Based on the three laws of motion, we can derive that the angular momentum of a system is conserved, if no external torque is applied. A crucial assumption while deriving this result is that the force between any two particles within the system must act along the line joining the particles only. I was trying to justify this assumption by considering the interaction between any two particles as the summation of the fundamental forces. If the assumption is true for all the fundamental forces individually, it will hold true for their summation as well. For gravitational and electrostatic forces, the assumption is true. Is it true for magnetic and nuclear forces as well? I noticed that magnetic force is perpendicular to the velocity of the particle, and may not conserve angular momentum of the system.
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$\begingroup$ As long as the net torque on the system about the axis is $0$, the angular momentum will surely be conserved. It doesn't really matter whether electrostatic, gravitational, nuclear or magnetic force is acting on the body. Just the net torque needs to be zero. $\endgroup$– user243267Commented Jan 3, 2020 at 8:20
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2$\begingroup$ For the torque of the internal forces to cancel each other, the force must act along the line joining the particles. Otherwise, angular momentum won't be conserved even if external torque is zero. How do you justify that all the internal forces are acting along the line joining the particles? $\endgroup$– NewbieCoderCommented Jan 3, 2020 at 8:34
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$\begingroup$ conservation of momentum (linear and angular) of an EM system must include the momenta of the field, when that is included magnetic forces also conserve momenta. $\endgroup$– hyportnexCommented Jan 3, 2020 at 13:36
1 Answer
OP is correct: mechanical angular momentum for a system of particles interacting with each other magnetically is indeed not conserved. However (as commenters already suggested), if we include into consideration electromagnetic field as an entity capable of carrying linear and angular momentum and exchanging it with particles, then the total angular momentum of the system “particles $+$ EM field” would be conserved. But analysis of fields generally puts such problems within the realm of field theory, outside of pure classical mechanics.
As an illustration we could consider nonrelativistic motion of a pair of dyons (hypothetical particles carrying both electric and magnetic charge). Though the existence of magnetic charges has not been established experimentally, they are predicted by many theories. More importantly, this problem could be solved by means of elementary classical mechanics (we could write down equations of motion for particles and find their solution).
Analysis of this problem could be found in the following paper:
- Dos Santos, R. P. Magnetic monopoles and dyons revisited: a useful contribution to the study of classical mechanics. European Journal of Physics, 36(3), (2015), 035022, doi:10.1088/0143-0807/36/3/035022, arXiv:1503.00499.
To summarize: the mechanical angular momentum, $\mathbf{L}=\mu \,\mathbf{r}\times \mathbf{v}$, in the center of mass frame ($\mu$ is reduced mass of the system) is not conserved. There is, however, the conserved Poincaré integral of motion: $$ \mathbf{J} \equiv \mathbf{L} + \frac{\mu_0 (e_1 g_2 - g_1 e_2 )}{4 \pi} \, \frac{\mathbf{r}}{r}, $$ where $\mu_0$ is vacuum magnetic permeability, $e_i$ and $g_i$ is an electric and magnetic charges of $i$-th particle.
The second term could be interpreted as the angular momentum of EM field, so that $\mathbf J$ is the total angular momentum of the system. For the field-theoretic perspective on this problem (including the calculation of field angular momentum) see, for example, the book J.D. Jackson, Classical electrodynamics.
