How to calculate the smallest possible faraday’s law based electromagnet component size to pow

Could anyone point me in the right direction to find the calculations  Required to find the  Correct Wire length gauge the and coil size and neodymium magnet combination To power the lowest voltage and smallest LED light bulb available. Magnet will be passing through coil at a high frequency and led needs to only be on while magnet is moving. Ideally I would like this setup to be no bigger than a box of tic tacs.  Any response is greatly appreciated.
Thanks!Mike

I don't have an answer to your question but here is a description of how do build a larger version.https://www.instructables.com/id/Shake-Flashlight-1/Looking at the explanation of how it works, I suspect that your size will be limited by a minimum length that allows the magnet to move in and out of the coil.It sounds like an interesting project. Please let us know how it goes.

Thanks for the comment!

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