This paper establishes some general properties of a two‐dimensional (z‐invariant) model describing smooth equilibria of a magnetized quasineutral plasma enclosed in a finite cylindrical box of arbitrary cross section &OHgr; and submitted to the following constraints: (i) the normal component of the magnetic field is fixed on the box boundary; (ii) the plasma is in contact with a heatbath at temperatureT; (iii) the plasma carries in thezdirection a current whose total intensityIis externally imposed. It is shown in particular that: (i) the free energy of the system is bounded from below if and only if the dimensionless control parameter &lgr;≤1 (&lgr;∝IT−1/2); (ii) a minimum free energy state (thus thermodynamically absolutely stable) exists for &lgr;≤1 and is actually the unique equilibrium that can be achieved if &lgr; is small enough; (iii) this state is absolutely stable with respect to all ideal magnetohydrodynamic perturbations; (iv) no equilibrium can exist if &lgr; exceeds a critical value &lgr;c≥1 (at least in the case where &OHgr; is ‘‘star shaped’’). The nonexistence of a minimum free energy state and even of any available regular equilibria with high current may have some consequences for understanding eruptive processes in plasmas, which are briefly discussed.