Pauthenier equation

The Pauthenier equation states that the maximum charge accumulated by a particle modelled by a small sphere passing through an electric field is given by:

$Q_{\mathrm {max} }=4\pi R^{2}\epsilon _{0}pE$

where $\epsilon _{0}$ is the permittivity of free space, $R$ is the radius of the sphere, $E$ is the electric field strength, and $p$ is a material dependent constant.

For conductors, $p=3$.

For dielectrics: $p=3\epsilon _{r}/(\epsilon _{r}+2)$ where $\epsilon _{r}$ is the relative permittivity.

Low charges on nanoparticles and microparticles are stable over more than 10³ second time scales.