📝 SummarXiv

Abstract

We study the distribution of point charges in a straight conductive needle and the electric field created by them. Starting from the bead model with $n$ point charges on the needle, we show the existence and uniqueness of an equilibrium state. We also study the differential system pertaining to the system and show that the system of point charges does not converge towards an equilibrium state. In order to move from a discrete to a continuous model we increase the number of the point charges and explain the paradoxical convergence towards a uniform distribution although the charges tend to accumulate towards the ends of the needle (both at equilibrium and for the differential system of equations describing the Newtonian motion). This convergence has to be understood as that of the distribution functions of a sequence of probabilities (whether it is at equilibrium or when the charges move). We also provide visual illustrations that help understanding the studied phenomena.