Electric Field Derivation from the Scalar Potential of a Filamentary Plane Uniformly Charged Ring at Any Point in Space

The growing popularity of technological devices has led to renewed interest in the electric fields generated by various configurations of charged elements. Early physicists calculated these fields, but only for the simplest cases. Several textbooks on electrostatics have calculated the off-axis positions of rings, and these solutions can be obtained using approximate expressions. The major difficulty in calculating the field for nearly all configurations is that the equation cannot be solved without using special functions, such as elliptic integrals. Althoug many of these are tabulated, the calculations are laborious. However, such calculations can be performed using computers, because software programs or subroutines can be written for many of these special functions. This study investigates the derivation of the electric field from the electric potential created by filamentary plane uniformly charged rings at arbitrary points in space. The field expressions were obtained as functions of the complete elliptic integrals of the first and second kinds. Our method provides a basis for studying the advanced electrostatics of uniformly charged rings