Studies of convection in the horizontal Bridgman configuration were performed to investigate the flow structures and the nature of the convective regimes in a rectangular cavity filled with an electrically conducting liquid metal when it is subjected to a constant vertical magnetic field. Under some assumptions analytical solutions were obtained for the central region and for the turning flow region. The validity of the solutions was checked by comparison with the solutions obtained by direct numerical simulations. The main effects of the magnetic field are first to decrease the strength of the convective flow and then to cause a progressive modification of the flow structure followed by the appearance of Hartmann layers in the vicinity of the rigid walls. When the Hartmann number is large enough, Ha > 10, the decrease in the velocity asymptotically approaches a power-law dependence on Hartmann number. All these features are dependent on the dynamic boundary conditions, e.g. confined cavity or cavity with a free upper surface, and on the type of driving force, e.g. buoyancy and/or thermocapillary forces. From this study we generate scaling laws that govern the influence of applied magnetic fields on convection. Thus, the influence of various flow parameters are isolated, and succinct relationships for the influence of magnetic field on convection are obtained. A linear stability analysis was carried out in the case of an infinite horizontal layer with upper free surface. The results show essentially that the vertical magnetic field stabilizes the flow by increasing the values of the critical Grashof number at which the system becomes unstable and modifies the nature of the instability. In fact, the range of Prandtl number over which transverse oscillatory modes prevail shrinks progressively as the Hartmann number is increased from zero to 5. Therefore, longitudinal oscillatory modes become the preferred modes over a large range of Prandtl number.