The aim of this paper is to analyse a finite-element method to solve an eddy current problem arising from the modelling of an induction furnace. By taking advantage of the cylindrical symmetry, the three-dimensional problem reduces to a two-dimensional one on a meridional section, provided that the current density, written in cylindrical coordinates, has only an azimuthal component. A mixed formulation in appropriate weighted Sobolev spaces is given. The existence and uniqueness of the solution are proved by analysing an equivalent weak formulation. Moreover, an additional regularity result is proved under suitable assumptions on the physical coefficients. The problem is discretized by standard finite elements and a priori error estimates are proved. Finally, some numerical experiments that allow an assessment of the performance of the method are reported.