📢Delta Functions Reshape Electric Potential

The sources collectively demonstrate the utility of the delta function as a flexible tool for mathematically modeling concentrated charge distributions and reveal how the physical dimensionality of that concentration affects the resulting electric potential. Specifically, modeling a line source (charge along the $z$-axis) requires using a two-dimensional delta function $\delta^{(2)}(x, y)$—defined as $\delta(x) \delta(y)$—within the volume charge density formula $\rho(x, y, z)=\lambda \delta^{(2)}(x, y)$, where $\lambda$ is the linear charge density. Conversely, modeling a surface source (charge on the $x-y$ plane) utilizes a one-dimensional delta function $\delta(z)$, leading to the density $\rho(x, y, z)=\sigma \delta(z)$, where $\sigma$ is the surface charge density. Physically, concentrating charge into lower dimensions progressively smooths the singularity: the most extreme singularity is the $\mathbf{1/r}$ potential characteristic reserved exclusively for the 3D point charge, while a line charge produces a milder logarithmic singularity, and a surface charge completely eliminates the singularity, resulting in a non-singular linear potential near the sheet.

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Delta Functions Reshape Electric Potential | Audiosviadean on Notion