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:

$$Q_{\mathrm{max}}=4\pi R^2\epsilon_0pE$$

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.