Background: Given that equilibrium solutions of charges in a spherical shell formed from a conductor satisfy the Poisson problem ∇2Φ=−ρϵ0 as a corollary of the differential form of Gauß's Law ∇⋅E=ρϵ0, would it be natural to model the events proceeding the equilibrium with the diffusion equation?
To make my question more rigorous, we prescribe the system with initial charge density ρ(r,θ,φ) that is not uniform. Without using a Galerkin method, does there exist an analytical solution for Φ(r,θ,φ,t)?
Pardon me if my intuition is grossly over simplified; however, could the transition to an equilibrium state be viewed as a progression to positive eigenvalues? Taking the diffusion pde in R3 (∂tu=k∇2u) to be parabolic with degenerate eigenvalues, would not the equilibrium in the Poisson equation represent an elliptic PDE (∇2Φ=−ρϵ0)? Additionally, the steady state implies that ∂tu=0, which enforces the requirement of being harmonic on the solution.
Question: Would the following initial value problem suffice for modeling the redistribution of the charge density and voltage to reach equilibrium. My system is underconstrained, which leads me to believe that I am neglecting some physical relationship.
∂tΦ=−kρ(r,θ,φ,t)ϵ0
ρ(r,θ,φ,0)=ρ0(r,θ,φ):R×[0,2π)×[0,2π)→R
I am struggling to find the evolution of ρ(r,θ,φ,t) in time. I understand that areas to which the →E-field lines converge will have increasing charge density, and the opposite will hold for the case of diverging lines. Could I take the divergence of the following expression for the →E-field? Would some clever application of Gauß's law suffice instead of evaluating the following integral?
→E(r,θ,φ)=∫π0∫2π0∫r1r0dq4πϵ0d((r∗,θ∗,φ∗),(r,θ,φ))32→d((r∗,θ∗,φ∗),(r,θ,φ))dr∗dθ∗dφ∗
d is the chordal distance and →d is the chordal displacement. They will correspond to the euclidean distance between the two points on the sphere written in Cartesian coordinates.
Meanwhile: I have found a solution for Laplace's PDE on the spherical shell. I am now attempting to use a Green's function to reconstruct a solution to the particular equation.