If I put a permanent magnet under a box and an energized electromagnetic coil under another box could you tell me which box was covering the permanent magnet? If the answer is yes what test would you use? Note: The coil is receiving a steady dc current.
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".
If the permanent magnet and the electromagnet are shaped so that the field in the box has precisely the same shape, there is no way to know which box covers which magnet, simply by measuring the field using a tiny probe magnet or tiny probe coil with very small current.
However, it may be possible to distinguish between the two cases by a more "intrusive" measurement: A metal detector, for example, would very likely respond slightly differently for one than the other. The electromagnet coil would act as a transformer secondary if driven by an electromagnet above the box, and should behave in a slightly different way than the permanent magnet.
Edited for clarity: "Driven" means, in this context, that the electromagnet above the box is energized by a time-varying current, which will produce a time-varying field, which in turn will "drive" a time-varying additional EMF in the coil below the box.
With an x-ray snapshot you should be able to see the difference :). However the magnetic fields can be made to be indistinguishable within certain intensity limits. There is a limit to what a permanent magnet can achieve in a certain volume. An electromagnet can produce a much stronger field.
Electromagnets can produce a stronger pulsed field, but not a steady-state field as the windings would burn up fast. Oddly enough the same intense magnetic pulse is used to magnetize neodymium-iron-born (NIB) magnets.
For a steady-state magnetic field (not superconducting) NIB alloys have the strongest field for now. Some NIB alloys will tolerate intense heat at the cost of total field strength.