If the electric field and boundary conditions are known exactly for a region of space, is it true that there exists only one charge distribution in that region of space that could have produced it? My understanding of the uniqueness theorem in electrostatics is that for a given charge distribution and boundary conditions for a volume, there exists only one (unique) solution to Poisson's equation, and thus the electric field in that volume is uniquely determined. Does the arrow point the other way, too? If we know the field and boundary conditions, is the charge distribution uniquely determined in the volume? Is there a simple example that illustrates why or why not?