The cross-section between the plates is discretized on a grid and solved for the real electrostatic potential by relaxing Poisson's equation for a spatially varying permittivity ε(x,y):
∇·(ε∇V) = 0 (no free charge between plates)
Gauss–Seidel / SOR update at each free node:
V = [εE·V_E + εW·V_W + εN·V_N + εS·V_S] / [εE+εW+εN+εS]
(εE etc. = harmonic mean of ε across each cell face)
E = −∇V, u = ½ε|E|², U = ∫u dV, C = 2U / V²
The plates are fixed-potential rows (+V/2 and −V/2); everything else — the bulging field lines beyond the plate edges, the non-uniform field near the corners and the dielectric boundary, the deviation from the textbook capacitance — falls straight out of the relaxation, not out of a decorative overlay. The idealized formula C = ε0·εr·A/d assumes a perfectly uniform field confined exactly to the plate footprint and ignores both the outward bulge at the plate edges (which adds capacitance) and the way field lines bend and redistribute at a dielectric-air boundary inside the gap (which can add or subtract, depending on geometry) — the "Δ vs. ideal formula" readout is that real difference, sign included, not a fixed correction factor.
- Slab insertion — slides a dielectric block horizontally into the gap between the plates from the left edge, like the classic lab demo; the field inside the dielectric weakens (E = D/ε) while the free charge needed to hold the same voltage rises, so C climbs as the slab goes in.
- Background heat-map — the solved potential V(x,y), red positive / blue negative.
- White streamlines — the electric field direction, integrated through the bilinearly-interpolated solved field, including the outward bulge at the plate edges.
- Drifting dots — positive test charges advected along the locally solved field (illustrative motion, not a literal drift-velocity model), moving faster where the field is stronger.
The whole domain is bounded by a grounded enclosure a short distance beyond the plates (a standard finite-domain approximation for this kind of solver), and the model assumes 1 m of depth into the screen — real capacitance also depends on that out-of-plane length, held fixed here so the cross-section stays comparable across settings.