Basic misconception about magnetism?

[jethomas3182]

Here's what I don't understand.

Start with two charges e1 and e2 on the plane of this screen, that are Y vector apart. e1 is moving at velocity V1 toward the top of the screen. e2, directly to the left, is moving at velocity V2 toward the right.

http://academic.mu.edu/phys/matthysd/web004/l0220.htm
The magnetic field produced by e1 at e2 is:

B1=(V1xY)*z1
where z1 is a fudge factor that lumps together everything I'm not currently interested in.

B1 is perpendicular to the screen.

http://en.wikipedia.org/wiki/Lorentz_force
The force on e2 is: e2*V2xB

So the magnetic force of e1 on e2 is perpendicular to B1 and to V2, so it's parallel to V1.

Meanwhile, the magnetic field produced by e2 is

B2=(V2xY)*z2 = 0

Since v2 and Y are parallel, there is no magnetic field from e2 on e1.

What happened to Newton's equal and opposite reaction? The way I have interpreted this, the system of two particles is getting a change in average velocity. Its average velocity toward the top of the screen is changing due to magnetism, and nothing else is changing to balance that. So I have misunderstood it.

 

[Dale]

Here's what I don't understand.

Start with two charges e1 and e2 on the plane of this screen, that are Y vector apart. e1 is moving at velocity V1 toward the top of the screen. e2, directly to the left, is moving at velocity V2 toward the right.

http://academic.mu.edu/phys/matthysd/web004/l0220.htm
The magnetic field produced by e1 at e2 is:

B1=(V1xY)*z1
where z1 is a fudge factor that lumps together everything I'm not currently interested in.

B1 is perpendicular to the screen.

http://en.wikipedia.org/wiki/Lorentz_force
The force on e2 is: e2*V2xB

So the magnetic force of e1 on e2 is perpendicular to B1 and to V2, so it's parallel to V1.

Meanwhile, the magnetic field produced by e2 is

B2=(V2xY)*z2 = 0

Since v2 and Y are parallel, there is no magnetic field from e2 on e1.

What happened to Newton's equal and opposite reaction? The way I have interpreted this, the system of two particles is getting a change in average velocity. Its average velocity toward the top of the screen is changing due to magnetism, and nothing else is changing to balance that. So I have misunderstood it.

Your understanding is correct.

Newton's third law is a statement of the conservation of momentum for mechanical systems. For more general systems, such as those involving fields, momentum is still conserved, but Newton's third law may be violated. That is because fields carry momentum and energy. So, although the forces on the charges are not equal and opposite the momentum of the charges + the momentum of the field is constant. The difference between the forces on the particles is exactly equal to the rate of change of the momentum of the fields.

 

[jethomas3182]

Newton's third law is a statement of the conservation of momentum for mechanical systems. For more general systems, such as those involving fields, momentum is still conserved, but Newton's third law may be violated. That is because fields carry momentum and energy. So, although the forces on the charges are not equal and opposite the momentum of the charges + the momentum of the field is constant. The difference between the forces on the particles is exactly equal to the rate of change of the momentum of the fields.

Ouch!

So, my next step must be to find out how to calculate the momentum of the field, and also how to measure it.

http://en.wikipedia.org/wiki/Momentum
Momentum is mass*velocity, so if I can find the velocity of the magnetic field I can find its mass. Woo!

http://en.wikipedia.org/wiki/Magnetic_moment
Now I'm even more confused. I was thinking of a field as a potential sort of thing -- it describes the force that would be put on a particle at any point in space, if there happened to be a particle there. Magnetic force is strange because you can set up points and velocities where one puts force on the other but not vice versa. What does it mean to say that the field itself is changed by the nonreciprocal force? Surely it's only changed if there happens to be something there to be forced.

People say that when e1 puts a magnetic force on e2 that it only changes its direction, not its velocity. That would imply that when it puts a magnetic force in direction V1 then V2 in a different direction should be reduced proportionately so the total velocity stays the same. But the formula doesn't say so.

Or maybe there's a force on e1 to slow it down to match the force on e2.

http://en.wikipedia.org/wiki/Momentum#Momentum_in_electromagnetism
This made no sense to me. Maybe something in the quantum mechanics section would have explained it, but isn't there some sort of classical understanding?

Thank you for exposing me to a new concept. If you have an explanatory link I'd like that a lot.

 

[Bill_K]

For more general systems, such as those involving fields, momentum is still conserved, but Newton's third law may be violated

Ouch, indeed. That's just not true.

This problem has a simple explanation, but it will require a little bit of special relativity, namely how E and B fields are seen in a moving rest frame.

First consider your two charges in a frame where q₂ is at rest and only q₁ moves. The forces on the two charges are

F₁ = q₁(E₂ + (v₁+v₂)/c x B₂) = q₁E₂ since B₂ = 0
F₂ = q₂(E₁ + v₂/c X B₁) = q₂E₁ since v₂ = 0

q₂E₁ and q₁E₂ are of course equal and opposite, so the forces are equal and opposite, and Newton's Third Law holds.

Now go to the original frame where q₂ moves with velocity v₂. In general the fields in a moving frame are

E' = γ(E + v/c x B)
B' = γ(B - v/c x E)

For the present case we have

E₁' = γ(E₁ - v₂/c x B₁)
E₂' = γ(E₂ - v₂/c x B₂) = γE₂
B₁' = γ(B₁ + v₂/c x E₁) = γB₁
B₂' = γ(B₂ + v_2/c x E₂) = 0

The forces are now

F₁ = q₁(E₂' + v₁/c x B₂') = γq₁(E₂ - v₂/c x B₂ + v₂/c x B₂) = γq₁E₂
F₂ = q₂(E₁' + v₂/c x B₁') = γq₂E₁

Still equal and opposite!

 

[Dale]

Ouch, indeed. That's just not true.

It is true. It is a well-known problem, and is the reason that the momentum of the fields was investigated in the first place.

While it is true that for a system of two particles you can find a frame where the forces do balance that does not save Newton's 3rd law. First, if Newton's 3rd law were a law of nature then it would apply in all frames, not just a specific frame. Second, my understanding is that with systems of 3 or more charges it can become impossible to find any frame where the forces balance. The first problem is the biggest: if (as the OP demonstrated) Newton's 3rd law is violated in any frame then it is not a law of nature.

Here is a lecture slide showing the OP's configuration and reaching the same conclusion as the OP:
http://www.uta.edu/physics/main/faculty/yuedeng/teaching/4324/Chap8_1.pdf

And another discussing this effect in a rather superficially:
http://farside.ph.utexas.edu/teaching/em/lectures/node28.html

 

[jethomas3182]

Ouch, indeed. That's just not true.

For myself, I want to try an operational view of physics. People do experiments etc and observe the results. They make up black-box theories that fit the results. I will figure that any theory which fits some of the results and has an esthetic or practical value, is not completely untrue. That way I get to explore whatever ideas look interesting and I hope not get too attached to any of them. This approach is not practical for an engineer, but it's OK for me.

This problem has a simple explanation, but it will require a little bit of special relativity, namely how E and B fields are seen in a moving rest frame.

I have an esthetic revulsion to relativity. It works in a Minkowski space where you can take two nonzero numbers A and B and multiply them together and get zero. So you can't in general do division because 0/A = B, etc. Minkowski space gives you singularities. Singularities make me nervous. And yet, if that's the way the universe works then I have to accept it.

I read a claim that special relativity can explain away all of magnetism. After all, whether a charge has a magnetic field or not depends entirely on your frame. Pick the frame which moves with the charge and there's no magnetic field there at all, and all the magnetic effects can be explained away by relativistic effects. No magnetism except the distortions that relativity provides, which can be interpreted as magnetism.

http://galileo.phys.virginia.edu/classes/252/rel_el_mag.html

To me, this only folds the unsatisfactory behavior of magnetic fields into the unsatisfactory behavior of SR. But what if it turned out that magnetism does behave as advertised, and somehow that creates SR? If magnetic behavior gives you an explanation for SR without Minkowski space that would be very nice. So to me it's worth looking at it, though the odds look low of finding something truly useful. But I'm not at all sure I understand what's going on. I may have misunderstood the experiments. And I'm not sure the experiments of the 1840's measured everything worth measuring...

 

[vanhees71]

Philosophical prejudices are never a good starting point to understand physics. First of all there are no singularities in SRT since Minkowski space time has no singularities.

Electromagnetism of moving media can only be understood properly within a relativistic framework that was the big breakthrough when Einstein wrote his famous article "On Electrodyanmiacs of Moving Media" in 1905, which has been the final interpretation of a lot of work done by physicists in the late 19th and early 20th century (Voigt, Lorentz, FitzGerald, Poincare...). It is no coincidence that relativity theory has been discovered when thinking about electromagnetics since the dynamics of electromagnetic fields cannot be explained non-relativistically since, in modern language, the em. field is a vector field of mass 0, and this makes no sense at all within Galilei-Newton space-time.

It's also clear that the magnetic field can only be tranformed away by a proper orthochronous Lorentz transformation if it is caused by uniformly moving charges, for which you can chose a reference frame, where they all are at rest.

Your cited website is not an exact relativistic solution of the problem of a DC carrying wire, but valid only in the limit of small velocities of the electrons (in the lab frame) or positive ions (in the moving-wire frame). Let's discuss this further in the Lab frame. The situation in any other inertial frame is then given by the appropriate Lorentz transformation of the four-currents and the field-strength tensor. The approximation given at the web page is, of course, a good-enough solution for everyday current because the drift velocity of the electrons in wires is indeed very small compared to the velocity of light, but for a matter of principle it's important to note that each electron in the wire does not only feel the electric forces from the external source (voltage drop along the wire) but also one from the magnetic field produced by all other electrons. The result is a static Hall-electric field in radial direction. This leads to precisely the electromagnetic field which makes the whole thing Lorentz covariant as it should be. This DC situation is pretty simple to understand since everything is about constant velocities of charges.

The original question is more complicated and not fully answered within classical electromagnetism. It boils down to the question of a completely self-consistent solution of the dynamics of a systems of point charges and electromagnetic fields, where the self-interaction of each charged particle with its own electromagnetic field has to be taken into account, and this makes a lot of trouble and leads to mass renormalization (in classical electromagnetics!). The best way to understand these issues is the modern way to formulate everything manifestly covariant, which you can find (along with the more old-fashioned formulation) in Jackson's Classical Electrodynamics textbook. A very good account is also the following paper:

V. Hnizdo, Hidden momentum and the electromagnetic mass of a charge and current carrying body, Am. J. Phys. 65, 55 (1997).

 

[Dale]

The original question is more complicated and not fully answered within classical electromagnetism. It boils down to the question of a completely self-consistent solution of the dynamics of a systems of point charges and electromagnetic fields, where the self-interaction of each charged particle with its own electromagnetic field has to be taken into account, and this makes a lot of trouble and leads to mass renormalization (in classical electromagnetics!).

As long as you stick with finite charge densities you don't need to worry about any of that and classical EM is fine. The point of the OP does not require a deep investigation into self-interaction, point charges, and renormalization.

All the OP is doing is expressing surprise about discovering a violation of Newton's 3rd law, and asking how that fits in with the physics he has learned. And the answer is that the field carries momentum and, considering the charges and the field together, momentum is conserved which is the general principle behind Newton's 3rd law.

 

[vanhees71]

Ok, maybe then I misunderstood the question, but are you sure that for a extended charge-current distribution there's no problem in the classical theory? I thought even in this case, you have trouble with radiation reactions, although there are no infinities like in the case with the self-energy of a point charge. Do you have a reference, where a fully self-consistent model of interacting macroscopic charge-current distributions + their own em. field is solved?

 

[jethomas3182]

Thank you everybody for the links and the explanations. I am fundamentally dissatisfied but that isn't your fault.

Relativity might be the only way to make this work, and I want to put it aside until I confirm that no other way is possible.

So far the only other alternative I've seen proposed is that force fields store whatever appears not to be conserved. I haven't yet managed to follow explanations of this.

It appears to me now that Dr. Deng's Chapter 8 part 2 first goes over the math about what is claimed to happen. Then it shows that momentum is not conserved, and then it states that the difference is stored in the fields. I hope I've missed the subtleties here, because what I saw was nothing like an explanation or even a description.

Operationally, a force field describes what would happen to charges at any particular location due to the known charges that create the field. (Magnetic fields could be created by pairs of magnetic monopoles instead of moving charges, but since nobody's noticed a magnetic monopole I'll leave that aside.) The mathematics for a force field describe what would hypothetically happen to hypothetical charges. And we cannot actually measure the force fields without actually observing what happens to charges in the fields (or maybe observe what happens to light in the fields, by observing the effect of the light on charges).

A mathematical description of a force field might happen to describe something that has actually distorted empty space to get the observed results. But it might not. It only predicts the observable results.

If momentum is conserved among charges, then each new force on a charge should have an equal and opposite force on some other charge. Electric fields do that. When like charges repel, both charged entities are repelled equally. Magnetic fields do not do that.

If momentum is stored in a field then the field should be changed in some way that can be observed. It should have a different effect on some charge at some time, that will balance out the effect which has not yet been balanced. The force field might become stronger or weaker in some direction, which changes its effect on other charges.

Alternatively, something could get stored in the particles. One or both might change their spin which would change their response to later forces. They might change their velocity in the old direction, in addition to getting a new velocity. Etc.

I would like it if there are actually forces on the charges which the Lorentz Force doesn't notice, which balance it out. Failing that, it would be very interesting if there are detectable effects on the forces themselves which balance the conservation laws. That might tell us something about the physical force fields that we don't already know. Failing that, maybe relativity makes it all come out right and there's no alternative that works.

 

[Dale]

So far the only other alternative I've seen proposed is that force fields store whatever appears not to be conserved. I haven't yet managed to follow explanations of this.

Here is a more in-depth presentation of EM momentum and its conservation that may be more to your liking:
http://farside.ph.utexas.edu/teaching/em/lectures/node90.html
http://farside.ph.utexas.edu/teaching/em/lectures/node91.html

And here is a similar presentation for EM energy:
http://farside.ph.utexas.edu/teaching/em/lectures/node56.html
http://farside.ph.utexas.edu/teaching/em/lectures/node78.html
http://farside.ph.utexas.edu/teaching/em/lectures/node89.html

If momentum is conserved among charges, then each new force on a charge should have an equal and opposite force on some other charge. Electric fields do that. When like charges repel, both charged entities are repelled equally. Magnetic fields do not do that.

Even electric fields do not do that in general. For accelerating charges even the electric field will generally not be equal and opposite.

If momentum is stored in a field then the field should be changed in some way that can be observed. It should have a different effect on some charge at some time, that will balance out the effect which has not yet been balanced. The force field might become stronger or weaker in some direction, which changes its effect on other charges.

Yes, this momentum in the fields may be converted back into mechanical momentum if the fields interact with other charges.

 

[harrylin]

[..]
If momentum is stored in a field then the field should be changed in some way that can be observed. It should have a different effect on some charge at some time, that will balance out the effect which has not yet been balanced. The force field might become stronger or weaker in some direction, which changes its effect on other charges.
[..]

Not sure what you mean, but I remember that Feynman gives an example of an observable effect of stored momentum in his lecture series. Citing an old post somewhere else:
In Feynman's vol.11 of his Lectures on Physics, in ch.17 is a thought experiment with a charged rotating disc with a magnetic coil around the axis. Feynman pointed out that according to standard electrodynamics the disc can be made to change angular velocity by changing the current. From that he concluded (ch.27) that there is angular momentum in the field ("there is really a momentum flow").

Thus angular momentum appears to be stored in the field, in a directly observable way.

Harald

 

[jethomas3182]

I remember that Feynman gives an example of an observable effect of stored momentum in his lecture series. Citing an old post somewhere else:
In Feynman's vol.11 of his Lectures on Physics, in ch.17 is a thought experiment with a charged rotating disc with a magnetic coil around the axis. Feynman pointed out that according to standard electrodynamics the disc can be made to change angular velocity by changing the current. From that he concluded (ch.27) that there is angular momentum in the field ("there is really a momentum flow").

Thus angular momentum appears to be stored in the field, in a directly observable way.

My question is whether there is actually any angular force on the magnetic coil in that case.

If you use an electromagnet to pick up a ton of scrap steel, there don't appear to be any unbalanced forces involved. The electromagnet lifts a ton of steel, and you can measure the mass it picks up by the force on the crane. But in that case the charges involved are all traveling in closed loops, and the ones in the steel are small closed loops, so there will be a tendency for anything unbalanced to be canceled out immediately.

I'd want to test whether there is an angular force on the coil before I believe that the angular momentum is somehow stored in the field.

Feynman was a tricky one.

 

[Dale]

If it (momentum, angular momentum, or energy) isn't stored in the field then it is not conserved in general.