I wrote a program to solve Laplace's equation using a finite difference over relaxation algorithm. A test case was setup to calculate the field between two concentric spheres where the exact solution is known. The program matched the exact solution to within 0.05 percent. The attached plot shows the E field magnitude for a 10 cm radius sphere with a 10 cm hole held at a voltage of 150 KV. The center of the sphere is at a height of 50 cm from the ground plane. The current hole edge is 1 mm thick and is not curved in to reduce the E field. I'm working on adding this inward re-entrant geometry to the program.