Electric field and the charge distribution on the surface of an insulator in a vacuum

The free charge steady-state distribution over the insulator surface that arises in a strong electric field in a vacuum can be found by solving the boundary-value problem for the electrostatic field strength if the angle between the field vector and vacuum-insulator interface is given. A general solution to this boundary-value problem is derived for the case of an in-plane field and rectilinear interfaces. Laws of charge and field formation that follow from the solution obtained are considered. Formulas for the electric field strength and charge density in terms of elementary functions are obtained for a number of particular cases. Power-type expressions for the electric field and a critical angle between the electrode and insulator surface that describe the field behavior and charge distribution near the vacuum-insulator-electrode contact are derived.