For a 10N-class, steady-state, self-field magnetoplasmadynamic (MPD) thruster using hydrogen as the propellant, the distribution of discharge current path has been investigated by means of axisymmetry two-dimensional magnetohydrodynamic (MHD) flow simulation including an electrode sheath model with cathode temperature distributions. The discharge current path concentrates in the downstream region of the thruster, particularly on the anode edge and the cathode tip, because the ionization of hydrogen occurs after the dissociation. This feature is so dominant that the cathode sheath voltage is determined mainly by the temperature at the cathode tip and the discharge current path is hardly affected by the temperature gradient of the cathode. These characteristics are quite different from those for argon MPD thruster.