According to classical Electrodynamics, the motion of a slowly accelerated charged particle in an electromagnetic field is ruled by the classical Lorentz Force equation (LFE), which is one of the fundamental equations in Mathematical Physics. Despite of this, it has been a remarkable lack of qualitative and quantitative results about its dynamics until recently. One of the main reasons is that the proper mathematical tools for it were developed during the last quarter of the last century. Moreover, such techniques were conceived in an abstract mathematical framework, and their application to LFE appears during the last 15 years. In this talk we will discuss different approaches from the nonlinear analysis (variational methods, topological degree) to the periodic problem associated to LFE, where general electromagnetic fields (continuous or with singularities) are considered. On the other hand, we will also address the dynamic of a charged particle in the electromagnetic field induced by a periodically time dependent current J along an infinitely long and infinitely thin straight wire. This field is obtained by solving the Maxwell's equations with the current J as data in the distributional sense. We shall see that many features of the integrable time independent case are preserved in both cases. In particular, introducing a cylindrical coordinates system, we will show the existence of radially periodic motions.