Electrostatic force acting on a spherically symmetric charge distribution in contact with a conductive plane

A spherically symmetric charge distribution is a useful, first-order model of a charged insulating particle, such as a toner particle used in electrostatic imaging technologies. The electrostatic force of adhesion between a spherically symmetric charge distribution in contact with a conductive ground plane is derived using the method of images. Finite element analysis is used, with the uniformly charged sphere of charge Q modeled with K charge points, each of which has a charge of magnitude Q/K. Both a closed form solution and a numerical calculation are used to derive the total force. It is found that the electrostatic image force acting on the few charge points located in proximity to the conductive plane (the proximity charge) is comparable to the electrostatic force acting on a single charge point of magnitude Q located in the center of the sphere. This is a surprising result since it is conventionally assumed that a spherically symmetric charge distribution can be modeled by a single charge point Q located in the center of the sphere. However, this assumption is only valid at distances far from any conductive planes. When all pairs of charge points and their charge images are considered, the total electrostatic force acting on a sphere of charge Q in contact with a conductive plane is larger than the conventionally assumed electrostatic image force by a correction factor (1+4/π). The additional force, that we call the proximity force which is primarily due to the proximity charge (the 4/π term), is independent of the number of K charge points assumed, suggesting it is model independent at contact. The proximity forceʹs functional dependence on the separation between the bottom of the charged sphere and the conductive plane depends on the distribution of the charge points near the contact point. Implications of the existence of the electrostatic proximity force are discussed.