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?
Answer
Yes, it is possible, but not in the way you suggest.
It's totally possible for Poisson's equation with non-continuous electric field; the charge density (and hence the second derivative of potential) could be non-finite. This is less esoteric than it seems; the electric field of a point charge is discontinuous at the location of the point charge itself.
There's another way to show that Poisson's equation is not enough, though. Consider an electric field which is constant, ie $\mathbf{E} = \mathbf{E_0}$ for some constant $\mathbf{E_0}$. We can deduce that the charge density is $0$ everywhere, but this is independent of $\mathbf{E_0}$! So if we are given that $\rho=0$ to start with and nothing else, there is no unique solution for the electric field.
This shows that in addition to Poisson's equation, we must have boundary conditions. In most cases it is implicitly assumed that the boundary conditions are that $E \to 0$ at large distances.
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