This article is a study of a family of nonlinear force-free magnetic fields (FFMFs), in Cartesian geometry under assumption of translational symmetry, as simple models of the magnetic fields in the solar corona. For this configuration all the physical quantities are invariant under translations in a fixed direction to be the direction Oz of a Cartesian coordinate system. Two classes of exact analytic solutions for the steady state are obtained. These solutions may be helpful in understanding the physics involved in the transition from the low-confinement to the high-confinement mode in tokamaks. In particular, they can be employed for stability investigations, which would be of relevance to magnetic confinement systems. Further, the obtained solutions may have several applications in the study of solar photosphere, the solar corona, as well as astrophysical plasmas.