Where,

L = length of the rectangular parallel plates

w = width of the rectangular parallel plates

d = dielectric thickness

Δt = increase in effective length and width due to edge effect

ε_r = relative permittivity

Edge Effect in a Capacitor

If d << √A, then on the length scale of d, near the edge of the capacitor, the capacitor looks like parallel semi-infinite plates separated by a distance d. Let us suppose that the upper plate is held at potential ϕ₀/2, and is located at a distance y = d/2 above the x-axis; the lower plate is held at potential −ϕ₀/2 and is located at y = −d/2. Suppose that δ₀ is the change in capacitance per unit length associated with such a scenario.

We can approximate

C/ε = A/d + Lδ
where, L is the perimeter of the capacitor plate.

To compute the edge effects, we use conformal mapping. Consider the transformation,

z=dw/ϕ0 + de2Πw/ϕ0/2Π

Here z represents the spatial coordinate, and w the complex electric potential.

Thus we can write

δ = 1/2Π * log(√A/d) + δ₀
where δ₀ is an unknown constant (independent of d or A).