First, I think there's a bit of confusion here between voltage and current. Moving the magnet through the coil will induce a voltage (not current), whose magnitude depends on the nature of the changing magnetic flux that the coils are subjected to as you move the magnet. I suggest taking that output, connect it to your rectifier and then measure the output voltage (without the capacitor) to see whether it's working as intended.
Second - you don't mention what value of resistor you are using, which should be in series with the voltage source (your rectifier) and the capacitor. You can take a stab at what value to use from the fact that using really thin wire means that you want to limit the current to perhaps 100 mA max. So if your voltage source puts out, say, 5 volts, R should be around 50 ohms or greater.
The value of the resistor R times the value of the capacitor C gives the time constant for the circuit, which is a rough way of measuring how quickly the capacitor will charge to about 63% of its final voltage. For C=2.5 ahd R = 50 this is 1250 seconds! So trying to power it up with a voltage source with a frequency much faster than that will not work well at all.
Finally, because the voltage wave form coming out of the rectifer is not constant, the capacitor wlll tend to discharge whenever the supply voltage is below the voltage of the capacitor. Again - you need to supply a steadier voltage and/or reduce the time constant of the circuit.
I would strongly encourage you to use a much, much smaller value of C. Not only might you get this to actually work, but it's much safer as well.
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