  We compare the magnetic field at the center of and the self-magnetic flux
through a current-carrying circular loop, with those obtained for
current-carrying polygons with the same perimeter. As the magnetic field
diverges at the position of the wires, we compare the self-fluxes utilizing
several regularization procedures. The calculation is best performed utilizing
the vector potential, thus highlighting its usefulness in practical
applications. Our analysis answers some of the intuition challenges students
face when they encounter a related simple textbook example. These results can
be applied directly to the determination of mutual inductances in a variety of
situations.
