Electric analysis of a conducting hemisphere

Abstract

Using Legendre polynomials, the boundary value problem of a charged, conducting hemisphere in an infinite space was reduced to the solution of an infinite system of linear, algebraic equations. Analytical solutions of electric charge and electric stress on the surface of the hemisphere were obtained. The numerical analysis revealed non-uniform distributions of the electric charge and electric stress over the surface of the hemisphere with local singularities at the edge of the hemisphere. Both the electric charge and electric stress distributions were expressed in terms of the power function with respect to the distance to the nearest hemisphere edge. The power index for the flat surface is larger than that corresponding to the spherical surface. Numerical result of the capacitance of the conducting hemisphere is the same as the result reported in the literature. There is no net force acting on the hemisphere.