= Capacitor gap aspect ratio
{title2=$\eta=d/W$}
The <capacitor gap aspect ratio> compares the gap $d$ with a lateral plate width $W$. For rectangular plates of widths $W_x,W_z$, require both $d/W_x\ll1$ and $d/W_z\ll1$ for the leading <parallel-plate capacitor> approximation. Large <area> alone is insufficient: a long narrow strip can have large <area> and still have $d$ comparable with its smaller width. In addition, the observation point must be many gap widths from an edge. Reducing $d/W$ improves the interior approximation without making the edge itself disappear.
In a two-dimensional straight gap, write the <electric potential> as the infinite-plate solution plus a perturbation:
$$
\phi(x,y)=\frac{V}{d}y+\delta\phi(x,y),\qquad -\frac d2<y<\frac d2.
$$
The perturbation satisfies the <Laplace equation> and vanishes on the two ideal plates. <Separation of variables> in a semi-infinite strip, with $s$ the distance inward from its entrance, gives transverse modes
$$
\delta\phi_n(s,y)=a_n e^{-n\pi s/d}\sin\!\left(\frac{n\pi(y+d/2)}{d}\right),\qquad n=1,2,\ldots.
$$
Thus edge disturbances have a natural penetration scale set by $d$. Modes forbidden by a particular <symmetry> have zero coefficients; this expression is an explanation of localization in an ideal strip, not a universal numerical correction to finite-plate <capacitance>. The outside <boundary conditions> determine the coefficients.
\Image[/capacitor-fringing-small-gap-aspect-ratio.png]
{title=<Capacitor gap aspect ratio> and the width of the nearly uniform <electric field> region}
{description=Three original solutions of the same <bounded two-dimensional capacitor field model> at $d/W=0.10,0.25,0.50$. The lower curves show $-E_y d/V$ on the midplane $y=0$; the dashed reference is the infinite-plate value inside the gap. The dashed value is not the infinite-plate prediction outside the plate footprint. All curves are calculated, with the same mesh and outer <Dirichlet boundary conditions>; they are neither measurements nor an exact open-domain three-dimensional <capacitance> calculation.}
There is no single aspect-ratio cutoff guaranteeing an assigned accuracy: the requested observable, plate shape and thickness, surroundings, <dielectric> distribution, and guard geometry all matter. The small-gap approximation is local and is not uniform across an edge. It also does not justify extrapolating $C\simeq\epsilon A/d$ to arbitrarily large $d$, where finite <electrical conductors> and their external <electric field> dominate. At fixed $V$, formally shrinking an ideal gap makes the interior <electric field> grow like $1/d$; real <electric currents> across an imperfectly insulating gap, surface roughness and contact eventually invalidate that ideal limit.
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