- #1

jarekduda

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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 [URL='http://en.wikipedia.org/wiki/Magnetic_monopole']magnetic monopole Wikipedia article[/URL] - full expression for Lorenz force in such case would be: [tex]\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)[/tex]

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: [tex] \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] [/tex]

where the last term would correspond to such eventual magnetic moment-electric field interaction.

Derivation: https://dl.dropboxusercontent.com/u/12405967/freefall.png

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?

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 [URL='http://en.wikipedia.org/wiki/Magnetic_monopole']magnetic monopole Wikipedia article[/URL] - full expression for Lorenz force in such case would be: [tex]\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)[/tex]

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: [tex] \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] [/tex]

where the last term would correspond to such eventual magnetic moment-electric field interaction.

Derivation: https://dl.dropboxusercontent.com/u/12405967/freefall.png

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?

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