Consider a finite set of point charges in three dimensions. These generate an electrical field F = (X,Y,Z). The field is zero when all its components X,Y, and Z are zero. What is the structure of the set of points (x,y,z) where X, or Y, or Z are zero? What can one say about the structure of the points (x,y,z) where in fact F(x,y,z) = (X(x,y,z),Y(x,y,z),Z(x,y,z)) = (0,0,0)? There are experimental, computational, and theoretical ways to consider these questions. One goal of this project is to study the long open question attributed to Maxwell, “If the point charges all lie in a plane P, is the set of zeros of F that lie in the plane P actually a finite set?”
Basic algebra and calculus (including power series). Computations can be done using Mathematica, MATLAB, or perhaps other software depending on the skills and interests of the participants.