Simple Solutions of Lattice Sums for Electric Fields Due to Infinitely Many Parallel Line Charges

Abstract

We present surprisingly simple closed-form solutions for electric fields and electric potentials at arbitrary position ( x , y ) within a plane crossed by infinitely long line charges at regularly repeating positions using angular or elliptic functions with complex arguments. The lattice sums for the electric-field components and the electric potentials could be exactly solved, and the duality symmetry of trigonometric and lemniscate functions occurred in some solutions. The results may have relevance in calculating field configurations with rectangular boundary conditions. Several series related to Gauss’s constant are presented, established either as corollary results or via parallel investigations conducted in the spirit of experimental mathematics.

Keywords

• lattice sums
• electric potential
• electric field
• dimensional analysis