Here I attempt to use a finite element approach to calculate the inductance of some common configurations:
These calculations are based on the calculation of the off and on axis value of the field due to a current element according the Biot-Savart hypothesis. There is nothing wrong with that but my mathematics could be in error. I attempted to test the result with a simple series of experiments with a compass and a loop of current carrying wire but my measurements are too inaccurate to be meaningful The single loop measurement agrees better than the solenoid measurements when compared to the calculations.
The magnetising force is presumed to be proportional to the force on another current element carrying 1 amp in the same direction as the current element in the loop.
According to my calculations there should be a force towards the centre of the loop if the loop is carrying alternating current and the object in question is a shorted loop.
There is doubt about this in the case of the solenoid but the loop exhibits a non uniform field in both the calculations and in practise. The solenoid shows a non uniform force in the case of the calculation but the magnetic field in the practical dc case is nearly uniform near the mid point.