Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Point Charge

A theoretical charge concentrated at a single point in space, simplifying electrostatic calculations and analysis.
📜

The statement of the theorem

Let QQ be the total charge, and let r⃗0∈R3\vec{r}_0 \in \mathbb{R}^3 be the location of the charge. The point charge is mathematically modeled by the charge density ρ(r⃗)\rho(\vec{r}) defined as:\n\nρ(r⃗)=Qδ(r⃗−r⃗0)\rho(\vec{r}) = Q \delta(\vec{r} - \vec{r}_0)\n\nwhere δ(r⃗−r⃗0)\delta(\vec{r} - \vec{r}_0) is the three-dimensional Dirac delta function, satisfying ∫R3δ(r⃗−r⃗0)d3r=1\int_{\mathbb{R}^3} \delta(\vec{r} - \vec{r}_0) d^3r = 1.\n\nFrom this density, the electric potential V(r⃗)V(\vec{r}) at any point r⃗≠r⃗0\vec{r} \neq \vec{r}_0 is given by the integral of the potential kernel 14πϵ01∣r⃗−r⃗0∣\frac{1}{4\pi\epsilon_0} \frac{1}{|\vec{r} - \vec{r}_0|}:\n\nV(r⃗)=14πϵ0∫R3ρ(r⃗′)∣r⃗−r⃗′∣d3r′=14πϵ0Q∣r⃗−r⃗0∣V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \int_{\mathbb{R}^3} \frac{\rho(\vec{r}')}{|\vec{r} - \vec{r}'|} d^3r' = \frac{1}{4\pi\epsilon_0} \frac{Q}{|\vec{r} - \vec{r}_0|}\n\nThe resulting electric field E⃗(r⃗)\vec{E}(\vec{r}) is the negative gradient of the potential:\n\nE⃗(r⃗)=−∇V(r⃗)=14πϵ0Q(r⃗−r⃗0)∣r⃗−r⃗0∣3\vec{E}(\vec{r}) = -\nabla V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \frac{Q (\vec{r} - \vec{r}_0)}{|\vec{r} - \vec{r}_0|^3}\n\nThis formulation rigorously defines the point charge as a singular source term in the Poisson equation, ∇2V=1ϵ0ρ\nabla^2 V = \frac{1}{\epsilon_0} \rho, whose solution is the Coulomb potential.