= Bounded two-dimensional capacitor field model
{title2=$\partial_x^2\phi+\partial_y^2\phi=0$}
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 $|x|\leq W/2$ at $y=\pm d/2$, with <electric potentials> $\pm V/2$. Fix $\phi=0$ on the entire rectangle $x=\pm2W$, $y=\pm1.5W$. 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>.<The Feynman Lectures on Physics, Volume II by Richard Feynman and others>[II.6–1, equations (6.3)–(6.6)]{ref}
On a square mesh of spacing $h=W/160$, the <finite difference method> imposes
$$
\phi_{i,j}=\frac14\left(\phi_{i+1,j}+\phi_{i-1,j}+\phi_{i,j+1}+\phi_{i,j-1}\right)
$$
at free nodes, while plate and outer nodes retain their prescribed <electric potentials>. There are $641\times481$ 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 $2\times10^{-10}$, and use <central finite differences> to approximate $\mathbf E=-\nabla\phi$ 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 $\pm2W\times\pm1.5W$ to $\pm3W\times\pm2W$ 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.
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