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 m a x = 4 π R 2 ϵ 0 p E {\displaystyle Q_{\mathrm {max} }=4\pi R^{2}\epsilon _{0}pE}
where ϵ 0 {\displaystyle \epsilon _{0}} is the permittivity of free space, R {\displaystyle R} is the radius of the sphere, E {\displaystyle E} is the electric field strength, and p {\displaystyle p} is a material dependent constant. For conductors, p = 3 {\displaystyle p=3} . For dielectrics: p = 3 ϵ r / ( ϵ r + 2 ) {\displaystyle p=3\epsilon _{r}/(\epsilon _{r}+2)} where ϵ r {\displaystyle \epsilon _{r}} is the relative permittivity. Low charges on nanoparticles and microparticles are stable over more than 103 second time scales.
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