An educational path for the magnetic vector potential and its physical implications

We present an educational path on the magnetic vector potential A addressed to undergraduate students and to pre-service physics teachers. Starting from the generalized Ampere-Laplace law, in the framework of a slowly varying time-dependent field approximation, the magnetic vector potential is written in terms of its empirical referent, i. e. the conduction current. Therefore, once the currents are known, our approach allows a clear and univocal physical determination of A overcoming the mathematical indeterminacy due to the gauge transformations. We have no need to fix a gauge, since for slowly varying time-dependent electric and magnetic fields, the natural gauge for A is the Coulomb one. We stress the difference between our approach and those usually presented in the literature. Finally, a physical interpretation of the magnetic vector potential is discussed and some examples of calculation of A are analysed.