Past exam of the advanced quantum physics course of the University of Cambridge / 2022 / Section B / 4 / a

Let the electric charge be $q=-e$, where $e>0$ is the elementary charge. Minimal coupling to a tangential magnetic vector potential givesA constant tangential magnetic vector potential is an available gauge on the ring, so $a=q{A}_x$. By Stokes theorem, its circulation is the oriented magnetic fluxThe orientation matters here. In the original diagram, increasing $x$ goes anticlockwise, whose associated positive normal points out of the page, whereas the indicated positive magnetic flux $\Phi$ points into the page. Hence ${\Phi }_C=-\Phi$ for that diagram, and its convention gives $\boxed{aL=e\Phi }$. If instead $\Phi$ is defined with the normal associated to $+x$, the relation is $aL=-e\Phi$.

There is no requirement that the magnetic field be nonzero on the ring: it can be confined to a solenoid inside it. The nonzero circulation of the magnetic vector potential still affects the Electron through the Aharonov-Bohm effect.