In this paper we consider the finite element approximation to an elliptic optimal control problem with boundary observations in $H^{-\frac{1}{2}}(\Gamma)$. This problem is motivated by certain applications in optimal control of semiconductor devices where the discrepancy of the current density on the boundary to the target one is the objective to be minimized. The observation in $H^{-\frac{1}{2}}(\Gamma)$ is realized through a Neumann-to-Dirichlet mapping which facilitates the theoretical and numerical analysis. A priori error estimate for the optimal control is derived based on the variational control discretization, whereas the state and adjoint state variables are approximated by piecewise linear and continuous finite elements. Second order convergence rate for the optimal control is theoretically proved and verified by numerical results.