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## action = reaction and lorenz force

 Quote by DaleSpam 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

 Mentor 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.
 Recognitions: Gold Member Science Advisor 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 realise 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.

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 Quote by Andrew Mason 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.

 Quote by Andrew Mason 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.

 Quote by Andrew Mason 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.