Monday, December 3, 2007

Relativity + Electrostatics = Magnetism

[Update: Given the popularity of this post, I made a clearer rewrite.]

There is another dramatic success of Special Relativity that is virtually unknown among laymen. Special Relativity is what causes magnetism.

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 9.4*10-5 [edit: corrected math] 3.3*10-5 meters per second. That doesn't even begin to approach the speed of light (3*108 meters per second). And yet, if you place two of these wires next to each other, there will actually be a measurable magnetic force. Not a significant force (about the weight of a paperclip per foot of wire), but not negligible either. We rely on this force for electric motors, generators, and countless other applications.

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.


Electrical engineer with a taste for physiscs said...

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?

miller said...

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).

Skeptic said...

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)

miller said...

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.

marlowgs said...

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.

miller said...

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".

Anonymous said...

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?

miller said...

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).

anonymous said...

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?

miller said...

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.

Anonymous said...

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.

miller said...

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.

miller said...

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.

miller said...

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

Anonymous said...

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.

miller said...

If you have a question, you could ask it.

Hugo said...

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...

miller said...

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.

miller said...

I meant to say Proxima Centauri, not Alpha Centauri. Mixing up my stars.

Hugo said...

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!

Anonymous said...

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.
I would really like to know how to explain magnetism (for example the force between two parallel wires) just with electrostatic and relativity.

miller said...

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.

pini said...

(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.

miller said...

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.

pini said...

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.

miller said...

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".

pini said...

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.

miller said...

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.

pini said...

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.

Jeo said...

Wow. That is amazing. Thanks Miller.

Scott said...

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?

Ryan said...

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?

Ryan said...

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.

miller said...

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.

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.)

Ryan said...



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?

miller said...

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.

Ryan said...

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?

Scott said...

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.

miller said...

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.

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.

Ryan said...

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.

Ryan said...

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.

Scott said...

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...

susan said...

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.

susan said...

Oh I got the magnitudes wrong. Should be 9000km/s for shockwave velocity and 100km/s for fermi velocity.

miller said...

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.

bob said...

"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."

This doesn't really sit well with me. Why would the electron density equilibrate in one reference frame and not another? It seems to violate the principle of relativity.

I think the only rest frame in which the wires are neutral is the one in which electrons and protons have the same speed in opposite directions, because in this case the length contraction is equal. Isn't this the only reference frame where the force between the wires can be seen as entirely magnetic?

miller said...

The mechanism I proposed for the wire to reach equilibrium is contact with the environment. If you look at a frame where the environment is moving, then the charge doesn't necessarily equilibriate because it's affected by the magnetic force.

It does not violate the principle of relativity to say that there is a special frame where the environment is at rest.

bob said...

Would you agree that the wires are electrically neutral in the frame where the electons and protons are going the same speed? What about in frames somewhere in between this one and the proton reference frame?

miller said...

No, they're not neutral in that frame, unless there is some other mechanism to equilibriate the electric charge of the wire.

bob said...

I think my confusion may be because I'm used to seeing the explanation more like this:

In this explanation, they start with neutral wires in the reference frame where electrons and protons are moving in opposite directions at the same speed. In this reference frame the length contraction for both protons and electrons are the same, since their speed is the same. So the density of electrons and protons are equal, and the wires are neutral.

Then if you look at either the proton or the electron's frame, the attractive force can be explained by an electric field.

miller said...

From the site you linked:
"In fact, in reality only the negative charges move. However, it is convenient to think of both the positive and negative charges moving in opposite directions as shown."

Well there's the source of our confusion. Their explanation and my explanation have different pedagogical choices; we've simplified the same problem in different ways.

If only the electrons are moving while the ions are staying still, the electrons will indeed experience more length contraction than the ions. But length contraction is just the ratio between the relativistic length and the proper length. As we increase the current in the wire, that doesn't change the density of electrons. It doesn't change the relativistic length between electrons. Instead, it just increases the proper length between electrons. (The "proper length" is the distance between electrons in the electrons' rest frame.)

bob said...

Okay, thanks. That makes sense now.

What I'm finding interesting is that in your case, it's pretty easy to show that the forces on the protons cancel in the electrons' reference frame. But in the case I linked to, I don't think that the forces cancel. Or if they do, the math is more complicated and I can't figure it out.

esther said...

hi Miller, I've been trying to figure out the connection between electricity and magnetism and your post explains it nicely. I'm wondering if you know of any sources that elaborate on the concept without going too much into too many technicalities? Like another commenter, I find there are few resources that treat magnetism, electrostatics, and relativity at a level suitable for a lay person interested in physics.

Thanks! I realize this is an old post but came upon your blog and found several of your other posts interesting and insightful as well.

miller said...

There are many other layman-level expositions of the same concept on the web (example), but most of them (mine included) are more or less the same. It's also difficult to find elaboration without technicalities, because at some point, to further elaborate is to explain the technicalities.

You're welcome to throw me any questions.

Anonymous said...

Hi Miller. Your explanation of the magnetism as a relativistic phenomenon was wonderfull. Is it an objective and generally accepted explanation??? (I mean it's not just your personal opinion... Right???...)

miller said...

Anonymous, it's a pretty standard explanation, and you can find very similar things elsewhere on the internet. It's based on well-founded physics over the last century.

But if we're speaking precisely, it's not really an "explanation" of magnetism. It just shows that in order for electric forces to make sense in relativity, magnetism must exist.

newuniverse said...

Very good explanation but you might want to correct it in terms of solid state physics. There would be no proton flow. This phenomenon would be better explained by incorporating electron 'hole' conduction.

miller said...


Yes, conduction is actually much more complicated than I've made it out to be in this post (and in my subsequent comments). In some conductors, electrical current is best described as electron transport; in others it's best described as hole transport; others have a combination of both.

In the case of hole transport, the conductor acts as if positively charged particles are carrying the current.

concretedonkey said...

There's one thing that still bothers me about this. Sorry if it was already explained. I didn't see it (or didn't get it).

So an electron is moving in wire A, and the contraction of the nuclei in wire B results in a higher density of positive charge, resulting in a net attraction to that wire. But what about the contraction of wire A? Doesn't the attraction to that wire increase too?

I figure it has to have something to do with the proximity, or the way the electron actually moves in the wire. The only simple explanation I can think of is that since the electron is "spread about" the nuclei of wire A as it passes along, the additional attraction to those nuclei produces no net force perpendicular to the wire. But I'm not sure that makes sense.

miller said...


In wire A, the protons do contract, leading to a higher density of positive charge. And yes, this positive charge would attract the electrons in wire A. But of course, they're already attracted to the wire, just as any electron is attracted to its own atom's nucleus.

This only causes internal motion, keeping the different components of the wire stuck together. It does not prevent overall motion of the entire wire.

Anonymous said...

Considering wires complicates things; it should be possible to explain magnetism by considering two isolated electrons side by side if relativity is indeed the explanation.

Let's start by comparing the situation for uncharged particles. Consider two stationary uncharged masses next to each other. They will begin moving towards each otherr by the force of gravity.

Now consider a frame of reference relative to which the two masses are initially moving side by side parallel to each other. There will be three relativistic effects:

(1) Time dilation should slow down the apparent rate at which the masses are moving towards each other.

(2) Relativistic increase in inertial mass should slow down the apparent rate at which the masses are moving towards each other.

(3) Relativistic increase in gravitational mass should speed up the apparent rate at which the masses are moving towards each other.

Effects (2) and (3) should cancel each other out.

Now consider two electrons next to each other. Effects (1) and (2) should still hold (except they are moving apart instead of together). However, there is no analogue of effect (3) for the electrical force. Magnetism operates in the wrong direction to cancel out (2). This seems to be a paradox because a clock moving with the electrons suggests they should only show the time dilation effect on their motion away from each other.

miller said...


Considering two charged particles doesn't really simplify over the two wires, because it introduces additional effects having to do with changing electric and magnetic fields. It's not at all straightforward, but the end result is that the electric field is stronger in the moving frame, thus resolving the paradox.

Anonymous said...

There actually *is* a secondary force to gravity, which is the gravitational analog of magnetism, usually called gravitomagnetism. It is part of an approximation to general relativity. The secondary gravity force is weaker than gravity by a factor of v/c, and gravity is already weak, so it's hard to notice the effect of this second gravity.

Two "currents" of matter in the same direction will repel by gravmag in addition to attraction by gravity, while two currents of matter moving in opposite direction will attract by gravmag added onto gravity.

There are no negative masses, however, so you can't have zero mass matterflow in the same way as you can have a neutrally charged wire.

Gravity also doesn't really act on mass, but on stress-energy, of which energy density is a part, of which mass is a part. And with full relativity gravity isn't a force at all, but a pseudoforce byproduct of the coordinate system.

But we could, if we wanted to, use a gravitational analog of the magnetic force in lieu of time dilation and lorentz contraction acting on masses, just as we usually talk about "magnetism" instead of "relativistic electricity".

Anonymous said...

Good post, but how is relativity define the magnetic force on a charged particle mooving perpendicular to direction of current.

miller said...

If you have a charged particle which is moving towards or away from the wire, you can shift to a reference frame where the particle is motionless. In such a frame, the only force acting on the particle is the electric force (because the magnetic force is proportional to the speed of the particle, which is zero). However, calculating the electric force in this frame would be very difficult because there isn't much symmetry in the problem.

boban said...

That means the lawrence contraction is not enough to explain. In reality it cannot explain anything except strict parallel motion.The resemblance with length contracted nothing but a factor of luck. and the length contraction is a reality, it happen with respect to gravitational reference, not with respect to the observer. and again it is questionable that the length contraction increases the charge density. if so it should increase charge when electrons are circulated inside a ring.

miller said...

boban, that's not the correct conclusion to leap to.

In the case where the charged particle is moving towards the wire, and we boost to the frame where the particle is motionless, the wire as a whole is no longer motionless. That means that we have to worry about changing magnetic and electric fields, in addition to Lorentz contraction. So you're correct that Lorentz contraction is not sufficient to explain all magnetic forces. You still need Maxwell's equations, and the rest of the Lorentz transformation. For simplicity's sake, I stick to the example that can be understood just by understanding Lorentz contraction.

When there is a ring of current, then Lorentz contraction would induce opposite charges on opposite sides of the ring (because the current is in opposite directions on opposite sides of the ring).

boban said...

miller, it is true that this is simple and popular.
but I've strong disbelief in this `explanation'.

here the protons are moving with respect to test charge, and electrons are motionless, then the moving protons will create enough complexity to this experiment like all other perpendicular motion, isn't it?, if that is considered the lawrence contraction may not be required solve this issue.

secondly, the current in the ring produce opposite charges on both sides require that this is an observation of moving observer(moving with respect to ring), there is no need for stationary observer to see opposite charge because of only opposite velocity.

miller said...

Bonan, your objection isn't clear enough to say anything. But I should say that I don't really consider this an "explanation". Rather, it's an example of internal consistency in the physical laws.

Do the math, Please! said...


I've posted a more extensive analysis for 2 moving point charges on:

To my opinion it is not the length contraction but the time dilution... (and the mass)...
Please review my article!

Clark said...

I just wanted to say thanks for a clear explaination for the case of two point charges in relative motion. It was clear how moving charge populations could increase through Lorentz contraction but not the case of a single point charge. Increase of the the field intensity itself for the moving particle through Lorentz contraction is an elegant visualization/explaination. This would seem to apply to individual charges in moving current also - do you think that in addition to population increases of charged particles through contraction, the field intensity of each moving particle increases too? I'm thinking not - that it's just two ways of viewing the same thing, but they do appear on the surface to be separate considerations....

miller said...

Hi Clark,
So there are two effects:
1. In the case of a population of moving charges, the density increases through lorentz contraction.
2. In the case of a single moving charge, the electric field increases through lorentz contraction.

You're asking, do these two effects stack together?

No, they do not. In the case of a single moving charge, the electric field gets stronger from lorentz contraction, but only in the direction perpendicular to motion. If you venture some distance parallel to the velocity, lorentz contraction actually makes the electric field weaker. When we consider a wire, the wire runs along the direction of motion, so the field from some parts of the wire will be diminished by effect 2. The net effect is only effect 1.

Clark said...

To clarify the sceanario, say we have a point charge at rest and and another point charge moving past (relatively speaking) in a linear trajectory. Lorentz contraction of the moving particle will increase it's field intensity (as percieved by the stationary particle) and create a force on the particle at rest. Now imagine a train of moving particles on the same trajectory rather than single one (no wire in this thought experiment). Lorentz contraction will make the number of moving charged particles per unit length increase. THe question is whether this effect is additive to the field intensity increase previously noted per moving particle....

miller said...

No, it is not additive. Consider this figure showing the electric field of a moving charged particle. The electric field is increased in directions perpendicular to the motion, but decreased in other directions. If you have a train of moving particles, then the electric field of different particles will be strengthened or weakened. The net effect is that of the increased density of particles.

valjok said...

> 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.

Why do you see Lorenz contraction (and electric field) only from electron observer but no Lorenz contraction (and electric neutrality) from the proton observer? Why is the asymmetry? Why do you ignore that there are forward-moving electrons from the protons' point of view? Don't you know that the whole idea of relativity is that we cannot distinguish between observers who is moving of them and who is at rest?

miller said...

In my rewrite linked at the top, I explain how the electrons are Lorentz contracted from the proton's frame of reference, and become un-contracted when we switch frames.

humoshi said...

"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."

I'm not sure your explanation here makes sense. Why doesn't the moving current cause a net negative charge density and repel a stationary electron? You are saying excess charge will be discharged, but this wouldn't happen if it was electrically insulated, such as a perfect vacuum. Yet, even if it was insulated in this way, observation tells us it would still not repel the electron.

miller said...

"Why doesn't the moving current cause a net negative charge density and repel a stationary electron? You are saying excess charge will be discharged, but this wouldn't happen if it was electrically insulated, such as a perfect vacuum. Yet, even if it was insulated in this way, observation tells us it would still not repel the electron."

I do photoemission research, which means looking at a lot of electrons in vacuum. I can attest that objects do in fact get charged if not properly grounded, and that this deflects the electron paths. So yes, to do this hypothetical experiment, you need to ground the wires, or the static charge will overwhelm the magnetic force.

Although what I said years ago may have been a bit misleading. I made it sound like the wire is grounded by physical contact with the air, but it's more effective to ground it by electrical contact with a larger object.

But for illustration's sake, suppose that you have a wire in vacuum that is electrically neutral (but not grounded). Suppose you start an electrical current through the loop. The wire should remain electrically neutral in its rest frame. In a moving frame, the total charge on the wire loop is still zero, but it now has an electric dipole moment.

humoshi said...

Thanks for the quick response. The physics here is definitely over my head, but my friends and I were discussing this and that led me here.

You wrote, "The wire should remain electrically neutral in its rest frame. In a moving frame, the total charge on the wire loop is still zero, but it now has an electric dipole moment."

I guess the first sentence is what I don't quite understand. If there is no current running through the wire, it remains neutral. Now we hook it up to a voltage source, and the current begins to flow. Wouldn't there be a length contraction, and increase in charge density, of the the distance between the electrons?

It seems you are saying it is neutral with no current and it is neutral with current.

But then when we increase the velocity of our stationary charge to match the current, the relative motion of the positive charges in the causes a length contraction and increase in positive charge.

Why does relative motion in the one frame create a change in net charge density but not the other?

miller said...

The loop has to remain electrically neutral, because no charge has moved into or out of the loop. So in all frames, it must be either neutral, or an electric dipole. The electrons will move around until they are evenly distributed in the rest frame, because that's the lowest energy configuration.

The infinite wire may be confusing from a charge conservation perspective, because there are an infinite number of electrons. In the real world, the wire would have to eventually loop back on itself. So if it appears there are fewer electrons in a moving frame, that's because they've moved to the other side of the loop.

Will Flannery said...

I just chanced on this .... there is a relatively :) simple explanation of the relativly-magnetism phenomena at ...