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The process of levitation melting of metals has some important advantages like cleanliness and superheating. In order to optimize this fusion process we present here a model of the coupled aspects of the magnetic field and of the equilibrium geometry of the liquid metal. In a rather simple configuration of a levitated charge in a conic wire we determine the equilibrium form of the charge by minimizing the energy of the system with an iterative variational method. In each step the crucial point is the calculation of magnetic values which is facilitated by simple discretization of the free surface. The resolution of the linear system with no discretization of all space reduces computer memory requirement.