# Dual Lorentz force for classical spin-orbit interaction?

1. Nov 21, 2015

### jarekduda

Electric and magnetic parts of Maxwell's equations are kind of similar, so physical effects relating these properties have many 'dual' analogues - with exchanged places.
For example in Aharonov-Bohm effect, the phase of charged particle depends on side of magnetic flux tube it comes through, while in its 'dual' analogue: Aharonov-Casher, the particle has magnetic moment and tube contains line of charge (it was used e.g. for neutron or fluxon interference).
Another interesting 'dual' effect (hypothetical) can be found in magnetic monopole Wikipedia article - full expression for Lorenz force in such case would be: $$\mathbf{F}=q_e\left(\mathbf{E}+\frac{\mathbf{v}}{c}\times \mathbf{B}\right)+q_m\left(\mathbf{B}- \frac{\mathbf{v}}{c}\times \mathbf{E}\right)$$
where q_m is magnetic charge - the last term corresponds to magnetic monopole - electric field interaction.

The question is if we should expect similar term for not only magnetic monopoles, but also for much more common: magnetic dipoles like electron or neutron ?
So imagine classical electron traveling in proton's electric field - let's change reference frame such that electron stops (for infinitesimal time) and proton is moving in also magnetic field created by quite large electron's magnetic moment - because of 3rd Newton's law, resulting Lorentz force should also work on electron ...
Here is Lagrangian for such electron's movement: $$\mathbf{L} = \frac{1}{2}m\mathbf{v}^2+\frac{Ze^2}{r}+\frac{Ze}{c}\left[ \mathbf{v}\cdot\left( \frac{\mu\times \mathbf{r}}{r^3}\right)\right]$$
where the last term would correspond to such eventual magnetic moment-electric field interaction.
Derivation: https://dl.dropboxusercontent.com/u/12405967/freefall.png [Broken]

While this dual Lorentz force seems important: classical analogue of spin-orbit interaction, I couldn't find any serious materials about it - have you met it anywhere?
Where it might be important? Some experiments with electrons?
What other dual effects seem important ... forgotten?

Last edited by a moderator: May 7, 2017
2. Nov 26, 2015

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Nov 28, 2015

### jarekduda

Hi, I have just seen your response.
Like here, it seems as a completely forgotten topic. I have recently talked with electron experimentalist and he told that he was aware of this effect, but couldn't point any source.
Such force for magnetic dipole traveling in electric field could be an essential contribution in many experiments involving electrons, neutrons (... fluxons like in Aharonov-Cahser).

One place where it seems crucial is (semi-)classical treatment of atom - this correction would make Bohr circular trajectories unstable, it classically explains why electron cannot fall into nucleus: because such Lorentz force would bend the trajectory, preventing electron from collapse.
Here is my notebook for simulations of such single electron atoms: https://dl.dropboxusercontent.com/u/12405967/freefall.nb [Broken]

It is interesting that zero angular momentum free falling electron bends exactly 120deg (I can show derivation, doesn't depend on physical constants!) - such electron would travel between vertices of equilateral triangles.
Surprisingly, even having zero angular momentum, this system rotates - it is allowed for complex systems, like in the falling cat problem.
For nonzero angular momentum there can appear all kind of hedgehog-like trajectories, it is an interesting question to classify the closed ones (there is supposed to be also tetraedric 109deg trajectory).
There are also possible back-scattering electron trajectories - proton is in 0:
https://dl.dropboxusercontent.com/u/12405967/traj.png [Broken]
If it is a realistic picture, imagine a proton approaching from the right side - electron's attraction electron screens the proton-proton repulsion, what could make fusion more probable (?)

Last edited by a moderator: May 7, 2017