The original Maxwell equations as formulated with quaternions lead to expressions in pic rel. (Notice the "Temporal Scalar-Field" T) [1].
What you learn in school is the Heaviside version which just alleges that "Electromagnetic potentials are not physical" and only the fields are a physical phenomenon. Potentials allegedly can be "gauged away".
There are many phenomena that prove the opposite (Aharonov-Bohm effect, Longitudinal forces in sparking cables, the "force wall" incident in the 3M plastic foil factory)

1.: https://arxiv.org/abs/math-ph/0307038v1