An approximate analytical solution is found of a quasi-static growing steady-state mode III crack in an elastic perfectly plastic solid. The analysis makes use of a close analogy between the mode III crack problem and the problem of the thin electrical conducting plate that is embedded in a medium of finite conductivity through which electrical current flows. Plastic strain is calculated from the motion of non-redundant infinitesimal screw dislocations. Screw dislocations are Eshelby's analogue of lines of electrical charge. Calculation of the dislocation crack tip shielding can be used to check the analysis. The plastic zone is a combined Dean & Hutchinson (DH) type unfocused fan plastic zone and a Chitaley & McClintock (CM) type focused fan plastic zone. The asymptotic solution at the crack tip is the CM one. The plastic zone found differs significantly in size and shape from the DH and the CM plastic zones. Novel features in the analysis are the use of a (dislocation density required) stress gradient boundary condition and the introduction of a shifting centre cylindrical coordinate system.