4 ms·

> "even though the average drift speed of electrons in a wire usually doesn’t exceed several centimeters per minute [...]". So on average, relativistic effects

by Eduard 2y ago

> "even though the average drift speed of electrons in a wire usually doesn’t exceed several centimeters per minute [...]". So on average, relativistic effects are negligible. Then how can relativistic effects then be used in the later arguments for where magnetic effects come from?

wrs 2y ago
There are a lot of electrons in a real piece of wire. As the author says, relativistic effects are usually negligible, but in this case the math does work out.
madhadron 2y ago
Because they're not negligible. It's worth calculating it out. Length contraction produces a very slight increase in charge density of the nuclei, but there are a lot of charges and electromagnetism is very strong.
AlotOfReading 2y ago
A very small force affecting the very large number of particles in a conductor balances out to the human scale phenomena we're familiar with.
ben_w 2y ago
Two things: 1) Lots of small things that interact constructively. IIRC, the conduction electrons in copper have something like -13600 coulombs of charge per cubic centimetre, but that all cancels out with the positive ions right next to them; the magnetic fields from each electron's motion don't cancel out*, but the motion of the ions is ~ isotropic*, so they do cancel each other*. 2) The strength of the electric (and by extension magnetic) force is also strong, which in turn also helps make this easier to notice. * in the lab frame
bee_rider 2y ago
I’m slightly confused by this question. The full quote is: > The effect is negligible in most contexts unless the objects are moving at very high (“relativistic”) speeds, but electric fields are a notable exception — even though the average drift speed of electrons in a wire usually doesn’t exceed several centimeters per minute. The point of the text you’ve quoted is to set up the seeming contradiction or unexpected idea. Then the hope is that the author explains it in the rest of the article. It is fine to say that the article doesn’t follow through with the implied promise of the quote, but the deficiency is in the answering if that is the case, not the quoted text.
- ahartmetz 2y ago
  The relativistic effect matters because the electrostatic force is very, very strong. You only don't notice it much usually because positive and negative charges are almost always balanced - it takes a lot of energy to separate them because the force is so strong.
amluto 2y ago
It's been quite a while since I thought about any of this carefully, so I could be off base here, but: lcamtuf is analyzing a somewhat odd scenario, and he seems to have carefully omitted all the math. In the lab frame, there's a stationary test charge and an uncharged current-carrying wire. Ignoring relativity for a bit, the magnetic field around the wire is proportional to the current I (and I is proportional to v_drift/c). The Lorentz force [0] on the charge is q·v_charge·B, where v_charge is the velocity of the test charge, which is, drumroll please, exactly zero. There's no force, at least to first order! A current carrying wire does not attract charges of one particular charge, nor could it without rather odd symmetry violations. If you want to analyze this in a moving frame, even at first order in the boost velocity v_boost (the velocity of the moving frame, which does not need to equal v_drift), it gets more complicated: the test charge is moving, so it will experience a Lorentz force. But the fields themselves transform too, like this: https://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity#Non-relativistic_approximations https://en.wikipedia.org/wiki/Classical_electromagnetism_and... And, of course, to first order everything will cancel out and there is still no force! (Otherwise there would be a massive inconsistency: a current-carrying wire would attract negative charges from the perspective of someone walking one way and would repel them from the perspective of someone walking the other way, and they would end up rather dramatically disagreeing with each other, and special relativity is not that weird.) The actual phenomenon that lcamtuf is talking about only shows up at higher order in the velocities involved, and one might argue that relativity does affect electromagnetic fields even at first order (see that link above!). FWIW, a useful heuristic for thinking about relativity is that the simultaneity-violating term in the Lorentz transformation (the -γβx term in t') implies that almost everyone's intuition for what's going on with multiple moving objects that are not in the same place (like multiple different electrons in a wire from the perspective of someone moving relative to the wire) is likely to be wrong. I haven't considered lcamtuf's explanation that carefully, and I'm not entirely convinced that it's right. But even that is a higher order effect, and lcamtuf's explanation seems like it has to be wrong even at first order! After all, at first order, there is no force on the test charge. But in a moving frame, as above, there is a Lorentz force on the charge unless the magnetic field disappears entirely, and v_drift is not going to be the boost velocity that makes the B field disappear. So I would apply large grains of salt to this model. [0] Wow, Lorentz got his last name all of this stuff!

by Eduard 2y ago

wrs 2y ago

There are a lot of electrons in a real piece of wire. As the author says, relativistic effects are usually negligible, but in this case the math does work out.

madhadron 2y ago

Because they're not negligible. It's worth calculating it out. Length contraction produces a very slight increase in charge density of the nuclei, but there are a lot of charges and electromagnetism is very strong.

AlotOfReading 2y ago

A very small force affecting the very large number of particles in a conductor balances out to the human scale phenomena we're familiar with.

ben_w 2y ago

Two things: 1) Lots of small things that interact constructively. IIRC, the conduction electrons in copper have something like -13600 coulombs of charge per cubic centimetre, but that all cancels out with the positive ions right next to them; the magnetic fields from each electron's motion don't cancel out*, but the motion of the ions is ~ isotropic*, so they do cancel each other*. 2) The strength of the electric (and by extension magnetic) force is also strong, which in turn also helps make this easier to notice. * in the lab frame

bee_rider 2y ago

I’m slightly confused by this question. The full quote is: > The effect is negligible in most contexts unless the objects are moving at very high (“relativistic”) speeds, but electric fields are a notable exception — even though the average drift speed of electrons in a wire usually doesn’t exceed several centimeters per minute. The point of the text you’ve quoted is to set up the seeming contradiction or unexpected idea. Then the hope is that the author explains it in the rest of the article. It is fine to say that the article doesn’t follow through with the implied promise of the quote, but the deficiency is in the answering if that is the case, not the quoted text.

ahartmetz 2y ago
The relativistic effect matters because the electrostatic force is very, very strong. You only don't notice it much usually because positive and negative charges are almost always balanced - it takes a lot of energy to separate them because the force is so strong.

amluto 2y ago

It's been quite a while since I thought about any of this carefully, so I could be off base here, but: lcamtuf is analyzing a somewhat odd scenario, and he seems to have carefully omitted all the math. In the lab frame, there's a stationary test charge and an uncharged current-carrying wire. Ignoring relativity for a bit, the magnetic field around the wire is proportional to the current I (and I is proportional to v_drift/c). The Lorentz force [0] on the charge is q·v_charge·B, where v_charge is the velocity of the test charge, which is, drumroll please, exactly zero. There's no force, at least to first order! A current carrying wire does not attract charges of one particular charge, nor could it without rather odd symmetry violations. If you want to analyze this in a moving frame, even at first order in the boost velocity v_boost (the velocity of the moving frame, which does not need to equal v_drift), it gets more complicated: the test charge is moving, so it will experience a Lorentz force. But the fields themselves transform too, like this: https://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity#Non-relativistic_approximations https://en.wikipedia.org/wiki/Classical_electromagnetism_and... And, of course, to first order everything will cancel out and there is still no force! (Otherwise there would be a massive inconsistency: a current-carrying wire would attract negative charges from the perspective of someone walking one way and would repel them from the perspective of someone walking the other way, and they would end up rather dramatically disagreeing with each other, and special relativity is not that weird.) The actual phenomenon that lcamtuf is talking about only shows up at higher order in the velocities involved, and one might argue that relativity does affect electromagnetic fields even at first order (see that link above!). FWIW, a useful heuristic for thinking about relativity is that the simultaneity-violating term in the Lorentz transformation (the -γβx term in t') implies that almost everyone's intuition for what's going on with multiple moving objects that are not in the same place (like multiple different electrons in a wire from the perspective of someone moving relative to the wire) is likely to be wrong. I haven't considered lcamtuf's explanation that carefully, and I'm not entirely convinced that it's right. But even that is a higher order effect, and lcamtuf's explanation seems like it has to be wrong even at first order! After all, at first order, there is no force on the test charge. But in a moving frame, as above, there is a Lorentz force on the charge unless the magnetic field disappears entirely, and v_drift is not going to be the boost velocity that makes the B field disappear. So I would apply large grains of salt to this model. [0] Wow, Lorentz got his last name all of this stuff!