In this paper, we present and analyze the adaptive finite element method for elliptic optimal control problem in H (curl) space. Firstly, based on Maxwell equation, we propose the elliptical optimal control problem in H (curl) space and establish an optimal control system. The optimal control model is equivalently transformed into the PDE systems and present the regularity property of the OCM. Then we apply the AFEM to solve the system. For the finite element approximation, we introduce the limitations of the posterior error estimator of the residual type and prove the convergence. The numerical examples are presented to verify theoretical results and indicate that the AFEM is more reliable and effective under the same conditions. Finally, our theoretical analysis and numerical algorithms can be promoted and applied to more complex problems.