Microstrip Capacitance for a Circular Disk Through Matched Asymptotic Expansions

The solution of the potential around two parallel circular disks separated by a dielectric slab is obtained by using the method of matched asymptotic expansions, asymptotic formula for the capacitance has been derived in the limit of small separation $2\delta $. The formula obtained includes terms of order $\delta $ as well. The mixed boundary value problem is solved by dividing the space around the parallel plates into three regions; the exterior region, the edge region, and the interior region. The solution of the edge region incorporating diegectric effects is obtained by using the Wiener–Hopf technique. The exterior solution of the circular disk problem is obtained by using Hankel transforms. The Hankel transform representation of the exterior solution facilitates the easy derivation of its edge expansion from the Lipschitz–Hankel integrals. The solutions are comiared with Shaw’s result for the free-space case [Phys. Fluids, 13 (1970), pp. 1935–1947] and her errors are corrected. Improvement of her approach to the free-space capacitor problem is also discussed.