# Force From Spin Magnetism

Q:
As I understand it magnetic fields arise from moving charges. This I understand and even know how to calculate using laws like Biot-Savart. My question doesn't concern that though. I've heard that electrons have magnetic fields while at rest. What I'm having a problem with is calculating that. Obviously if an electron is at rest putting something like "qv" into Biot-Savart will lead to a zero field because "qv" is equal to zero. Is there an equation that would allow me to calculate the force between two electrons at rest from their inherent magnetic fields? And if so can you describe the terms and units to be used in the equation? What I'm looking for is an equation that will give me a force value between two electrons at rest separated a certain distance apart. To be clear I'm not looking for an equation like Coulomb's for the electrostatic force or one that involves motion from one or the other electron. What I want is an equation that will allow me to just calculate the force from two stationary electrons due to their inherent magnetic fields.
- John (age 44)
Phenix City, AL U.S.
A:

Sure. The force depends on how the intrinsic magnetic dipole moments of the spins are oriented with respect to each other and to the direction between them. For a given set of directions, the force falls off as the inverse of the fourth power of the distance.

The magnetic dipole-dipole interaction is described in detail here: https://en.wikipedia.org/wiki/Magnetic_dipole%E2%80%93dipole_interaction in conventional but irritating Standard International units. Here's a link that gives the interaction energy in nicer CGS units: http://http://userpage.chemie.fu-berlin.de/~tolstoy/chapter3.pdf. To get the force you just take minus the derivative with respect to the distance between them. In CGS units, the magnetic moment of an electron spin is about 10-20 erg/Gauss (https://en.wikipedia.org/wiki/Bohr_magneton). So, depending on the orientations, the magnitude of the force between two such spins at distance r is around 10-40/(r/cm)4 dynes. (The notation means to give r in cm.)

The electrical force is around 25*10-20/(r/cm)2 dynes. So even on atomic-scale distances (10-8 cm) the electrical force is much bigger. The reason you can sometimes notice magnetism is that materials are usually about electrically neutral, but in some materials the electron spins tend to line up so that their fields add up.

Mike W.

(published on 07/17/2015)

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