# Newton's Third Law Exceptions?

Lately I've been brushing up on some of my old Physics texts from college. Most recently, I've been rereading parts of "Classical Dynamics of Particles and Systems (5th ed.)" by Thornton and Marion.

In the second chapter, the authors discuss Newton's Laws of Motion, and point out about the Third Law that

We must hasten to add, however that the Third Law is not a general law of nature. The law does apply when the force exerted by one (point) object on another (point) object is directed along the line connecting the objects. Such forces are called central forces. (Pg 50)

I know that gravity and the electrical attraction between static charges are central forces, and so I know that these forces certainly obey Newton's Third Law. However, aside from briefly mentioning that velocity-dependent forces do not generally follow the Third Law, the authors do not elaborate. So my question is two-fold.

First, is Newton's Third Law not universally true? It was my understanding the the Third Law was a necessary consequence of the homogeneity of space, and conservation of momentum.

Second, what would be a familiar example of a non-relativistic force that does not follow the Third Law. Am I correct in understanding that drag would not necessarily follow the Third Law, because the magnitude of the force is velocity-dependent?

## 2 Answers

I have Marion-Thornton 4th ed. around here somewhere. It is an older book and presents some material differently than we are used to in more modern books (for instance they even use the old imaginary time method when discussing some things in special relativity, which I personally dislike). However I agree with DanielSank, different pedagogy does not equal "nonsense".

Newton's laws are presented slightly differently by different books. For instance, it can be argued Newton meant his second law to be $F=dp/dt$ (although he didn't write it in this modern notation), although many books present it as $F=ma$. Some people go even further and try to extract a modern meaning, as I've seen some people say Newton's third law is the conservation of momentum. This may be pedagogically useful, but not historically accurate. It is worth reminding that some debate over the exact statements translated to modern language is understandable. Even though Newton invented calculus, some concepts in mechanics still took long after Newton to come into their modern understanding, such as the concept of kinetic energy was put in its modern form much later.

Thus to answer this question requires agreeing on a statement for Newton's third law. I don't have Marion-Thornton handy, so using wikipedia

When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

The force between two particles in electromagnetism can violate this. For a concrete example consider a positive charged particle A pulled along the x axis at a constant velocity in the positive direction, and another positive charged particle B pulled along the y axis at a constant velocity in the positive direction. If it is arranged such that when A is at (0,0), B is at (0,1), then we can calculate the fields and find:

• the electric forces on the particles will be in opposing directions
• the magnetic force on A is zero
• the magnetic force on B is in the -x direction

Does this mean momentum is not conserved here? No.

If we include the person or device pulling these charges along as part of the system (so there are no external forces), then we should expect the momentum of the system to be conserved.

Where is the missing momentum then? It is in the fields!

I constructed this scenario specially to also help break a bad habit of some descriptions of this phenomena. Because the charges are moving at a constant velocity, there is no radiation. We don't need radiation to provide a force back on the partices or something to solve this. Momentum can be stored in the fields themselves. (While not shown in this example, even static fields can have non-zero momentum.)

• For curious readers, here is an extended discussion of how unclear some classical mechanics concepts were back then: nature.berkeley.edu/departments/espm/env-hist/articles/2.pdf – liuwei Oct 9 '14 at 5:17
• the magnetic force on A is zero the magnetic force on B is in the -x direction Could you please show me how to perform the evaluation of the field values you've gotten using the Savart Law? @liuwei – Kugelblitz Aug 15 '15 at 3:52

The Lorentz force

$$\vec F = q \vec E + \frac{q}{c} \vec v \times \vec B$$ Doesn't obey Newton third law and is one of the fundamental forces of nature (unlike drag for example). The magnetic part of the force satisfy that two charged particles exert a magnetic force with equal magnitute to each other, but the direction is not along the line that join the two particles. This have as a consequence that the angular momentum of the system is not conserved in presence of such forces in general, so one has to add the angular momentum of the electromagnetic field to get a conservation law.

• I've deleted a comment discussion with some arguably inappropriate comments from multiple people involved. I'd suggest starting over, and remember to be nice. (If you really want, I can put the comments in chat, but still, our rules on civility are in force.) – David Z Oct 1 '14 at 4:11
• @DanialSank The magnetic force between two particles is not necessarily "of equal magnitude and opposite direction". I included an explicit counter-example in my answer if you'd like to learn more. – liuwei Oct 2 '14 at 0:06
• @liuwei: Yep, my comment was completely wrong. Deleted so as to not confuse hapless future readers. Thanks. – DanielSank Oct 2 '14 at 7:02