Application of Computer Algebra System and the Mean-Value Theory for Evaluating Electrostatic Potential and Its Associated Field for Nontrivial Configurations

摘要

Evaluation of electrostatic potential at an arbitrary point within a two dimensional region free of electric charge containing geometrically dispersed nontrivial configurations electrified to constant potentials applying the standard classic approach, i.e. Laplace equation is challenging. The challenge stems from the fact that the solution of the Laplace equation needs to be adjusted to the boundary conditions imposed with the configurations. Numeric solution of the latter is challenging as it lacks generalities. There exist an alternative numeric method which is based on the application of The Mean-Value Theory. The latter is a pure numeric approach; although the output of its iterated refined version is successful, it is cumbersome. In this investigation utilizing the powerful features of Computer Algebra Systems, specifically Mathematica by a way of example we show an innovative approach. Our approach is based on a combination of numeric aspect of The Mean-Value Theory on one hand and Mathematica features on the other hand. This semi numeric-symbolic approach not only provides the desired output, but it also generates information beyond the scope of the standard classic method. By way of example we present the intricacies of our approach, showing (1) how the potential is evaluated and (2) how corollary essential information not addressed in classic cases such as electric field is calculated as well. Our method is applied to a two-dimensional case; its three dimensional version may easily be applied to cases of interest.