It is shown that a weakly ionized plasma column in an axial magnetic field and an inward‐directed radial electric field—a Penning discharge at higher pressures—is unstable against a macroscopicm= 1 perturbation. The instability is analyzed by dimensional analysis. The applicability of this method is discussed. A steady‐state model of some experimental Penning discharges is developed. On the basis of this model, the critical magnetic field for the onset of instability as predicted by the theory agrees well with the values observed. The instability discussed here is caused by the presence of a third fluid, the neutrals, in an electron‐ion plasma. The same mechanism is also applicable to the initial phase of some rotating plasmas of the homopolar type.