A Wiener-Hopf solution to the triple integral equations for the electrified disc in a coplanar gap

The axisymmetric potential problem for a plane circular electrode of radius a in a concentric hole of radius b in a coplanar earthed sheet is formulated in terms of triple integral equations for the Hankel transform of the potential, and reduced to a single Fredholm equation by use of the Erdélyi-Kober fractional operators. In the limit of small gap width (b − a)/b, the equation takes the form which is solved by applying the Wiener-Hopf technique to the Mellin transform of f(x). This leads to the asymptotic expression for the capacity of the disc; for the opposite limit the expression is derived. Numerical integration of the governing Fredholm equation has been carried out for a range of intermediate values of b/a.