Distribution of Charges on a Flat Disk as the Application of Far-Ranging Mathematics

Abstract

The foremost grail of this academic indagation is to delineate a mathematical expression of the normalised charge density over a flat disk. Aiming to ensue, firstly, 2 different frameworks have been dealt with to formulate the potential distribution which allows stability of a non-uniform charge distribution. At first, a logical but mathematically toilsome integral method has been approached. Out of the unyielding territory, we reduced the expression into algebraical functions using the Bessel coefficient and Green’s theorem, eventually inferring a new mathematical equivalence. Subsequently, this paper explores beta function as a solving tool of complete elliptic integral so that the normalization of charge apportion leads to 0 gradients of potential. Finally, the article deduces an integral equation whose implicit solution brings into the required charge distribution. The write-up also encounters finding a proximate graphical illustration of the assortment following the CAS system and direction fields. Beyond the conventional approach of real analysis, it facilitates proving the convergence of an acclaimed series. Consequently, it conceives a discussion on image charges for a flat disk. Even a short view of the article’s impact on practical fields of biology and engineering sciences has been included as the denouement. So, it might be of interest to the wide-ranged audience of research scholars in both the fields of physical and mathematical sciences.
 HIGHLIGHTS
 
 Expression of potential at any arbitrary 3D space point due to uniformly charged disk
 Natural unconstrained normalisation of charge density
 Implicating trigonometric solution in complete elliptic integral
 Proving the convergence of a bivariate divergent series
 Results can be of interest to electrostatic and condensed matter physicist including mathematicians of real analysis
 
 GRAPHICAL ABSTRACT