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A novel error estimator for adaptive mesh refinement in finite element analysis (FEA) of magnetic field using quadratic finite elements is presented. The method uses a novel heuristic a posteriori error estimator, which is easy to compute and simple to implement, as an indicator of the numerical errors of the computed solution. The proposed error estimator is the L2 norm of the difference between the computed quadratic finite element solution and the interpolated linear solution. Throughout the time-stepping process for problems excited by periodic sinusoidal excitations, this error estimator can also be used to efficiently compute the numerical error at each time step and guide the adaptive mesh refinement in transient FEA. A multi-time-step adaptive mesh refinement method is also proposed in this paper for transient problems. The proposed method does not need to interpolate the solution from old mesh to a new adaptively refined mesh in transient FEA and hence there is no interpolation error. The effectiveness of the proposed error estimator is illustrated through several numerical examples being reported in this paper.