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 δ(2)(x,y)—defined as δ(x)δ(y)—within the volume charge density formula ρ(x,y,z)=λδ(2)(x,y), where λ is the linear charge density. Conversely, modeling a surface source (charge on the x−y plane) utilizes a one-dimensional delta function δ(z), leading to the density ρ(x,y,z)=σδ(z), where σ 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.
