I met a statement, that Lorentz force disobeys the third Newton's law.

Is this true?

Also I met some setups, where it was stated, that third law is violated. Also I met some explanations about WHY it is violated, saying that momentum is transferred to the field, like in "duplicate" question: Apparent violation of Newton's 3rd law and the conservation of angular momentum for a pair of charged particles interacting magnetically

But my question is different. I am disputing the fact itself. Is this really true, that it is possible to observe third law violation (i.e. momentum leak) in table experiments?

For illustration suppose we have two conducting parallel horizontal plates and two perpendicular horizontal rods between them. Rods have contacts with plates, so that one end of each rod is contacting one plate and another end is contacting another. Contacts go vertical, but they are infinitesimally small and we can ignore forces, induced by them.

So, two rods are perpendicular and have current flowing through them.

enter image description here

Magnetic field of red rod is shown by red ellipse. It reaches blue rod in the center and since it's perpendicular to the current, it applies a force to blue rod, shown by black arrow.

Magnetic field of blue rod is not shown. But it's geometry implies that red rod falls into the center of symmetry. So, there is no magnetic field, applied to red rod.

The question is: will the momentum lack be observable here?

This setup is taken from "duplicate" question setup, where two charged particles were moving. But I am not insisting on this setup.

Please suggest ANY other setup. I have my laboratory table, have wires, batteries, have Wimshurst machine etc.

Can I observe third law violation? If "yes" then "how"?


1 Answer 1


You've probably heard that photons are massless, and that they are quantums of electromagnetism. So even if you aren't going to do quantum mechanics you might worry that forces aren't going to cut it, there is no easy meaningful way to have $F=ma$ for a massless particle. But you can talk about momentum, e.g. $F=dp/dt$.

So momentum conservation is what Newton's third law is really about if you want it to be general. The other thing is that the electromagnetic fields themselves have energy and momentum and angular momentum. So what happens is that the fields can exchange momentum with the charged particles. And in that sense momentum is conserved again.

edited to respond to the edited question

If you've never heard of fields, but you have equipment then almost anything violates conservation of mechanical momentum if you have some dynamics beyond steady currents and steady charge distributions as long as you can measure things really quickly and precisely. The principle reason is relativity, if some charges move near $\vec{r}=\vec{0}$ at $t=0$, then charges over at $\vec{R}=\vec{a}$ won't react until later, specifically not until $t=|\vec{a}/c|$. So move one charge or wire about, then wait long enough for the other to react and at the moment that the other reacts there is a violation. Even if you set up a very nice symmetric situation in one frame, if there are unsteady currents or unsteady charge distributions, then there is a frame where some of those effects are currently being large while others are being small so at that moment they aren't equal and opposite. But this has nothing really to do with electrodynamics except that electrodynamics is naturally a relativistic theory. Any relativistic theory is likely going to need fields that carry momentum to save conservation of momentum.

If you want a nonsteady but simple example you can have charge one move in the x direction at a steady rate, charge two move in the y direction at a steady rate, put forces on them to make it so. Then notice that the electromagnetic forces they exert on each other do not point in opposite directions. That example is nice in the sense other forces are involved to make them move at a steady rate (but move in a way without a steady current) so there is no radiation (or radiation reaction) which makes it much simpler.

  • $\begingroup$ No, no, I need no explanations, I need proofs. Let's take macroscopic charged bodies and attach dynamometers to them. Will wee observe third law violation? $\endgroup$
    – Dims
    Feb 5, 2015 at 17:43
  • $\begingroup$ Explain what you think a third law violation is. Are you referring to the observed accelerations and inferred F=ma not being opposite and equal? $\endgroup$
    – Bill N
    Feb 5, 2015 at 19:05
  • $\begingroup$ "No, no, I need no explanations, I need proofs." You're likely to discover that approach has little purchase here. $\endgroup$ Feb 6, 2015 at 2:27
  • $\begingroup$ @BillN I am sure third law is never violated; I met a statement, that it is false. So I wish a sample, where it is violated. I think there are NO EXPERIMENTS, where it is violated. I may be wrong. Where I am wrong? $\endgroup$
    – Dims
    Feb 6, 2015 at 11:52
  • $\begingroup$ @Dims : Wow! I indeed was surprised why you drew the wires so ugly, now I understand. As to the rods, I didn't pay attention. I am so sorry. I did a nice proof, but not for your question. Anyway, there is no such thing as violating Newton's third law. Please tell me, the answer in the former question doesn't satisfy you? $\endgroup$
    – Sofia
    Feb 6, 2015 at 14:11

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