In the ideal case the magnetic field lines of a toroidal coil are entirely contained witnin tne coil and = 0 outside. This can be checked by using a flip-coil, Hall probe, or compass needle near the toroidal coil.
However, the lines of vector potential circle the flux lines:
![](/sites/default/files/demomanual/electricity_and_magnetism/electrodynamics/vector_potential1.gif)
So A is not zero outside the coil.
![](/sites/default/files/demomanual/electricity_and_magnetism/electrodynamics/vector_potential2.gif)
If a wire is threaded through the hole of the toroid and returned through the hole, case 1, no galvanometer deflection is seen as B is switched on and off. But if the wire is threaded through the hole and returned around the coil, case 2, a galvanometer deflection will be seen from = -d/dt.
![](/sites/default/files/demomanual/electricity_and_magnetism/electrodynamics/vector_potential3.gif)