Action = reaction and lorenz force

[0xDEADBEEF]

Newtons third law states that there is a counter force to every force. Unfortunately this doesn't seem to work for moving point charges. The Coulomb force cancels out but
the B-Field of a moving point charge is:
$$\mathbf{B}=\frac{\mu_0}{4\pi}q \frac{\mathbf{v}\times\mathbf{r}}{\left|r\right|^3}$$
And the Lorenz force is
$$\mathbf{F}=q\, \mathbf{v}\times \mathbf{B}$$

Lets assume that the two charges have velocities $\mathbf{v}_1,\mathbf{v}_1$

Therefore the two Lorenz forces are
$$\mathbf{F}_1=k(r)\, \mathbf{v}_1 \times (\mathbf{v}_2 \times\mathbf{r})$$
and
$$\mathbf{F}_2=k(r)\, \mathbf{v}_2 \times (\mathbf{v}_1 \times (- \mathbf{r}))$$
Due to the Jacobi identity the sum of the two forces is not zero
$$\mathbf{F}_1+\mathbf{F}_2=- k(r)\, \mathbf{r}\times(\mathbf{v}_1\times \mathbf{v}_2)$$

What is the solution here? The Pointing vector? Relativity? I think that the basic formulas must be correct for slowly moving charges. So it shouldn't be due to non linear trajectories, neglected acceleration or some such thing.

 

[Simon Bridge]

Solve each force in the rest-frame of the other charge.

 

[jtbell]

Indeed, Newton's Third Law does not apply to the Lorentz force on charges, in general.

Conservation of momentum (from which the Third Law can can be derived for mechanical forces) is more general. We can make conservation of momentum work in electrodynamics by associating a momentum density with the electromagnetic field: $\vec g = \epsilon_0 \vec E \times \vec B$. The sum of the mechanical momenta of the particles, $\vec p_i$, and the volume integral of $\vec g$ is conserved in a closed system.

 

[Andrew Mason]

Indeed, Newton's Third Law does not apply to the Lorentz force on charges, in general.

Conservation of momentum (from which the Third Law can can be derived for mechanical forces) is more general. We can make conservation of momentum work in electrodynamics by associating a momentum density with the electromagnetic field: $\vec g = \epsilon_0 \vec E \times \vec B$. The sum of the mechanical momenta of the particles, $\vec p_i$, and the volume integral of $\vec g$ is conserved in a closed system.

Interesting. If one interprets the third law as a statement about the reciprocity of changes in motion rather than reciprocity of forces there would be no third law problem. The reciprocity of forces holds only if the interacting bodies measure time the same.

Physics texts often downplay Newton's "action/reaction" version of the third law and state it in terms of forces. Newton uses the term "force" (Latin: vis) in his first two laws but in the third he refers to "action" as in: "the mutual actions of two bodies upon each other are always equal, and directed to contrary parts." In his explanation of the law Newton talks about equal and opposed changes in motion:

"If a body impinges upon another, and by its force change the motion of the other, that body also (because of the equality of, the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of bodies; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes of the velocities made towards contrary parts are reciprocally proportional to the bodies."​

AM

 

[Dale]

Interesting. If one interprets the third law as a statement about the reciprocity of changes in motion rather than reciprocity of forces there would be no third law problem.

No, there would still be a problem since the change in momentum of one charge is not equal and opposite the change in momentum of the other. The key is to consider both the charges and the field. Then, to rescue Newton's third law you still need to express it in terms of forces since the field is interacting with both bodies.

In his explanation of the law Newton talks about equal and opposed changes in motion:

Only after talking about equal and opposite forces twice and only with the caveat that while talking about changes in momentum he is specifically referring to a pair of isolated bodies with no other interaction forces. Your interpretation of Newtons third law is unsupported by any reference that I have seen.

 

[stevendaryl]

No, there would still be a problem since the change in momentum of one charge is not equal and opposite the change in momentum of the other. The key is to consider both the charges and the field. Then, to rescue Newton's third law you still need to express it in terms of forces since the field is interacting with both bodies.

Well, Newton's laws really were not made to take into account momentum in fields. Interpreting his 3rd law in terms of forces, you need to consider a "force" on the field due to the particles. I guess you could do that, but it's a very different notion of "force'; it's certainly not equal to $m a$.

 

[sophiecentaur]

Am I reading this correctly? Are we saying that momentum, here, is not conserved? Is it not true to say that the force on the deflector mechanism (magnetic or electric) of a crt is equal and opposite to the force that deflects the electron beam?

 

[stevendaryl]

Am I reading this correctly? Are we saying that momentum, here, is not conserved? Is it not true to say that the force on the deflector mechanism (magnetic or electric) of a crt is equal and opposite to the force that deflects the electron beam?

Momentum is certainly conserved in electrodynamics, but only if you take into account momentum in the field itself. It's not conserved if you only consider the momenta of particles.

 

[sophiecentaur]

Momentum is certainly conserved in electrodynamics, but only if you take into account momentum in the field itself. It's not conserved if you only consider the momenta of particles.

But wouldn't that be a daft thing to do? It would be as unrealistic as to ignore the effect of (and on) a trampoline with two little boys fighting on it.

 

[jtbell]

Am I reading this correctly? Are we saying that momentum, here, is not conserved? Is it not true to say that the force on the deflector mechanism (magnetic or electric) of a crt is equal and opposite to the force that deflects the electron beam?

With electric forces, there's no problem. With magnetic forces, on the other hand...

Consider charged particle A traveling along the x-axis towards the origin (but not located at the origin), and charged particle B traveling along the y-axis and passing through the origin at a certain time. The magnetic field produced by A is zero along the x-axis, so there is no magnetic force on B at that instant. The magnetic field produced by B is non-zero at the location of A, so there is a magnetic force on A.

 

[stevendaryl]

But wouldn't that be a daft thing to do? It would be as unrealistic as to ignore the effect of (and on) a trampoline with two little boys fighting on it.

Yes, it would be a daft thing to do. The question was whether we can interpret conservation of momentum in electrodynamics as consistent with Newton's laws of motion. I think that it's a mistake to even try. The modern view of conservation of momentum is more general than what Newton considered.

 

[Dadface]

B can be defined with reference to the force on a current carrying conductor in a magnetic field and it is standard practise in A level physics to measure this force by measuring the equal and opposite force on the magnet arrangement. In other words, in this example, it is the deflecting system that can be considered as the particle that the moving charges interact with. Is this not basically the same as the example given by sophiecentaur in post 7?

 

[stevendaryl]

On the other hand, Feynman developed a theory of electrodynamics in which there were no extra degrees of freedom in the electromagnetic field; the theory could be interpreted as a pure particle theory with binary particle interactions (although strange ones--unlike Newton's action-at-a-distance, the interactions aren't instantaneous).

His theory is equivalent to the usual classical electrodynamics if one assumes that all electromagnetic radiation is eventually absorbed by charged particles. If there is "free" electromagnetic radiation that spreads out forever never being absorbed, then Feynmann's theory doesn't work.

 

[Dadface]

On the other hand, Feynman developed a theory of electrodynamics in which there were no extra degrees of freedom in the electromagnetic field; the theory could be interpreted as a pure particle theory with binary particle interactions (although strange ones--unlike Newton's action-at-a-distance, the interactions aren't instantaneous).

His theory is equivalent to the usual classical electrodynamics if one assumes that all electromagnetic radiation is eventually absorbed by charged particles. If there is "free" electromagnetic radiation that spreads out forever never being absorbed, then Feynmann's theory doesn't work.

But what about the points made previously regarding the deflector systems?

 

[sophiecentaur]

With electric forces, there's no problem. With magnetic forces, on the other hand...

Consider charged particle A traveling along the x-axis towards the origin (but not located at the origin), and charged particle B traveling along the y-axis and passing through the origin at a certain time. The magnetic field produced by A is zero along the x-axis, so there is no magnetic force on B at that instant. The magnetic field produced by B is non-zero at the location of A, so there is a magnetic force on A.

OK. I can see that you get a zero force with that model. That situation only applies at one instant, doesn't it (when the one electron is dead ahead of the other) and there will be the electric force, also, which needs to be considered and which is increasing to a maximum at some point during the near-miss. I can't say I have 'explained away' any apparent paradox but there is more going on than the initial description of the situation implies.
I get the strong message that "it can't be as simple as that" - situation normal then!

 

[0xDEADBEEF]

So the solution lies in the Poynting vector. Does someone know if/where the momentum leaves the system? Where is the largest portion of the field's momentum?

 

[Dale]

Well, Newton's laws really were not made to take into account momentum in fields. Interpreting his 3rd law in terms of forces, you need to consider a "force" on the field due to the particles. I guess you could do that, but it's a very different notion of "force'; it's certainly not equal to $m a$.

Yes, it would be a daft thing to do. The question was whether we can interpret conservation of momentum in electrodynamics as consistent with Newton's laws of motion. I think that it's a mistake to even try. The modern view of conservation of momentum is more general than what Newton considered.

I agree.

 

[Andrew Mason]

No, there would still be a problem since the change in momentum of one charge is not equal and opposite the change in momentum of the other. The key is to consider both the charges and the field. Then, to rescue Newton's third law you still need to express it in terms of forces since the field is interacting with both bodies.

I think I am correct in saying that the field contains momentum because there is an energy exchange between the charges. If I am wrong on this I expect someone will enlighten me.

Newton obviously had no idea about electromagnetic interactions and charge but his corpuscular model of light involved light carrying momentum. The difference is that in Newton's model, light had rest mass and one could imagine forces between light and matter. That really does not apply to interactions between photons and charges because photons lack rest mass and do not accelerate.

Only after talking about equal and opposite forces twice and only with the caveat that while talking about changes in momentum he is specifically referring to a pair of isolated bodies with no other interaction forces. Your interpretation of Newtons third law is unsupported by any reference that I have seen.

See Feynman Lectures, Vol. 1 Ch. 10-1 entitled Newton's Third Law. Feynman seems to equate the principle of conservation of momentum with the third law.

It is not clear why Newton used the term "action" instead of "force" and it is not clear exactly what "action" means. He uses "action" to relate the concept of force to the movement of bodies. But what Newton's commentary shows is that he concluded from the Law III and Law II that "quantity of motion" (which is the term he uses for "momentum") is conserved in an isolated system. Take, for example, this Corollary to his Laws of Motion.

The quantity of motion, which is collected by taking the sum of the motions directed towards the same parts, and the difference of those that are directed to contrary parts, suffers no change from the action of bodies among themselves.

For action and its opposite re-action are equal, by Law III, and therefore, by Law II, they produce in the motions equal changes towards opposite parts. Therefore if the motions are directed towards the same parts, whatever is added to the motion of the preceding body will be subducted from the motion of that which follows; so that the sum will be the same as before. If the bodies meet, with contrary motions, there will be an equal deduction from the motions of both; and therefore the difference of the motions directed towards opposite parts will remain the same.​

AM

 

[D H]

Andrew, I see from your profile that you're a lawyer by profession, which is a field where going back to original sources (precedence) is very important. Going all the way back to Newton to understand Newtonian mechanics is not a particularly good idea. Going all the way back to any original writing in the sciences is in general not the best way to gain understanding of that topic. This is one of the key differences between science and liberal arts. Those original writings almost inevitably are obtuse and verbose, and are occasionally mistaken. They are not viewed as authoritative in the sciences.

Newton was no different. He intentionally conflated force and impulse, and he intentionally avoided using algebra and calculus in his Principia. He didn't use vectors. (How could he have? Vectors came 200 years after Newton wrote his Principia.) You're much better off using a modern formulation of Newtonian mechanics.

As far as the primacy of Newton's third law versus the conservation laws, yes, one can derive conservation of momentum from Newton's third law. However, one can also derive Newton's third law from the conservation laws with the additional assumptions that forces act instantaneously and can be attributed to pairs of objects. What if forces those assumptions are violated -- forces that don't act instantaneously (e.g., electromagnetism) or forces that cannot be isolated to pairs (e.g., the chiral three body forces in ³He nuclei)? You don't get Newton's third law. The conservation laws are now seen as more the basic concept (and Noether's theorems being even more basic), with Newton's third law being a special case where those limiting assumptions are valid.

 

[sophiecentaur]

There is surely another point to be considered here. A charge (or mass) in a field will be distorting and affecting that field. You can't pick which of two charges is going to be 'the charge' with the other one the 'field provider', without a bit of a risk. I know that's what is done when we talk in terms of fields but aren't there some questions to be asked about this?
If we acknowledge the conservation of momentum then shouldn't we be bending this 'particle in a field' treatment to include it - rather than the other way round, and trying to demonstrate that momentum isn't always conserved? I realize it's all only a model but if we can discard momentum conservation in a trivial example like two electrons going past each other, then we may as well forget about the principle altogether.
Newton may have been a massive, grumpy ego but I'm sure he wouldn't have been too surprised to be told that, hundreds of years later, his very simple laws may not be adequate for all cases.

 

[Dale]

The difference is that in Newton's model, light had rest mass and one could imagine forces between light and matter. That really does not apply to interactions between photons and charges because photons lack rest mass and do not accelerate.

I agree, which is why I think it is a bad idea to try to apply Newtons third law to EM.

See Feynman Lectures, Vol. 1 Ch. 10-1 entitled Newton's Third Law. Feynman seems to equate the principle of conservation of momentum with the third law.

Sure, the conservation of momentum can be derived from the third law together with the second.

It is not clear why Newton used the term "action" instead of "force" and it is not clear exactly what "action" means. He uses "action" to relate the concept of force to the movement of bodies. But what Newton's commentary shows is that he concluded from the Law III and Law II that "quantity of motion" (which is the term he uses for "momentum") is conserved in an isolated system. Take, for example, this Corollary to his Laws of Motion.

The quantity of motion, which is collected by taking the sum of the motions directed towards the same parts, and the difference of those that are directed to contrary parts, suffers no change from the action of bodies among themselves.

For action and its opposite re-action are equal, by Law III, and therefore, by Law II, they produce in the motions equal changes towards opposite parts. Therefore if the motions are directed towards the same parts, whatever is added to the motion of the preceding body will be subducted from the motion of that which follows; so that the sum will be the same as before. If the bodies meet, with contrary motions, there will be an equal deduction from the motions of both; and therefore the difference of the motions directed towards opposite parts will remain the same.​

Excellent. I think that this corollary clearly demonstrates that the third law "action/reaction" refers to forces, not momenta. If the third law itself was about momenta then conservation of momentum would not be a corollary, it would be the law itself. Also, if the third law's "action" referred to momenta then there would be no need to use the second law to establish conservation of momentum.