Electrical damping from Inductor

I work in low temperature physics, and as part of my cell I have an inductor (which is placed to measure the displacement of a rod moved up and down). I need to know the damping on the system caused by this inductor. I'm not very good at electronics so I figured you guys might be able to help. I know it will be electrical damping, the inductor is a solenoid made from coiled copper wire.

I know the energy stored in an inductor is 1/2LI^2. The change in inductance, due to the movement of the rod, is on the order of 10µH. The resistance of the solenoid has been measured as 2.5Ω. The current supplied is 1mA.

Thank you for your help!

How does the solenoid current measure the displacement of the rod?
How do you monitor the solenoid current?
Is the 1 ma solenoid current a constant quantity?

The change in inductance is measured (using an inductance meter) and related to the position of the rod. The 1mA is a constant quantity. The rod will have slight oscillation due to the movement being produced by another superconducting coil (using the Meissner force).

When you change the position of the rod .... do you observe a decaying oscillation in the rod before it comes to rest?

Is this related to the damping effect that you are trying to quantify?

If no oscillation of the rod is obvious, when its position is moved, then the relevant damping is probably taking place within the 2.5 Ω solenoid coil.

It is possible ... likely ... that there is a super-imposed damping current in the coil that exists only when the rod is moved. The damping is due to the Joule Heating Effect, and is found in terms of I²R. You could devise some method of measuring the transient current in the coil, which would be separate from the 1 ma current that is a constant value.

There would be no other way for damping to occur. If the damping is is not mechanical, due to rod oscillation, the energy dissipation of the system has to be due to an independent current of some sort, as it passes through the solenoid coil resistance.