Derivation of exact and approximate formulas for the force on a conducting particle standing on a ground plane, and comparison with finite element computations

The force on a conducting particle standing on a ground plane in a uniform background field is important to a range of technical areas, such as particle motion in gasinsulated substations. The force and lifting field for such a particle is normally evaluated using an approximate formula in an obscure paper published over 40 years ago. Software technology now facilitates the solution of many such problems exactly, which allows evaluation of (i) the published approximation and (ii) the range of parameters over which the approximation is accurate. In the present contribution, we provide an exact solution to the field-induced force semi-spheroid standing on a ground plane, derive the commonly used approximation from the exact solution, and find that the commonly used approximate solution for the force on a rod-like particle agrees poorly with finite element computations of the force. We provide both "exact" and approximated formulas which agree well with finite element computations of the force on a rod-like particle for asperities from 2 to 100.