The equipotential conducting disk: An analytical and computational study

Abstract

Abstract

Abstract We present a pedagogical analytical and computational study of the classical problem of an isolated conducting disk maintained at a constant electric potential. Although the physical statement of the problem is simple, the determination of the surface charge density is nontrivial and provides a natural introduction to electrostatic integral equations. Starting from the potential of an arbitrary axisymmetric charge distribution on a disk, we discuss both the elliptic-integral and Bessel-function representations of the problem. We then impose the equipotential condition and obtain the corresponding integral equation for the unknown surface charge density. The exact solution is reviewed, emphasizing the inverse-square-root divergence at the rim and the associated edge effect. A direct numerical implementation based on a Chebyshev-weighted Nyström discretization is then developed. The logarithmic singularity of the elliptic kernel is treated by a local diagonal correction, without imposing the known analytical solution. In addition, we introduce a general algebraic edge factor and show numerically that the regularized density remains finite only when the trial exponent is s=1/2. Thus, the characteristic edge singularity is recovered from the numerical solution itself rather than assumed in advance. The numerical density, capacitance, and edge exponent are compared with the analytical results, providing clear tests of convergence and physical consistency. The conducting disk therefore serves as a compact educational example connecting elementary electrostatics, special functions, boundary conditions, singular charge distributions, and the numerical solution of integral equations. The approach is suitable for advanced undergraduate or beginning graduate courses in electromagnetism, computational physics, or mathematical methods in physics.