## FANDOM

148 Pages

The analysis presented here is base on this note.

Conservation of momentum for electrons in a thermal plasma is given by

$n m \frac{d \mathbf{v}}{dt} = n q \left(\mathbf{E} + \frac{1}{c} \mathbf{v} \times \mathbf{B} \right ) - \nabla p$

where $n$ is the number density of the electron, $m$ is the electron mass, $q$ is the electron charge, $\mathbf{v}$ is the velocity, $\mathbf{E}$ is the electric field, $\mathbf{B}$ is the magnetic field and $p$ is the pressure. We assume that the inertia of the electrons is so small that it can by neglected. Hence, the left hand side vanishes, and we are left with a modified Ohm's law

$\mathbf{E} + \frac{1}{c} \mathbf{v} \times \mathbf{B} = \frac{\nabla p}{n q}$

Applying curl and using Faraday's law $\nabla \times \mathbf{E} = - \frac{1}{c} \frac{\partial \mathbf{B}}{\partial t}$ yields

$\frac{\partial \mathbf{B}}{\partial t} = \nabla \times \left(\mathbf{v} \times \mathbf{B} \right) - c \frac{\nabla n \times \nabla p}{q n^2}$

Therefore, a misalignment between the gradients of the pressure and density can generate magnetic fields. This effect is called the Biermann battery.