A Symbolic-Numeric Approach to an Electric Field Problem

A combination of symbolic and numerical methods is used to extend the reach of the purely symbolic methods of physics. One particular physics problem is solved in detail, namely, a computation of the electric potential in the space between a sphere and a containing cylinder. The potential is represented as an infinite sum of multipoles, whose coefficients satisfy an infinite system of linear equations. The system is solved first symbolically by using a series expansion in a critical ratio, namely the ratio of the sphere radius to cylinder radius. Purely symbolic methods, however, cannot complete the solution for two reasons. First, the coefficients in the series expansion can only be found numerically, and, second, the convergence rate of the series is too slow. The combination of symbolic and numerical methods allows the singular nature of an important special case to be identified. Mathematics Subject Classification (2000). Primary 35C99; Secondary 68W25; Tertiary 65B99.