Pauthenier equation explained

The Pauthenier equation states ^{[1] [2]} that the maximum charge accumulated by a particle modelled by a small sphere passing through an electric field is given by:

Q _{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.

Notes and References

1. Levitated electromechanics: all-electrical cooling of levitated nano- and micro-particles. Daniel. Goldwater. James. Millen. Quantum Science and Technology. 2019. 4. 024003. 10.1088/2058-9565/aaf5f3. 1802.05928. 119244999.
2. K. Adamiak, "Rate of charging of spherical particles by monopolar ions in electric fields",IEEE Transactions on Industry Applications 38, 1001-1008 (2002)