  We study 4-dimensional asymptotically flat electrostatic electro-vacuum
spacetimes with a connected black hole, photon sphere, or equipotential photon
surface inner boundary. Our analysis, inspired by the potential theory approach
by Agostiniani-Mazzieri, allows to give self-contained proofs of known
uniqueness theorems of the sub-extremal, extremal and super-extremal
Reissner-Nordstr\"om spacetimes. We also obtain new results for connected
photon spheres and for connected photon surfaces in the extremal case. Finally,
we provide, up to a restriction on the range of their radii, the uniqueness
result for connected (both non-degenerate and degenerate) photon surfaces in
the super-extremal case, not yet treated in the literature.
