It is shown that the self magnetic field of a high current discharge plays an important part in determining the radial electron density distribution The effect of this magnetic field can be treated by adding a `magnetic potential' term V_m=f^r₀wH dr to the electrostatic potential V_s in the Boltzmann equation for the radial electron density distribution, w being the longitudinal electron drift velocity. The drift of electrons and ions to the tube walls is equalized at low arc currents by a radial electric field. Since the magnetic feld reduces the drift of electrons to the wall whilst having little influence upon the ions, increasing the arc current leads to a progressive reduction of the radial electric field Probe measurements in an arc at low pressures show that at currents over 50 amperes, the radial electric field is zero except for a narrow sheath at the wall The axial density is then mainly determined by the self magnetic field A theoretical expression is derived for the equilibrium density distribution assuming a balance between the diffusion rates due to the concentration gradient and the magnetic forces. Experiments in a mercury vapour arc at high currents support this theory