I intend to model the capacitance, energy storage capabilities, and fringing fields of capacitors. The assumption of infinite area will be discarded in favor of using analytic approaches whenever possible; for example, one could start with Coulomb’s Law for a surface charge density:
where is the separation vector between a source point and the field point. From here, one can theoretically model the electric field of a sheet of arbitrary shape and charge density. I will start with a uniform charge density and a simply shaped sheet and work up to more complicated, realistic arrangements of positive and negative plates (i.e. capacitors, though not necessarily parallel plate capacitors only).
When a capacitor with a vacuum between the plates has been satisfactorily described, the effects of dielectrics placed between the plates could be modeled. Exotic dielectric media could be explored. If feasible, these realistic simulated capacitors could be incorporated into an RLC circuit model. As stated above, analytic solutions will be preferred, with approximations used only when the exact answer is computationally impractical. Approximations can, however, be used often as reality checks, as well as to compare the accuracy and computational requirements of approximations with those of the analytic solution.