# Why does Newton's 3rd law holds macroscopically?

1. Apr 20, 2015

### ELB27

Hi all,

I recently learned about the breakdown of Newton's 3rd law in electrodynamics and this got me thinking. The forces we consider in classical mechanics like friction and normal forces are microscopically electromagnetic interactions (repulsions?) of the atoms of two surfaces. If Newton's 3rd law doesn't hold for those interactions (which are certainly not static), how come it holds macroscopically? Perhaps it holds just as a good approximation, "on average"?

2. Apr 20, 2015

### Orodruin

Staff Emeritus
It actually does not break down. You simply have to consider the momentum carried by the electromagnetic field as well.

3. Apr 20, 2015

### Vatsal Sanjay

I hope this isn't out of on going discussion domain.

4. Apr 20, 2015

### Staff: Mentor

As Orodruin mentioned, the key is to recognize that while the 3rd law may not hold, it generalizes to the conservation of momentum, which does hold provided that you include the momentum of the fields as well. Then the reason that the 3rd law holds macroscopically follows from the fact that there are no "macroscopic" fields carrying away momentum.

5. Apr 20, 2015

### Staff: Mentor

6. Apr 20, 2015

### Orodruin

Staff Emeritus
It does hold if you consider the force to have a corresponding (local) force acting on the electromagnetic field, i.e., imparting momentum on it. The SR version is simply the conservation of the total energy momentum tensor, i.e., if you have two components of energy and momentum, $\partial_\mu (T_1^{\mu\nu} + T_2^{\mu\nu}) = 0$. Each term can be considered to be a 4-force density.

7. Apr 20, 2015

### Staff: Mentor

Agreed completely. That is why I used the word "may". I tend to favor exactly the interpretation you mentioned where the matter exerts a "force" on the field which gains momentum, but I haven't seen that as an "official" interpretation so I didn't want to push it.

Even if you take the contrary view and consider the "force" on a field to not be a valid force (so that Newton's 3rd is violated in some cases) you still have conservation of momentum and, if no macroscopic fields carry momentum, then Newton's 3rd is recovered.

8. Apr 20, 2015

### ELB27

Thank you for the replies! I need to think about it a bit more from this point of view of momentum conservation.

By the way, Orodruin, is your name a reference to Mount Doom?

9. Apr 20, 2015

### Orodruin

Staff Emeritus
Yes, but that is off-topic.