This theoretical paper deals with a fully developed free convective flow of an incompressible viscous and electrically conducting fluid in an infinitely long rigid vertical channel in the presence of an externally applied magnetic field. By retaining the induced magnetic field, we have carried out a detailed analysis of the field equations and obtained a host of interesting results corresponding to both open-circuit and short-circuit arrangements. The governing PDEs, which under the chosen physical configuration get transformed to a set of ordinary differential equations together with appropriate boundary conditions, have further been subjected to non-dimensionalisation. Using the theory of simultaneous ordinary differential equations, the analytical solutions for velocity, temperature field and induced magnetic field were obtained. These solutions were used to obtain important quantities of engineering interest such as current density, wall skin friction and volumetric flux for both open and short circuits.The effect of Hartmann number on the velocity, induced magnetic field and induced current density were shown quite extensively. Furthermore, the results for the symmetric and asymmetric heating of the vertical walls of the channelfor open- and short-circuit arrangements were compared. It is found that for the open-circuit arrangement, the velocity, induced current density and induced magnetic field are higher than that for the short-circuit arrangement.