It is not possible that both the input voltage and current are given as constant. Therefore this answer assumes that:
- The 12V power supply is a stable DC power supply such as a car-battery.
- The current of 2A mentioned in the question is the measured current.
Under these assumptions, the question becomes:
If we increase the weight of the objects being lifted by that electromagnet, will we measure more current?
The answer is no, because:
When we connect 12V to the winding, the current will raise very fast while the magnetic field builds up, and once the magnetic field is stable, the current will only depend on the resistance of the winding (assuming the 12V DC is stable).
When we add some weight to the load of the magnet, there may be a change in the magnetic field and that may cause a very short change in the current, but once the magnetic field is steady again, the measured current will be the same as before because it depends only on the resistance of the winding.
The resistance of the winding will not be affected by the load, it depends only on the material (such as copper), the cross section, the length and the temperature.