Permanent Magnetic Field vs Electromagnetic Field
The box containing the coil will heat up more and more due to the joule effect. So measuring the temperature you could tell.
If this is a normal coil wound with resistance wire, then around the coil with current will be the electric field and magnetic field, and around the permanent magnet only magnetic field. If the boxes do not shield the electric field, it will not be difficult to detect the coil with current by measuring the electric field.
Warm it up near or beyond the Curie temperature, it will make no difference to the coil but the "permanent" becomes "impermanent".