Condition (i) is of Dirichlet type since potential is given on a part of line z = 0 and (ii), which arises from the symmetry of the potential about plane z=0, is of Neumann type since potential derivative is given on the remaining part. From this circumstance the name of ‘mixed’ boundary problem. The most general solution of (1), limited at the origin and vanishing as r and z tend to infinity, can be expressed as an integral
where J0(z) is the ordinary Bessel function of zero order.
Inserting this solution in the conditions (i) and (ii) we obtain the integral equations which determine the unknown function f(p)
On the plane z = 0 the potential is
and on the upper/lower surface there is a charge density
Integrating over the disc surface and doubling (for the lower surface), we obtain the capacitance of the disc, C = 8e, and for a disc of radius R, C = 8eR.