Direct current discharges between two or several electrodes submerged into an electrically conducting viscous incompressible liquid are discussed. When the electrical conductivity of electrodes is much higher than the liquid's conductivity, the vector of the electric current density in the liquid has horizontal components only. With this, only the z-component of the electromagnetic force field is non-potential and generates electrically induced vortical flow in a conducting liquid. A general theory of such quasi-two-dimensional electromagnetic fields is described with examples for cylindrical, hyperbolic electrodes and horizontal dipole. Complex functions of complex variables are used to describe quasi-two-dimensional potential fields. Analysis of electromagnetic force fields allows approximate estimation of 3D patterns of related electrically induced vortical flow. Figs 15, Refs 7.