According to the Biot Savart Law and the Lorentz Force Law, if electron 1 is moving directly at electron 2, and 2 is moving at right angles to 1, then the 2nd particle will exert a magnetic force on the first particle with no instantaneous concurrent magnetic reaction force from 1.
Does this mean that Newton’s third law does not always hold?
Newton’s laws break down on a quantum scale. But for most everyday situations they are still true.
You can’t trust Newton when you’re dealing with quantum stuff or really fast speeds or really big masses.
Blalron is correct but to expand on it a bit…
Newton’s laws aren’t detailed enough to apply at the extremes (quantum or relativistic). They work just great for day to day stuff. A neat one I saw was using Relativity to calculate something simple (IIRC we were figuring how to orbit a satellite in class). Relativity worked just great but the calculations were a bit thick. The professor then simplified the relativistic equations and poof…out popped Newton’s equations.
Kinda cool.
But Biot-Savart and Lorentz Force are formulae of classical electrodynamics. Also, they don’t just apply to electrons, but also to larger charges. Is it just that this is a macroscopic effect of quantum goings-on?
Exactly. This question doesn’t have anything to do with quantum mechanics or relativity. The two particles could be charged up baseballs moving at meters per second, but there still wouldn’t be any instantaneous reaction force from baseball 1. So for every action there is not necessarily a reaction?
:rolleyes:
Biot-Savart’s law only holds for a steady current, which the situation described is not.
Electrodynamics is much more complicated than electrostatics.
No, but it’s a good question. The Biot-Savart law applies to “current elements” (which are different from moving electrons) in a circuit with steady current flow. It’s true that the magnetic force from one current element on another may not be “equal and opposite” to the reaction force of the second on the first. However, when you sum over the forces acting on all current elements in a circuit (which you have to do to find the actual forces acting on the two circuits) the two forces are equal and opposite as they should be. This should be discussed in a good EM text (Griffith’s book, for example).
The Biot-Savart law only applies to steady currents (which single electrons are not), so you can’t apply it directly in this case. Newton’s third law (conservation of momentum) is not violated by Maxwell’s equations. Achernar is right, though, that this is not a “quantum” phenomenon; it’s just a question of dealing with nonsteady current flow, which is beyond the scope of Biot-Savart.
[sup][sub]Caveat: Momentum can also be carried away by the electromagnetic fields, so the sum of the momenta of just the two electrons need not be conserved.[/sub][/sup]
Thanks, Omphaloskeptic, for a) having the patience to write a more detailed explanation than I, and b) writing a long enough post that mine appeared first, as it would have looked pretty stupid coming under yours. 
Heh.
From Goldstein’s Classical Mechanics third edition, page 8.
I will again recommend David Griffith’s EM book (Introduction to Electrodynamics) as a very good EM text. You should read section 7.5, which talks about exactly this problem. Or, if you don’t want to wade through the algebra, get out a magnifying glass and read my caveat above; the “missing momentum” is carried in the fields.
(He says that this problem violates Newton’s third law, which I think is a bit alarmist. At any rate momentum is still conserved (the law underlying Newton’s third); whether the third law is violated or not depends on how literally you want to interpret it in the context of electromagnetism, which (obviously) wasn’t what Newton was talking about at the time.)
I have already finished Griffith’s text, and I certainly never doubted that momentum was conserved. Both Griffith and Goldstein agree that the mechanical momentum plus the momentum stored in the fields is a conserved quantity.
On the other hand I’m beginning to think that that Newton’s third law only applies in the case of contact" interactions, or instantaneous action-at-a-distance.
The last post was by Sacroiliac not Ring. I better start being more careful.
If you’re thinking only of the forces on the two electrons, this is true. But since the electromagnetic fields can also carry momentum, the electrons aren’t the only things in the picture. It’s kind of hard to point to a nebulous “field” and say that this is what’s pushing on the electron, but that’s the only way to rescue Newton’s Third Law in these cases.
It’s something like pulling on an object attached to a rope. Since the rope has mass, not all the force will be transmitted through it to accelerate the body at its other end; some of it will do work accelerating the rope instead. If you just look at the forces at the two ends of the rope, they’re different; some momentum has gone into the “field” carrying the force.
(To borrow from an Einstein quote, the only difference is that there is no rope.)
I fail to visualise this. How is it possible that point A can move directly at (towards) point B while point B moves at right angle to point A?
UrbanRanger-the OP refers to an instantaneous state of the system. Electron A and electron B are moving in such a way that, at some particular point in time, the velocity vector of A points directly at the current location of B, while the velocity vector of B is at a right angle to the velocity vector of A.
Here’s a simple visual: imagine that both electrons are moving on straight-line paths that intersect with a right-angle at point X. (They couldn’t actually be moving in straight lines in the described situation, but anyway …) Both electrons are moving toward X, but their starting positions and speeds are such that electron B gets there first. Now, at the instant B arrives at X, A is moving directly towards point X (and therefore B), while B continues to be moving along its own orthogonal path.
OK?
This sort of stuff fascinates to such an extent that I’m willing to risk ridicule by asking for the sort of explication real physicists, scientists, and mathematicians just hate.
Can someone supply a much simpler, “visual” model of what’s happening in a situation like the one cited–ie, how, moment by moment, these laws are altering what one would expect to occur?
You have these two balls gliding along, frictionless space, steady pace. Just for simplicity, maybe we can visualize the field around them (electrostatic? electromagnetic?) as a sort of spherical halo around each ball with a definite surface, not something that just attenuates out to infinity.
Then what happens? Where, in the halo, is the missing momentum, and where does it go? Does the halo “have” the momentum itself, or does it somehow convey it without having it?
Is this sort of effect true of the other three fundamental forces?
Does this have to do with the fact that, apres Einstein, we cannot speak in broad and general terms about simultaneity, but rather about individual spacetime points where some denoted interaction takes place?
Basically looking for a visualizable, non-numerical model. Heh, so who isn’t.
I’m trying to figure out a simple model to explain where the momentum goes, but I’ll reply to a couple of your other questions now.
The only way we have of measuring the momentum of an object is, basically, by letting it run into something else and watching the recoil. So “conveying” momentum (applying force) and “having” momentum are, empirically, indistinguishable. Photons cause a recoil when absorbed (or emitted), so we can say that they carry momentum. If we say this we get the added bonus that now momentum is conserved, and physicists just love a good conservation law.
This is true of the other fundamental forces (and, as long as momentum conservation remains a working law, must remain true for any as-yet-undiscovered force too), since force is just a momentum transfer, F = dp/dt. It is certainly true that some strange, nonlocal force law would run into problems with momentum conservation in a relativistic theory, and this is one reason physicists like to stick with local-interaction theories.
(Oh, I think I have to disagree with your assertion that this is the sort of question physicists hate. Actually, in my experience physicists love to think, and talk, intuitively about physics. The math is often just a way of “proving” what physicists think, from their intuition, that they already know. The math is important, though, because sometimes the intuition is wrong.)