Charge Singularity at the Corner of a Flat Plate

In the present work we calculate the strength of the singularity in surface charge density at the vertex of a sectorial conducting plate in an electrostatic field, including the case in which the media occupying the half-spaces above and below the plate have different dielectric constants. The singularity strength is calculated numerically for different vertex angles $\chi $. In addition, for a “spike” $( {\chi \ll 1} )$, a “nearly straight edge” $( {| {\chi - \pi } | \ll 1} )$ and a “slit” $( {2\pi - \chi \ll 1} )$ , analytic expressions for the singularity strength are found by singular perturbation techniques. The results are relevant to the calculation of circuit board stray capacitances, where one must determine the charge densities on rectilinear sheet conductors, mounted on a dielectric board, for constant conductor potentials. For this case, it is found that the charge singularity, inversely proportional to the square root of the distance from the edge for points far from the vertex, is proportional to $r^{ - 0.7034} $ at an outside corner $( {\chi = {\pi / 2} )$ and to $r^{ - 0.1854} $ at an inside corner $( {\chi = {{3\pi / 2}} )$, where r is the distance from the vertex.