  An ongoing debate in the first-principles description of conduction in
molecular devices concerns the correct definition of current in the presence of
non-local potentials. If the physical current density ${\bf
j}=(-ie\hbar/2m)(\Psi^* \nabla \Psi- \Psi \nabla \Psi^*)$ is not locally
conserved but can be re-adjusted by a non-local term, which current should be
regarded as real? We prove that the extended Maxwell equations by Aharonov-Bohm
give the e.m.\ field generated by such currents without any ambiguity. For an
oscillating dipole we show that the radiated electrical field has a
longitudinal component proportional to $ \omega \hat{P}$, where $\hat{P}$ is
the anomalous moment $\int \hat{I}(\mathbf{x})\mathbf{x} d^3x$ and $\hat{I}$ is
the space-dependent part of the anomaly $I=\partial_t \rho+\nabla \cdot
\mathbf{j}$. In the case of a stationary current in a molecular device, a
failure of local current conservation causes a "missing field" effect that can
be experimentally observable, especially if its entity depends on the total
current.
