In this Communication, we consider in the half-space {z > 0} a fully 3-D force-free magnetic field B˷ embedded in a perfectly conducting plasma and its quasi-static evolution driven by motions imposed to the feet of its lines on the boundary {z = 0}. Assuming that the field lines of B˷ have a simple topology - i.e. that it is possible to choose a (non-unique) set of nested magnetic surfaces which are either “arcade-like” or “tube-like” - we first establish a global representation of B˷ in terms of two Euler potentials u and v (with v multivalued) and derive new formulae giving in particular the relative helicity end the energy of B˷ as functions of the values of u and v (and of their derivatives) on {z = 0}. We thus establish analytically some general qualitative features of the behaviour of B˷.