 Bounded two-dimensional capacitor field model

The diagrams solve an explicitly bounded Dirichlet problem in a uniform charge-free medium of constant permittivity. Take a width scale $W=1$ and terminal electric potential difference $V=1$. Two zero-thickness ideal plate segments occupy $|x|\le W/2$ at $y=\pm d/2$, with electric potentials $\pm V/2$. Fix $\varphi =0$ on the entire rectangle $x=\pm 2W$, $y=\pm 1.5W$. This boundary is an additional enclosing equipotential, rather than a condition at infinity. The model is invariant along the omitted third direction. The connection between the electric potential, its gradient, and the charge-free Laplace equation follows from electrostatics.^[ref]

On a square mesh of spacing $h=W/160$, the finite difference method imposesat free nodes, while plate and outer nodes retain their prescribed electric potentials. There are $641\times 481$ nodes. Each plotted gap aligns exactly with the mesh; the smallest gap has 16 mesh intervals. The generators solve the resulting linear system by the Conjugate gradient method, stopping when the maximum dimensionless five-point residual is below $2\times 10^{-10}$, and use central finite differences to approximate $\mathbf{E}=-\nabla \varphi$ away from plate nodes.

The discrete maximum principle keeps the computed electric potential between $-V/2$ and $+V/2$. Reflection checks require evenness in $x$ and oddness in $y$; halving $h$ tests discretization effects, while enlarging the outer rectangle tests sensitivity to its artificial boundary conditions. These are different checks: a small residual only establishes that the discrete equations were solved. Differentiating near zero-thickness endpoints is especially sensitive to the mesh, so the plots do not report a converged edge peak. No total electric charge, exact open-domain three-dimensional capacitance, or measured material property is extracted from this model.

For a bounded numerical check, compare $h=W/80$ with the plotted $h=W/160$ at midplane positions $x/W=0,0.25,0.40,0.50,0.60,0.80$. At $d/W=0.10$, the largest change in $-E_{y}d/V$ is approximately $0.0265$, at $x/W=0.50$; the other five sampled changes are below $0.00472$. At $d/W=0.25$, moving the outer rectangle from $\pm 2W\times \pm 1.5W$ to $\pm 3W\times \pm 2W$ while retaining $h=W/80$ changes these samples by at most approximately $0.00622$. These calculated sensitivities support the qualitative comparison and expose its limits; they are not experimental uncertainties or evidence that the artificial boundary has converged to infinity.