The Influence of the Self-Magnetic Field on the Steady-State Current Distribution in an Axially Flowing Conducting Medium

The steady-state electric current distribution in a multicathode-spot vacuum arc was determined by a solution of the magnetic transport equation subject to various boundary conditions. The inter-electrode region of the arc is modeled as a uniform plasma flowing from the cathode to the anode. Dimensional analysis shows that three parameters determine the magnetic field, and hence the current density which is derived from it: AR-the ratio of the electrode separation to the electrode radius, Rmm-magnetic Reynolds number of the axial material flow, and Rme-magnetic Reynolds number of the axial electron flow. While the anode side of the conducting medium is described as an equipotential surface, the following three cases of boundary conditions for the cathode side are examined: 1) a known current density distribution is assumed over the entire cathode side of the plasma surface; 2) the cathode side is an equipotential surface; and 3) the current is allowed to cross the cathode surface only through a finite number of ring shaped regions. Numerical solutions of the nonlinear magnetic transport equation show a constriction of the current at the anode side for all boundary conditions mentioned. On the other hand, the current moves to the perimeter of the cathode for boundary condition 2). When AR, Rmm, and Rme equal 0.72,-0.16, and 1.73, respectively, and a uniform current density flows at the cathode side, the on-axis current density at the anode is six times larger than its value at the cathode.