Numerical Solution to Poisson?s Equation for Estimating Electrostatic Properties Resulting from an Axially Symmetric Gaussian Charge Density Distribution: Charge in Free Space

oisson?s equation frequently emerges in many fields, yet its exact solution is rarely feasible, making the numerical approach notably valuable. This study aims to provide a tutorial-level guide to numerically solving Poisson?s equation, focusing on estimating the electrostatic field and potential resulting from an axially symmetric Gaussian charge distribution. The Finite Difference Method is utilized to discretize the desired spatial domain into a grid of points and approximate the derivatives using finite difference approximations. The resulting system of linear equations is then tackled using the Successive Over-Relaxation technique. Our results suggest that the potential obtained from the direct integration of the distance-weighted charge density is a reasonable choice for Dirichlet boundary conditions. We examine a scenario involving a charge in free space; the numerical electrostatic potential is estimated to be within a tolerable error range compared to the exact solution.