A point charge sits at the origin

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Partial differential equations and boundary-value problems · Partial differential equations and boundary-value problems

Mathematical Methods for Physics, Ch The classical partial differential equations of physics and their boundary conditions

Problem

A point charge qq sits at the origin. Its potential for r≠0r\ne0 is φ(r)=q/4πϵ0r\varphi(\mathbf{r}) = q/4\pi\epsilon_0r, and Poisson's equation with this point source is ∇2φ=−q δ3(r)/ϵ0\nabla^2\varphi = -q\,\delta^3(\mathbf{r})/\epsilon_0. Using ∇2(1/r)=−4πδ3(r)\nabla^2(1/r) = -4\pi\delta^3(\mathbf{r}), verify that φ\varphi satisfies this Poisson's equation.

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