1. The Electric Field Disc Problem

The simplest problem which can be reduced to a pair of dual integral equations is the determination of potential field generated by an electrified disc. We suppose that a circular disc of unit radius, the centre of the disc being at the origin of a cylindrical coordinate system, is at potential unity. We want to find the potential u(r,z) in all the space. The potential satisfies the axis-symmetric Laplace equation

with the boundary conditions

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)

This pair of integrals is of type (I) and (II) and they represent a first example of dual integral equations with Bessel type kernel. Looking at the tables of Bessel integrals it is immediate to find that f(p)=(2/pi)sin(p) satisfies the above integrals. Inserting this solution in (2) the problem of electrified disc is completely solved

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.

Dual Integral Equations

Definition


The pair of integral equations


where the kernel K(p,r), the auxiliary kernels G(p), g(r) and h(r) are given functions and f(p) is the unknown function to be determined, are called (by a rather ambiguous term) dual integral equations. They mostly appear in the solution of boundary vale problems where the conditions on a boundary line are of mixed type. Therefore we begin our notes giving some simple problems in cylindrical simmetry that historically were the firsts to involve dual integral equations with Bessel type kernels : K(p,r)=Jn(pr). Usually h(r)=0.