# Is Newton's 3rd law valid in Electrodynamics?

1. Feb 13, 2016

### PhiowPhi

I'm really conflicted with this one, I found many sources that state that Newtons 3rd law does not hold in electrodynamics, reasonable sources with mathematical proof(example). And as a student who just finished Physics I and taking Physics II course, this is really confusing... but when I think about it, like a simple set-up of a solenoid and a wire placed in front of the solenoid, and allow current to flow in both conductors to create their magnetic fields, the magnitude of action/reaction pair are not equal(correct me if I'm wrong please!). The magnetic field of the solenoid is greater than the magnetic field created by the single wire, and the only forces that would represent the action/reaction pairs is the Lorentz force acting on both. And when I write down this example and work it out, they are not equal. Am I getting this right? Or is there something I'm missing?

2. Feb 13, 2016

Each loop/turn of the solenoid experiences an equal and opposite force that it puts on the single external loop of wire. Thereby the solenoid is pulled towards the single loop with the same force that the single loop experiences and thereby Newton's 3rd law is obeyed.

3. Feb 13, 2016

### Staff: Mentor

Newton's third law is not obeyed any time that momentum is transferred to or from the field.

4. Feb 13, 2016

A google of the topic indicates there are apparently some exceptions to Newton's 3rd Law in Electrodynamics. In many cases, I think Newton's 3rd Law still applies, but it appears the exceptions are numerous enough that electrodynamics calculations can not employ this law, but need to be worked on a case by case basis.

5. Feb 13, 2016

### PhiowPhi

Let's use the example I studied, assume 4 loops creating a solenoid, each having current $I$ flowing within, they add up their magnetic fields to create one net field $B$ now, those loops experience a Lorentz force on themselves similar to this diagram:

But when their magnetic fields add up like so:

And the net magnetic field $B$ is perpendicular to the wire it's magnetic field $B$ > $B_w$(aside from their currents($I_s , I_w$), might not be equal). Their forces are not the same, hence the conservation of momentum would be accurate for this system.

I agree, many sources so far made me focus into the conservation of momentum more.

6. Feb 13, 2016

### PhiowPhi

Just to be sure @Dale , @Charles Link the example I just posted falls in that same category? Where Newton's 3rd does not apply?

7. Feb 14, 2016

### vanhees71

Of course, momentum conservation holds in relativistic physics, and electromagnetism is entirely relativistic. If you interpret Newton's 3rd law as momentum conservation, then it holds true for electromagnetism too, but not for the involved point charges alone but only together with the electromagnetic field, which is part of the dynamical system as are these point charges. So you have momentum exchange between the pointcharges and the electromagnetic field. That's one of the deeper reasons to introduce fields in the relativistic theory, because the 3rd law cannot be valid if you only consider interacting point charges since you cannot have instantaneous action at a distance. Also note that point charges are somewhat problematic in classical relativistic physics. They are only treatable as approximations like with the (Landau-Lifshitz modified) Abraham-Lorentz-Dirac equations of motion. Google for "radiation reaction". You'll find tons of material ;-)).

8. Feb 14, 2016

### Staff: Mentor

What do you think? Is the momentum of the field changing?

9. Feb 14, 2016

### PhiowPhi

my intuition says yes, but haven't reached the point we're I can explain it.

10. Feb 14, 2016

### maline

The fundamental point, in my opinion, is that a strictly Newtonian force is an interaction between two bodies. Thus it makes sense to talk about it acting equally on the two. Post-relativity, this is simply not the case! Bodies never exert forces upon one another! That would be action at a distance, because there is always some distance between any two bodies. Instead, fields exert forces on bodies, and are affected by motions of those bodies. The new definition of momentum, which is conserved in electrodynamics, must be derived from Noether's theorem and/or Maxwell's equations themselves. It has very little to do with the Third Law ar understood by Newton.

11. Feb 14, 2016

I think one part of Newton's 3rd Law is the idea of equal and opposite forces. In an E&M class in college, the instructor presented a detailed and accurate vector calculus proof that when two (stationary) loops of wire are each carrying a current, the force on one is equal and opposite the force on the other. From the discussions that followed, along with the idea that the total momentum is still conserved when the fields are included, we really have not discredited Newton here. The electromagnetic field is a little more complex than anything Newton considered, but his inputs are still good science.

12. Feb 14, 2016

### vanhees71

You must be careful in the case of relativistic interactions. Here it can be misleading to look at static situations and then conclude something for the general dynamical situation. There cannot be an action at a distance, and thus there's only energy-momentum conservation if you take into account both the matter and the field-degrees of freedom. The example posted in #1 is almost correct. Only at the end it's a mistake to use the non-relativistic approximation for the mechanical momenta, because this easily leads to confusion and the introduction of socalled "hidden momentum". There is no such thing as hidden momentum but just relativistic momentum. There are no ambiguities when everything is done correctly without non-relativistic approximations. If you take into account the electromagnetic field momentum you also have to use the full relativistic expression for the momentum of the charged particles or charge-current distributions.

13. Feb 14, 2016

### maline

This is a situation with constant fields, so you can show momentum conservation while ignoring the fields. But even there, the forces are not exerted by the wires themselves, but by the fields, which depend on what the current in the other wire was a moment ago. "Coincidentally", the current is still the same, so it becomes much easier to make the calculations as if the current is exerting the force (Biot- Savart).
To make clear that "equal & opposite forces" is not a correct idea in electrodynamics, let me simplify the example given by the OP. Let particles A & B have equal positive charges- say 1 Coulomb. A is at the origin moving downward (in the negative Y direction) at 1 m/s. B is at (1,0) moving to the left at 1 m/s. Both particles have been moving at constant speed for infinite time in the past. What are the forces on the particles?
There is no need to make any calculations! Simply note that B feels a magnetic force in the upward Y direction, while A feels no force at all in the Y direction!
Also note that this does not require "relativistic" speeds. We mention relativity here because Maxwell's equations are inherently relativistic, and this fact makes many of their implications- such as lack of direct particle interactions- very natural and elegant.

14. Feb 14, 2016

Thank you. A very good example. One minor correction, if I got my cross products correct, is that the magnetic force on particle B is in the downward (negative Y) direction. The magnetic field lines from particle A go clockwise when looking downward, and Qv x B for particle B points downward... On subsequent review, you have particle B moving radially inward, thereby, yes, the force is upward, and you are completely correct. Also, the above example seems to suggest that the magnetic forces on two current loops may not necessarily be equal and opposite. The proof that I saw in my E&M class, (it was quite some time ago), may in fact have been in regards to mutual inductances and not about forces or torques.

Last edited: Feb 14, 2016
15. Feb 14, 2016

A google of the subject gives problem 5.49 of Griffith's E&M textbook, that the magnetic forces on two current loops are indeed shown to be equal and opposite. The above example of the two moving point charges that don't have equal and opposite forces is quite interesting, and also appears to be completely correct. It seems to ensure a correct answer, the calculations need to be worked out in detail and it is difficult to assess whether a simple application of Newton's 3rd Law is going to yield the correct result.

16. Feb 14, 2016

### maline

The (sufficient) condition is fairly simple to check: are any EM fields changing with time?
But conceptually, even in the static case, I think it's important to realize that Newton's model of a force acting between two bodies is not really what's going on.

17. Feb 15, 2016

### maline

Just to be clear: this is the same condition as "all charge distributions & currents are constant". This is the only case where momentum conservation implies equal & opposite forces. There are a few other cases where the forces are equal & opposite for reasons unrelated to momentum, for instance if there is 180o rotational symmetry around the point between two charges.

18. Feb 15, 2016