Bender

The OP might not be thinking about a perpetuum mobile, but that is what the proposed device is. As is any machine that supposedly produces its own energy. I think it is only fair to warn the OP that the device violates the laws of thermodynamics. Using oxygen to make oxygen is always going to result in less oxygen.

Maxwell's pulling force: F = B² A / (2 μ0) B = 0.5 Tesla (at best) A = 9²pi/4 = 60e-6 m² µ0=4pi e-7 max force is about 12 N, so enough for your purpose However: - B decreases quickly with distance, so at 8 mm, only a fraction of the force will be left - I doubt you can make a coil this small that will generate such a magnetic flux - the flux from the coil could demagnetise the magnet my suggestion: use a pneumatic or hydraulic cylinder.

No. 1) The Earth's magnetic field is barely strong enough to turn a nearly frictionless needle. 2) The board will pull the dynamo down with an equal force, so the net upward force would be zero.

Sorry, no free energy. You cannot have excessive hydrogen and oxygen, because they will always produce less energy than was used in the electrolysis. When you seen something like this (which is essentially yet another perpetuum mobile), always remember the laws of thermodynamics; all other calculations are redundant (although perhaps interesting for academic purposes).

If the magnetic flux through a coil changes, this creates a voltage in the coil. Same as in an alternator really, but here the changing flux is caused by the special magnetostrictive material rather than a rotating magnet.