Distributing charge removes voltage singularities

In electrostatics, the fundamental relationship between the electric potential ($V(x)$) and the volume charge density ($\rho(x)$) is established by Poisson's Equation ($\nabla^2 V(x)=-\frac{\rho(x)}{\epsilon_0}$), which is derived directly from Maxwell's Equations, specifically the differential form of Gauss's Law. This equation utilizes the Laplacian operator ($\nabla^2$) and the permittivity of free space ($\epsilon_0$). The resultant physical field is critically dependent on how the charge is mathematically modeled: when a point charge is modeled using the Dirac delta function, which is the sole mathematical source of the singularity, the system's response is a potential field characterized by a singularity and the iconic decay. Conversely, when the charge is modeled as a distributed source (like a hollow sphere), the singularity is eliminated, proving that the potential remains finite and constant inside the charge layer, resulting in a physically smooth field at the origin.

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