with the same velocity.
See Wikipedia
College Physics fascinated me up to Relativity and Quanta, when the professor pretended to teach why those wires attract each other. As I now come back and try to get it, I again find ridiculous and fallacious explanations. Either physicists don't understand relativity or they just never learned to write. I've read everything I can find on the origin of magnetism and found no single bullet-proof explanation written anywhere by anyone. By the way, "string theory," (omg what bs) even though it's now about dead, revealed the obvious lack of credibility in today's Physics yet I'd still like to know why the wires with current attract.
If you have a question, you could ask it.
Aw drats. Had a long comment written, but I didn't plug in my laptop. Doh! (Or did I submit, but it failed, or ended up in a moderation queue? Hmmm... good question, as I have trouble submitting this one, and can't really remember how the previous interaction ended.)
In short: thanks for a lovely post! What confuses me about it is just the Lorentz contraction, which I expect you discussed elsewhere, based on "if you don't recall". Is Lorentz contraction "shorter distances in both directions along the axis of motion"?
Based on the solution someone mentioned to me to my question "if you travel near the speed of light, and time slows down, distance divided by time gives greater-than the speed of light" -> "distances also shorten as you speed up", I'd expect the distances are shorter ... in the frame of reference of the traveller. The protons in this case. So I still don't understand why they seem more dense from the perspective of the electron.
Anyway... that sums up my lack of understanding for now.
Drats, I'm going to re-ramble about Anonymous as well, because I enjoyed that ramble:
I'd point out "I don't understand it therefore it is ridiculous and fallacious" is precisely an example of fallacious reasoning. I can't seem to find fallacies in miller's post above, though I do not understand everything. And I also ponder: the "why" question means different things to different people: in science we're not talking about a "meaning", the scientific "why" seems to me more of a "how", in some sense. And the question is about how things fit into our current best theories, because we *must* understand their implications, to be able to use them for predictions and engineering, as well as understand how to empirically verify their validity. Especially as we go to the very big and the very small, where we don't really have any valid intuitions, since we don't live (evolve to live) on those scales.
What's bullet-proof anyway? Why does relativity do this? Why does it all exist in the first place? There's always a limit to how much we can know. In the end, the only way to know if something is bullet proof, is to shoot bullets at it. Evidence would be those bullets. And at some point, we discover new evidence, a stronger or faster or heavier bullet, that shatters our previous bullet-proof vest, and has us running about looking for a better theory.
Um, anyway, apologies for using your blog for babbling forth. ;) We all know Anonymous aint comin' back. /me is just having some fun expressing some ideas. Now if only I can keep channelling that fun into directions that mean something to someone at all times...
No, I don't have comment moderation turned on.
I had an earlier series on special relativity, and there's a relativity category.
But I don't know if I ever really explained Lorentz contraction particularly well. Basically, anything which is in motion becomes contracted. It becomes skinnier in the direction of its motion. If we change our reference frame to the electrons', then the entire wire (except for the motionless components, the electrons) will be in motion, and will therefore appear shorter, and more dense.
"distance divided by time gives greater-than the speed of light"
This is in a certain sense true. If you wanted to travel to Alpha Centauri (4 light-years from earth), you could potentially get there before you age four years. This is possible because in your frame of reference, the distance to Alpha Centauri becomes smaller than 4 light-years. However, to any Earth-observers, it would appear as if your journey took at least four years, and that you were merely aging slowly.
I was intentionally curt with the aforementioned anonymous commenter, because he was obviously talking more to himself than to anyone else.
I meant to say Proxima Centauri, not Alpha Centauri. Mixing up my stars.
WTH, I just lost a comment again.
In short, it required a bit more thought, and the penny dropped. Thanks, while these things are best explained with a couple of sketches over a cup of coffee, I can probably write a Lorentz Contraction post now — if only I had the time. ;) Much appreciated!
Ok, so we have two parallel wires with current in the same direction. If we observe this from electrons' reference frame, then, because of Lorentz contraction, both wires appear to have an overall positive charge. Therefore the two wires should repel each other. How come the reality is different?
I read somewhere that if you consider the electrostatic force as the only "real" interaction between charges, then the nonaxial force that we chosed to call magnetic force is solely and exclusively the result of the electrical interaction transformed through relativity (Lorentz) due to their movement.
(source)
I would really like to know how to explain magnetism (for example the force between two parallel wires) just with electrostatic and relativity.
Anonymous,
If you boost into a particle's own reference frame, it has zero velocity, by definition. In this frame, it cannot "see" the magnetic field, because the magnetic field only affects moving charges. Therefore, all forces on the particle must be explained by electrostatic forces, if they are to be explained at all.
But this only works if you consider each particle in its own reference frame. For instance, in your own example, if you consider the moving electrons' reference frame, then the protons in the wire are no longer stationary. Therefore, the magnetic force affects the moving protons. The magnetic force is what prevents the protons from repelling in this frame.
I would not say that magnetic forces are not "real". The magnetic force is real, unless you consider a particle's own reference frame to be the only valid reference frame. I think that rather than saying the magnetic force is caused by the electrostatic force, it is perhaps more accurate to say that it is implied by the electric force; otherwise, physics is not invariant under Lorentz transformations.
(sorry about anonymous on previous comment)
Well it’s hard to say what is real... But if we take, for example, two stationary electrons somewhere in space then the electric force between them is something that all observers would agree about. But on the other hand, they would disagree about the existence of magnetic force (a moving observer detects it whereas the stationary observer doesn’t). Do you see what bothers me? There is no inertial frame from which the electric force is not detected, but there is one where magnetic force isn’t.
Actually, in the rest frame of the wire, there is no electrostatic force, because both wires are electrically neutral.
Different observers would disagree about the electric field and magnetic fields. But they would also disagree about velocity. They would disagree about the total energy in a system. I don't care if you consider these to be "real" quantities or not, but I don't think it should bother you either way.
Well I am not really bothered about any of that stuff, I was just interested in getting some insight. Afterall we are talking about theories that try to fit reality inside a few mathematical formulations. Maybe it can be done, maybe it can't.
Well, you may also be interested in reading a bit about Quantum Electrodynamics. QED, as it is abbreviated, also takes into account the fact that the electromagnetic field is quantized in photons. I can't say I know much about it. However, I know that it causes such observable effects as the "lamb shift", a small shift in the energy spectrum of a hydrogen atom.
I suppose QED renders moot all our discussion of "what is real".
Yes qunatum mechanics is quite interesting... I've read some basics a few years ago. If I remember correctly, there are times when we treat for example electrons as if they were waves and sometimes as if if they were particles. This particle-wave duality can be (it cerainly was for me) hard to accept until you see that particles and waves are just two models or concepts imposed upon reality. As it became clear that neither of them is sufficient by itself scientists started talking about duality and some about new theories like strings.
Hmmm... You know, I got an entire category devoted to Quantum Mechanics (nothing on QED though). You're confusing it a bit with String Theory. Quantum Mechanics is the solution to the particle-wave duality which we observe (and yes, both particles and waves are just models which have been imposed on something different). String Theory is something else entirely.
Great I'll take a look at this category.
Well about string theory... Wikipedia says that it combines quantum mechanics and general relativity.
I meant to say that when wave-particle duality in quantum mechanics wasn't enough, scientists came up with a new approach - strings. That means we no longer have particles and waves but instead we have elementary strings vibrating in different manners.
Anyhow, in my opinion this approach is a bit funny. The spacetime in some of theories is 11-dimensional (which is totally acceptable) but there probably weren't 11 dimensions as they came up with the idea of strings. Then it became clear that a few dimensions are not enough and they introduced 10 or 11 dimensions and so on. It looks to me as a mathematical construct. Don't get me wrong; I still think that this theory is a great achievement in theoretical physics. But it seems to be based on really "general" assuptions. By that I mean that starting from the same assumptions, an entirely different outcome could be predicted.
Wow. That is amazing. Thanks Miller.
Actually, in the rest frame of the wire, there is no electrostatic force, because both wires are electrically neutral.
From the electrons' frame of reference, the protons are moving, undergoing Lorentz contraction; this contraction yields a net positive charge in the other wire.
From the proton's frame of reference, the electrons are moving. But you say that from their frame of reference, the wires are electrically neutral. Why do the electrons not also undergo Lorentz contraction, becoming denser and yielding a net negative charge?
Interesting stuff. I'm thinking pretty hard on this but I have a couple initial questions.
1. Does it bother you that electrons in a wire don't move but rather quantum-leap from energy level to energy level?
2. For an electron in its own reference frame we have v= 0 +/- 0. We also have an electron at its rest mass which we can call m = 1 +/- 0 in new units where one is the rest mass of an electron. With p = m*v we arrive at Delta(p)*Delta(x) = 0, and thus violate the uncertainty principle. How does that figure into your analysis? If we have a current of co-moving electrons billions of electrons would all see stationary electrons at rest mass and we'd have billions of violations, right?
Perhaps we resolve the paradox of magnetism by remembering that there is no such thing as a non-moving charge?
Third question, and thanks by the way for such an interesting idea.
You wrote:
Let's consider the wires with Relativity in mind. Of course, from the protons' motionless frame of reference, the wire is electrically neutral. But what happens if we consider the frame of reference of a moving electron? From the electron's point of view, the other wire contains a bunch of motionless electrons and a bunch of backwards-moving protons. Since the electrons and protons are moving relative to each other, we must take into account Lorentz contraction. If you don't recall, Lorentz contraction makes all distances in the direction of motion smaller. Lorentz contraction causes the protons to be closer together, more densely packed. As a result, the other wire has an overall positive charge, creating an electrostatic force. The electron will be attracted by this force.
Is there any necessity to say that electrons in current carrying wires are attracted to the other wire? The lab data verifies that the wires attract, but the wires are almost entirely protons.
Scott:
The wires must be (approximately) electrically neutral in the rest frame. The wires are in contact with the surrounding environment, and any rest-frame charge will be discharged. Lorentz contraction or not, the density of electrons will equilibrate so that the wire is electrically neutral.
Ryan:
1. No, not really :)
2. I never meant to say that all the electrons are moving at an exact speed. In fact, the random thermal motion of the electrons is much larger than their average velocity. But I use the average velocity, because that is what's relevant to this calculation.
3. Yes, we can experimentally prove that it is the electrons which attract each other, and not the protons. This creates the so-called "Hall effect". Because the electrons are gathered towards one end of the wire, there will be an observable voltage across the wire. (I hope I understood your question correctly.)
@Miller
Neato.
1. Perhaps maybe it should? As I understand the relativity/QM unification problem, there isn't really a way to reconcile the movement of relativity with the energy level transitions of QM. They're incompatible notions. Energy level transitions do not consist of an pebble shaped object moving from point A to point B as would be necessary to conceive of a special relativity based analysis.
2. Does it bother you that the average speed is not a speed of any particular electron? Doesn't that undercut your analysis of the reference frame? It seems to me that some particular electron needs to be stationary for your analysis to make sense. Wouldn't a stationary electron violate the uncertainty principle? Could it not be that the equation for magnetic force does not depend on the velocity of any particular charge, but rather the average "velocity" of a group of charges, where the average "velocity" is but an abstraction of a series of quantum leaps?
3. Is the hall effect not an attraction between electrons in a wire and protons in a wire, and not an attraction between electrons of one wire and protons of another? Also, doesn't your point about the attraction of electrons in wire one to protons in wire two rely on electrons in wire one perceiving themselves to be stationary compared to electrons in wire two? Woudln't that violate the uncertainty principle?
Ryan:
1. You are mistaken. In fact, Special Relativity and Quantum Mechanics have already been unified, though I won't pretend it's simple enough to explain in a single comment. Perhaps you were thinking of unifying General Relativity and Quantum Field Theory? That's an unrelated problem.
2. No, it doesn't bother me that the average velocity is not the velocity of any particular electron. All everyday objects are the same way. I could, in principle, do a more sophisticated calculation which takes into account the statistical uncertainty in electron velocity, but this would be a lot more work to get a very small correction, if any correction at all. So yes, we are basically considering the average velocity as a sort of abstraction, but it's a very accurate abstraction.
3. You need to clarify this question. Also, note that being stationary is not the same as having definite momentum. The electrons are neither stationary nor have a definite momentum.
Hi Miller, same Ryan, just no blogger account atm.
1. I had no idea special relativity and QM had been unified. I'd be glad to read about it from a first authority and by no means expect you to explain it. But I would greatly appreciate a point in the right direction.
2. If the velocity of some electron is not zero then the magnetic force does not disappear. The average "velocity" can be zero without any electron being stationary. If your point is about magnetic forces going to zero in a frame of reference, but that frame of reference doesn't exist, I mean, I'm just having trouble understanding why this isn't a problem for your argument.
3. Your original post said "From the electron's point of view, the other wire contains a bunch of motionless electrons and a bunch of backwards-moving protons." Your reply on this point said "The electrons are neither stationary nor have a definite momentum." I think those positions contradict each other.
4. After sleeping on it I came up with another question. If protons in a nucleus are undergoing length contraction how could they do so relative to stationary electrons? Since the electrons are in orbitals in atoms doens't the atom move as a unit? Wouldn't the wire remain neutral from the electron's frame of reference?
The wires must be (approximately) electrically neutral in the rest frame. The wires are in contact with the surrounding environment, and any rest-frame charge will be discharged.
Ok, that makes sense. So in effect, the fundamental asymmetry is between the rigid structure of the protons and the "fluid" structure of the electrons; from the electrons' frame of reference, the net positive charge caused by Lorentz contraction cannot dissipate because the protons occupy (approximately) fixed positions with respect to one another.
If I'm understanding you correctly, that's very interesting. Let me know if I'm mistaken though.
Ryan:
1. An intuitive way of explaining it: Though the particles may switch instantaneously between states, it always jumps only between states which are overlapping. That is, it stays at least partly in the same location, so it never needs to move faster than light.
If you want to really look deep into Relativistic QM, you might try looking up the Dirac equation.
2. There is a particular reference frame where the average velocity is zero. In this frame, the average magnetic force is zero. And yet, in this frame, there appears to be a net force on the electrons which must be accounted for by electrostatic forces.
3. If you consider the air around you, unless you are in a windy location, the air is stationary. And yet the molecules of air are constantly moving. It's a difference of average velocity vs individual velocity.
4. Good question. I couldn't say for sure what's happening on the atomic level, but first you should note that the electrons are not necessarily attached to particular atoms. Because there is current running through the wire, some of the electrons must be constantly jumping from atom to atom. In a moving reference frame, it will appear as if a small fraction of the atoms are missing an electron because it had jumped away.
Scott:
I'd say that it's mostly because the wire is grounded in the rest frame of the protons. On the other hand, what happens if the wires are held in a wind? I suppose then it would be grounded to a moving frame? I'd have to think about it.
I thought about this for a while and had another thought.
Suppose you are an electron. From an observer's perspective your life consists of being motionless and quantum leaping from location to location. From your perspective your life consists of watching the universe of objects quantum leap around you while doing nothing yourself.
Suppose now you're in wire 1 and an electron directly parallel to you quantum leaps linearly away from you. You are now further away from the electron, changing the electric field at your location, creating a force toward the wire - your distance to all the protons in the wire has not changed, but your distance to all the electrons in the wire has increased slightly.
Bah, never mind that last one. Not even relevant because the the test electron would by definition have to experience the same exact quantum leap somehow.
I'm not sure I understand why the frame in which the wire is grounded should matter. It simply seems to me that the protons and the electrons should both undergo Lorentz contraction in the direction of their relative velocity. In the case of the protons, this translates to higher charge density; but in the case of the electrons, you're quite right that it must not translate to higher charge density. The electrons are free to move about, so any excess of electrons quickly dissipates. So instead, it must translate to higher voltage and resistance.
Come to think of it, this might make some sense, because from the protons' perspective, the foreshortened electrons have more wire to go through, and therefore face higher resistance...
Hi Miller, Ryan's made the correct point about average velocitys of the electrons.
The relativity cause of magnetics cannot be explained by the average aka drift velocity of electrons in conductors.
You should consider the behaviour of the actual velocity of a typical electron in the conductor.
Which broadly is either fast speed (about 1000km per sec) in random direction ( fermi velocity ), or relativistic speed in direction of electron flow - about 90000km/s for 240V about which is about 0.03c
Typically, a free electron in a active conductor isn't affected by the current much. It bounces around in a local enviroment at the fermi velocity, it's net\average velocity is effectively zero. Occasionally it will be hit by current carrying particle, a high energy photon or electron. This electron then absorbs the energy of the previous current carrier, and therefore has high KE, and zips off in the direction of the current. But it doesn't get far at all before it collides with a boundary of it enviroment and passes on it's high energy, so the electron now reverted back to fermi velocity, which is non relativistic.
Most of time, the said energy is a photon rather than as KE of an electron.
This way, electrical energy is transfered quickly while the conducting electrons stay pretty much stationary. This energy transfer behaviour is typical of many common mediums, e.g. shockwave through water, water stays stationary, wave moves quickly through, water particles spend most of there time at there average random velocity and then suddenly get a high KE transfer which they pass on quickly.
With electrons in a conductor the shockwave KE is high enough for relativistic effects to be significant, magnetism.
Hope this explains magnetism for you.
Oh I got the magnitudes wrong. Should be 9000km/s for shockwave velocity and 100km/s for fermi velocity.
Some electrons will be moving much faster than the others. These electrons might even be traveling at relativistic speeds. However, I don't think this is a necessary condition for magnetism. Even in the impossible situation where all electrons were moving at constant nonrelativistic velocity, neither emitting nor absorbing electromagnetic radiation, we would still have magnetism.
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