Can Magnetic Attraction Between Electrons Overcome Electrical Repulsion?

[gildomar]

Since the magnetic force between dipoles goes as 1/r^4 and the electrical force goes as 1/r^2 for charges, would electrons be able to theoretically get close enough for their magnetic attraction to be greater than their electrical repulsion? If so, what then?

 

[tiny-tim]

hi gildomar!

(try using the X² button just above the Reply box )

i don't understand the question

stationary objects feel no magnetic attraction

 

[gildomar]

Thanks for the tip tiny-tim!

And for the magnetic field there, I was referring to the intrinsic magnetic field of an electron due to it's magnetic moment interacting with a nearby electron's magnetic field.

 

[bahamagreen]

Does the electron as a point particle have two poles?

 

[gildomar]

As far as I'm aware, since the electron has a magnetic moment, that that means that it is a magnetic dipole.

 

[Enthalpy]

Of course the electron has a magnetic dipole, and two electrons interact by their dipoles, sure.
If one dislikes the name "dipole" he may say "magnetic moment creating a dipolar-like field".
And despite being point-like, the electron has a rotation momentum as well.

And the interrogation you're suggesting is:
Magnetic attraction energy can be as 1/R^3
The kinetic energy resulting from their proximity hence confinement is as 1/R^2
The electrostatic repulsion energy is as 1/R^1
So a pair of electrons could stick together without any distance limit. Is that it?

Pauli's exclusion wants the magnetic dipoles to be anti-parallel, in which case the electrons attract an other by their magnetic dipole.

You made me uneasy... I suppose this would have happened only at distances so tiny that confinement implies important relativistic corrections that let the kinetic energy increase faster than 1/R^2 and faster than 1/R^3 as well. What is the necessary distance?

 

[Bill_K]

The electron's magnetic moment is a Bohr magneton, μ = eħ/2mc, so the dipole-dipole potential energy is roughly μ²/r³ = (1/4)(e²/ħc)(ħ/mc)³(1/r³)(mc²). The Colulomb energy is e²/r = (e²/ħc)(ħ/mc)(1/r)(mc²). The attraction and repulsion can balance when these are the same order of magnitude, namely at r = ħ/mc, the Compton wavelength, when they will both be of order (e²/ħc)(mc²).

But the kinetic energy of a particle confined within a Compton wavelength is of order mc², which is 137 times as great as the potential. So yes there will be attraction, and at sufficiently short distances it can equal or overcome the Coulomb repulsion, but due to the much greater kinetic energy there's no possibility of a bound state.