I can follow the math of how you derive the electric and magnetic fields from it okay, but I’m sometimes confused as to how a vector potential is the same kind of idea as a scalar potential.
So, scalar potential I have a philosophical intuition for. When a charge moves from an area of high potential to an area of low potential, this corresponds to a proportional change in potential energy (the change proportional to the potential drop and the amount of charge). The reason this works is because both potential and potential energy are scalar values.
When I hear about a vector potential, my mind imagines that a potential change therefore cannot correlate with a change in a scalar value (potential energy) and should instead correspond to a change in some vector quantity. As a charge moves from a position with (x,y,z) magnetic potential to a position with (x’,y’,z’) potential, then some vector quantity should also change. And obviously this can’t be potential energy because potential energy is a scalar.
So, assuming my reasoning here isn’t flawed, what is this vector analogue to potential energy? If there is none, for what reason am I wrong here? And how does this all apply to electromagnetic 4potential?