Closed-form expressions of the electrostatic potential close to a grid placed between two plates

The potential distribution between a grid and two plates is an electrostatic problem already solved for various applications such as MultiWire detectors used in Nuclear Physics, or electrostatic precipitators in Engineering. Since references and notations for this analytical solution are ancient and rather bewildering, the first part of this paper presents a revisit and a discussion of the formulations that assume a line charge on the wire. This is completed by establishing a ready-to use closed-form expression valid for the general configuration where the plates are not grounded. The second part is about the investigation of the line model accuracy close to the wires, using both analytical and numerical approaches. In the symmetric case where the grid is placed at equal distances between two grounded plates, it is shown that the error can be modelled using a quadrupole charge. For the asymmetric case, a larger discrepancy of the line model is brought to light, with an error featuring a dipole-like distribution. In order to cope with this small but not negligible error, a classical dipole model is implemented, leading to an accurate theoretical expression of the potential.