Can the Electric field and/or electric potential be determined (uniquely) given only boundary conditions? For example I was wondering if we have coaxial cylinders with inner radius $a$ and outer radius $b$ with $a<b$. The voltage difference between a and b is V. Is this sufficient to determine (uniquely) an electric field? Or are there many electric fields that satisfy that the potential between the coaxial cylinders' radius a and b is V. If yes. How much information do we need to do so?
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Assuming electrostatics here for simplicity as it is hard to define unique potentials otherwise.
There is a uniqueness theorem that guarantees the solution of these sorts of boundary value problems is unique in the sense that if a solution to Laplace's equation $\nabla^2 V=0$ can be found that satisfies the boundary conditions, then the solution is unique.
I haven't been able to find a proof succinct enough to include as an answer here (but I am interested to learn of one.).
There is one here How do I show that the Laplace equation has a unique solution under the Dirichlet closed-surface boundary condition? but I don't think it covers the general case. There's also one at Wikipedia: Uniqueness theorem for Poisson's equation. I can transcribe that one here, but I think it makes more sense as a reference.
