Analyses of a floating zone (FZ) crystal growth system with a radio frequency (RF) induction heating are carried out. The electromagnetic and temperature fields, and surface interfaces are solved for numerically. Finite element methods are used for the calculation of the temperature fields and interfaces and the hybrid finite element and boundary element methods are used for the calculation of the electromagnetic field. The calculation domain is divided into eleven regions, each of which, except for the RF coil region, require a coordinate transformation. In the present study, the silicon FZ growth system ,where the diameter of both feed-rod and single crystal were set to be 1cm was computed. The Lorentz force is found to play an important role in determining the melt free surface shape and the molten zone length. The effect of the current density, frequency of RF coil and crystal growth rate are investigated.