Proof of Biot-Savart's law for a discontinuous current-or another example of displacement current

When a current is discontinuous the use of Biot-Savart's law appears to produce results which contradict those found using Ampere's circular law. Time-dependent charges are introduced to account for the current discontinuities. These, in turn, produce displacement currents, which when included in the current used in Ampere's law resolve the apparent discrepancy.