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.
Highlights
The study of lattice sums has a long history
We targeted lattice sums arising from the consideration of electric fields and electric potentials set up by the crossing of infinitely many and infinitely long line charges through horizontally and/or vertically equidistant co-ordinates of the Gaussian plane
The equation presented in [2] allows changing the spacing between line charges, so it is essentially only a matter of complex algebra to arrive at our Equation (10b) via the superposition principle
Summary
The study of lattice sums has a long history. They are frequently encountered when studying physical and chemical systems (e.g., [1] and references therein). We targeted lattice sums arising from the consideration of electric fields and electric potentials set up by the crossing of infinitely many and infinitely long line charges through horizontally and/or vertically equidistant co-ordinates of the Gaussian plane. T. McDonald’s Notes on Electrostatic Wire Grids [2] where in turn a reference is made to Maxwell’s famous book A Treatise on Electricity and Magnetism. In solid-state physics, stress fields generated by screw dislocations seem to be mathematically analogous to the electric field due to line charges (see Section 2.8 of [3] and references therein)
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