Violation of conservation of angular momentum

Question

In this video professor V. Balakrishnan talks about conservation of angular momentum and it's implications, etc. (See from 2:23 to 9:25) Then he goes on to show how Newton's third law guarantees the conservation of angular momentum.

But a quick search over here gives the following question:

• Apparent Violation of Newton's $3^{\text{rd}}$ Law and the Conservation of Momentum (and Angular Momentum) For a Pair of Charged Particles

In the answer there Emilio Pisanty says (a portion of full answer:

You are correct in your assertion that pairs of charged point particles can interact magnetically in ways that seemingly violate Newton's 3rd law, and therefore also seem to violate the conservation of both linear and angular momentum. This is a fundamental result and it is the decisive (thought) experiment which forces us to change our viewpoint on electrodynamics from something like

charged particles interact with each other

to a field-based one that says

charged particles interact with the electromagnetic field.

What this means, and the key point here, is that

• the electromagnetic field should be considered as a dynamical entity of its own, on par with material particles, and it can hold energy, momentum, and angular momentum of its own.

So:

• Since he says "electromagnetic field should be considered as a dynamical entity of its own". Does that mean that the two particle system isn't a closed one? If not why not?

• Can this happen for a single particle i.e., to say that the particle spontaneously transfers it's momentum to the EM field and hence after some times lose the momentum? If not why not?

I just want to know how this single particle setup might be different from two particle ones?

Answer 1

You cannot consistently consider a charged particle without considering its electromagnetic field as well. A system of charged particles plus their field is “closed” and its energy, momentum, and angular momentum are conserved. Without the field it is not closed and these quantities are not conserved.

A single non-interacting charged particle does not transfer energy, momentum, or angular momentum to the field. This is calculable from the equations for its field. In its rest frame, the field is just an electrostatic Coulomb field. Only fields with both electric and magnetic components transfer energy, momentum, and angular momentum from place to place.

Interacting particles accelerate, causing the field to have a magnetic part, as seen in the Lienard-Wiechert potentials. Then the Poynting vector is nonzero and energy (and also momentum and angular momentum) flows through the electromagnetic field.

Answer 2

The case of two charges that magnetically interact can be discussed in a much simpler manner. We are all familiar with the concept of potential energy and know that kinetic energy is not separately conserved : only the sum of kinetic and potential energy is. In special relativity energy is the time component of four momentum. For the same reason kinetic energy is linked to kinetic momentum , potential energy is connected to potential momentum. The electromagnetic vector potential should thus be seen as producing potential momentum. While kinetic momentum, $m\bf v$, is not conserved, the sum of kinetic and potential momentum, $m{\bf v} + q\bf A$, is conserved in the case of two magnetically interacting charges.