In 1882 Lord Rayleigh developed a model related to the study of the (lack of) equilibria of liquid droplets, provided of an electric charge. In the seminar we provide a precise variational description of such a model. We will then discuss about the existence of minimizers and their regularity. Eventually we will provide a full solution of the problem in the bidimensional setting. The talk is based on collaborative works with M. Goldman (Paris Diderot university), M. Novaga (Università di Pisa) and C. Muratov (New Jersey Institute of Technology)