Using newton's third law in electromagnetism and special relativity problems

Question

I am confused with using newton's third law in electromagnetism problems which includes Lorentz transformation. In what conditions does this law holds? for example, if there is an infinite charged plate that moves in the +x direction with speed V1, and a charge q that moves in the -x direction with speed V2, and we want to know what force does the charge apply on the plate in the plate's frame of reference. I have been told that I can't use force transformation from the lab to the plate's frame (after using Newton's third law), but that I can use this law in the plate's frame.

What is the reason behind this saying? I searched for topics related to it in this forum but I couldn't find something that gives a direct answer to my question.

Thanks

Answer 1

Newton's Third Law has some tricky subtleties in electrodynamics. This is because Newton's Third Law is really a statement about momentum conservation of a closed system: if $\vec{F}_{12} = - \vec{F}_{21}$ in an isolated two-particle system, then $\dot{\vec{p}}_1 = - \dot{\vec{p}}_2$ and so the total momentum is a constant. The problem with applying this concept in electrodynamics is that the "system" you have to take into consideration is the charges and the electromagnetic fields, and electromagnetic fields can carry momentum.

The standard example where Newton's Third Law fails to hold (considering only the charges) is when we have one charge moving along in the positive direction along the $x$-axis, and another charge moving in the positive direction along the $y$-axis. Even in the limit of low velocity, both charges will exert magnetic forces on each other. We can still see that while the electric forces that one charge exerts on the other will be equal and opposite to each other, the magnetic forces will be in completely different directions. If I've done my right-hand rules correctly, the force on the charge on the $x$-axis will be in the positive $y$-direction, while the force on the charge on the $y$-axis will be in the the positive $x$-direction. This means that I would have to apply a net force to the charges (viewed as a system) to keep them moving at a constant velocity, which would appear to violate Newton's First Law.

The paradox is resolved by assigning the name "field momentum density" to the quantity $\epsilon_0 \vec{E} \times \vec{B}$, and (via heroic doses of vector algebra) showing that if we account for the change in this quantity integrated over some volume, the flux of this quantity into or out of this volume, and the mechanical momentum of the charges, then this "combined notion" of momentum is conserved. But this means that if the integral of $\vec{E} \times \vec{B}$ over space is changing with respect to time or has a non-zero flux out of the volume we're considering, we cannot expect the mechanical momentum of the system to be constant, and Newton's Third Law will not hold. Conversely, if the field momentum is constant in some volume and the momentum flux out of this volume is zero, then Newton's Third Law will hold.

Answer 2

You can use Newton's laws as long as speeds are non-relativistic ($v \ll c$, where $c$ is the speed of light). The full relativistic corrections will be of order $v/c$ and higher powers. For instance, if you use Newton's third law and then Lorentz transform to another frame, then in the new frame Newton's third law won't hold in general because of some $v/c$ differences that appear. When $v \ll c$ these differences are negligible and this is the reason why Newtonian mechanics is a valid description of Nature only at low speeds.

Answer 3

I guess this a question about when the Newton's third law can be used.

It can be used when time delays do not cause problems. So it can be used with constant forces. And it can be used with short distances.

For example in this situation Newton's third law can be used: There is an infinite charged plate that moves in the +x direction with speed V1, and a charge q that moves in the -x direction with speed V2, and we want to know what force does the charge apply on the plate in the plate's frame of reference.

In the plate's frame the plate exerts a force F on the charge, and in that same frame the charge exerts an opposite force on the plate.

In any frame that force pair is in balance.

Different frames may disagree very much about how large plate area is feeling the force from the charge at one moment. (Because of the length contraction)