the magnetic field inside the torroidal core is essentially confined to the core.
there is some leakage of field lines, but it is minimal.
now you put it another coil around the same core and rotate the original
if the current in the original coil is DC, the flux is constant and you'll get nothing
if the current in the original coil is AC, the flux changes and you'll get AC voltage and
current, depending on the load, etc, as with any transformer
i do not see any advantage to rotating the original torroid, and I see a mechanical nightmare.