This paper describes a finite element formulation of full Maxwell's equations in terms of a vector potential and a scalar potential and its application to eddy current problems. The vector potential and the scalar potential are approximated by vector basis functions and scalar basis functions, respectively. A gauge fixing of the potentials is translated into the regularization of the indeterminate linear equation finally obtained by the Galerkin procedure. Two convenient gauge conditions for three dimensional eddy current problems are presented. By applying the computer code to test problems, this finite element formulation and the gauge conditions are proved to be applicable to linear electromagnetic problems.