Summary: Enforcing Gauss’ law in numerical simulations of Maxwell’s equations, specially in plasma physics, is a well-known key issue. Although the set of Maxwell’s equations used for simulations do not include divergence laws, augmented systems which incorporates Gauss’ law while retaining an overall hyperbolic form are frequently used to cope with charge conservation defects. Focusing on bounded domains and usual boundary conditions, existence and well-posedness of associated augmented systems have already been established. These extensions are non-unique and, among them, some surprisingly do not model properly the solutions of Maxwell’s equations anymore. This matter is illustrated by a constructive example of waves propagating in a rectangular waveguide. A result providing both existence and uniqueness of an extension from Maxwell’s equation to a proper augmented problem is proved by a perturbation argument. The main idea is that the solutions of Maxwell’s equations must be a ground state, whenever charge conservation holds, for the solutions of any proper associated augmented system.