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.