# How the delta function is used to model charge distributions concentrated on a line or a surface ins How the delta function is used to model charge distributions concentrated on a line or a surface instead of a single point The analysis of lower-dimensional delta functions shows a direct correlation between the dimensionality of the charge concentration and the severity of the potential singularity. While the 3D point charge (modeled by $$\delta^{(3)}$$ ) creates the most extreme field, characterized by the singular $$1 / r$$ decay, distributing that charge across a line ($$\delta^{(2)}$$) or a surface ($$\delta^{(1)}$$) progressively smooths the singularity. The line charge yields a milder logarithmic ($$\ln (r)$$) singularity, and the surface charge completely eliminates the singularity, resulting in a non-singular, linear potential ($$|z|$$) near the sheet. This confirms that the $$\delta$$ function is a flexible tool for modeling concentrated charge, but the unique $$1 / r$$ behavior is a signature reserved specifically for point sources in three dimensions. ### Brief audio {% embed url="" %} ### Condensed Notes {% embed url="" %}