Explanation of force amplification inside a solenoid For a system being actuated by a motor, the force can be amplified by gearing. The energy is being used for force instead of distance, so it produces more torque but moves slower.
For a system being actuated with hydraulics, the force can be amplified by having a larger cross-sectional area for the compression piston than the load piston. The small area of the compressor piston means high pressure, then this can be multiplied by the area of the large piston creating larger forces. The key to the conservation of energy is that the volume of fluid in and out the pistons is the same, so if the load piston has a larger cross sectional area it will move up less. Energy is again being used for force instead of distance.
For a system being actuated by an electro magnet (a magnetized material allowed to move inside a coil with DC current). I.e a solenoid. The force can be amplified by increasing the number of windings or by wrapping the coil around an iron tube.
In both cases (increasing windings or adding iron) how is energy conserved? I've been trying to figure this out for ages and think that the only way I will understand it is if it can be related to the mechanical leverage principles of the first two force amplification examples I gave. In both of these force is being increased at the expense of distance, hence conserving energy. When you amplify the force in a solenoid using more windings or adding iron, what is this at the expense at? i.e. how is energy conserved?
 A: Remember that in a solenoid work is done because something moves.
The motion gives rise to a change in flux which results in an e.m.f. - and the product of this and the current flowing is the work done.
For the same motion, there will be a larger e.m.f. when the inductance of the coil is greater. Greater e.m.f. times same current = more work done.
When the solenoid is not moving, there will be energy dissipated by the coil as heat (current times voltage); in a superconducting solenoid, there would be no heat dissipation. But when the solenoid moves, the change in magnetic flux causes a back e.m.f. across the coil - and the current source would have to do work against THAT voltage. This is where the electrical energy is converted to mechanical energy.
Now when you have more turns in your solenoid, the relative change in flux for a certain motion of the core will be greater - the flux coupled is proportional to the number of turns. Greater change in flux = more induced voltage = more electrical work done (and more mechanical force available).
