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> Make the loop polygonal (piecewise straight) if straightness is important
> to you. The same argument applies.
If (as I think you are proposing) the E field accompanying the changing
magnetic flux is everywhere azimuthal ("circular") in direction,
it will drive charge carriers into the sides of any straight wire. A
static surface charge will build up so that the NET field drives the
current along the straight wire.
the field inside the current carrying wire is
sourced by surface charges on the wire.