After studying the definition (& derivation) of the potential to an electric field and the Poisson equation I'm currently wondering whether the following is possible:
- Can one give an example of a physical setup where the Poission equation fails to provide the electric field?
I have tried to come up with an example but failed. What I thought of was, that considering a valid solution of the Poisson equation, the electric field derived from the solution would have to be differentiable (since the second derivative of the potential appears in the differential equation). If one were to find a setup with a discontinuous electric field, would the Poisson equation fail?