A study is made of the flow engendered in a semi-infinite expanse of an incompressible non-Newtonian fluid by an infinite rigid plate moving with an arbitrary velocity in its own plane. The fluid is considered to be fourth order and electrically conducting. A magnetic field is applied in the transverse direction to the flow. The nonlinear problem is solved for constant magnetic field analytically using reduction methods as well as numerically and expressions for the velocity field are obtained. Limiting cases of interest can be deduced by choosing suitable parametric values.