Previous work on conditional amplitudes and on the exact electron factorization already discussed the features of various contributions of the total effective scalar potential that are important to retain in the design of approximations in relation to orbital-free or Kohn-Sham density-functional theory. In this work, we focused on the vector potential of the theory, which is essential when describing current-carrying states. Since, so far, not much work has been devoted to the electronic vector potential, we found it interesting to analyze it in a model case of non-interacting electrons, highlighting its behavior and that of its curl (related to an effective magnetic field generated by the conditional amplitude on the marginal wavefunction).