Time-varying magnetic fields are applied to living tissues for different purposes in medicine. For all applications, the induced current distributions must be accurately estimated in order to obtain the desired medical results. In this study a new 3D finite element method (FEM) formulation is derived. A code is prepared to implement these formulations using 20 noded isoparametric cubic elements. The accuracy of solutions is tested for three simple geometries and series expansion method is used to derive analytical expressions whenever it is necessary. It is shown that, for the selected geometries error in FEM solutions are less than 1%, and can be improved by increasing the number of nodes. The new 3D FEM solver runs in a 166 MHz PC using 128 Mbyte memory and is capable of solving a problem with 100.000 nodes in 80 minutes.