# Biermann battery

In astrophysics, the Biermann battery is a process by which a weak seed magnetic field can be generated from zero initial conditions.[1] The relative motion between electrons and ions is driven by rotation. The process was discovered by Ludwig Biermann in 1950.[2][3]

A simple derivation of the effect starts with the momentum equation for the free electron fluid, keeping only the electric field and pressure force:

${\displaystyle m_{e}{d{\vec {v}}_{e} \over dt}=-e{\vec {E}}-{\nabla p_{e} \over n_{e}}}$.

For phenomena on sufficiently slow time scales, the left-hand side of the above equation can be neglected. That leads to the Ohm's law for the electric field:

${\displaystyle {\vec {E}}=-{\nabla p_{e} \over en_{e}}}$.

The other terms in the general Ohm's law are neglected here since they typically vanish for zero magnetic field and do not spontaneously generate field. Inserting the above electric field into Faraday's law of induction gives an equation for the magnetic field:

${\displaystyle {\partial {\vec {B}} \over \partial t}={1 \over en_{e}}\nabla T_{e}\times \nabla n_{e}}$

This shows that magnetic fields develop "spontaneously", that is, from initially being zero, when there are non-parallel gradients in electron temperature and density.

## References

1. ^ Xu, Hao; O'Shea, Brian W.; Collins, David C.; Norman, Michael L.; Li, Hui; Li, Shengtai (2008). "The Biermann Battery in Cosmological MHD Simulations of Population III Star Formation". The Astrophysical Journal Letters. 688 (2): L57. arXiv:0807.2647. Bibcode:2008ApJ...688L..57X. doi:10.1086/595617.
2. ^ Biermann, L. "Uber den Ursprung der Magnetfelder auf Sternen und im interstellaren Raum (mit einem Anhang von A. Schluter)". Zeitschrift für Naturforschung. 5: 65.
3. ^ Khanna, Ramon (1998). "Generation of magnetic fields by a gravitomagnetic plasma battery". Monthly Notices of the Royal Astronomical Society. 295: L6. arXiv:astro-ph/9803240. Bibcode:1998MNRAS.295L...6K. doi:10.1046/j.1365-8711.1998.29511447.x.