Electric Charge and Electric Field Modeling Classical Theory, Computational Methods, and Multiscale Challenges

Abstract

Electric charge and the associated electric field are foundational to classical and modern physics and underpin electrostatics, electromagnetism and a wide range of engineering applications. This critical review synthesizes peerreviewed literature published between 1990 and 2026 with an emphasis on classical and continuum electrostatic theory, computational and numerical modelling, experimental characterization, and emerging AI-assisted and physics-informed methods. The analysis shows that classical formulations remain valid at continuum scales, but in heterogeneous, interfacial, multilayer-dielectric and nanoscale systems they require hybrid continuum-particle, coupled-physics or surrogate-aided extensions. Numerical methods such as the finite-element method, boundary element method, finite difference method and particle-in-cell approaches are widely used, but their reported accuracy depends strongly on geometry, mesh resolution, dielectric contrast, and boundary specification and solver convergence criteria. Physics-informed neural network surrogates offer efficiency but remain limited in interpretability and out-of-distribution generalization.  Experimental uncertainties arising from calibration drift, probe geometry, environmental noise and numerical post-processing propagate through the validation chain. Future progress will depend on the integration of rigorous physical theory, transparent uncertainty quantification, standardized benchmarks and physics-informed data-driven methods within coherent multiscale frameworks suited to electrostatic analysis in next generation technologies.