Electrostatic potential of a uniformly charged annulus: A study of dimensional crossover

We present a comprehensive analysis of the nature of the electrostatic potential generated by a uniformly charged annulus. The annulus, namely, the annular disk is characterized by an outer radius, $a$ and an inner radius, $b=\eta a$ where $0\le \eta \le 1$ is a dimensionless parameter which enables a continuous interpolation between a disk ($\eta =0$) and a ring ($\eta \to 1$). The goal is to investigate the geometric transition between two-dimensional (disk-like) and one-dimensional (ring-like) electrostatic behavior. The electrostatic potential created by the uniformly charged annulus is expressed in terms of complete elliptic integrals of the first kind and we explicitly analyze its behavior on its plane. We show that for all nonzero values of parameter $\eta$ the resulting electrostatic potential develops a pronounced maximum at a given radial position located within the annular region. As $\eta$ increases, this peak becomes sharper and shifts outward towards the outer radius. To quantify this geometric transition, we analyze the potential at the edge as a function of the annular thickness parameter, $\delta =a-b=(1-\eta )a$. Our analysis reveals a dimensional crossover at a critical thickness ${\delta }_c=0.59a$ identified by looking at the features of the second derivative of such potential with respect to $\delta$. The potential exhibits positive curvature for $\delta <{\delta }_c$ reflecting a concentration of charge near the edge, typical for a uniformly charged ring. Conversely, the potential displays zero curvature for $\delta >{\delta }_c$, with features that are more akin to a uniformly charged disk. This curvature-based approach offers a physically transparent and computationally accessible criterion for distinguishing between ring-like and disk-like electrostatic behavior.