🎬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.

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How the delta function is used to model charge distributions concentrated on a line or a surface instead of a single point | Animated Resultsviadean on Notion