Ask HN: have you built an electromagnetic levitator?

Theoretically, what's the biggest gap you could create between the levitator and the object? Why is it that all examples of levitation I've seen on youtube lift an object just an inch or two?

The specific challenge is to suspend a Canadian penny from the ceiling; it's primarily steel, and strongly influenced by magnets. My current strategy is to:

- use an arduino for the controller

- drive an electromagnet with a sinusoidally modulated voltage

- use a hall effect sensor to detect the distance of the penny

- in the arduino, change the maximum voltage of the electromagnet, based on the voltage coming from the hall effect sensor

I don't know how the penny will behave when it's in the field, so I don't know if the hall effect sensor will be able to detect the penny's distance. Obviously, when the penny isn't magnetized, it will be invisible to the sensor, but I hope it will behave differently when it's actively being levitated.

I am not sure why this system will become less stable as the distance from the electromagnet increases, but it seems like it must. It also seems like the amplitude of the sine wave is going to need to increase, based on the distance from the penny, to cause grosser adjustments to the penny. I'm assuming that because in the opposite case, when the penny is close to the electromagnet, it seems like you will only need to make small adjustments to keep the penny a specific distance. I assume the relationship between distance and the amplitude of the sine wave will be non-linear, which will determine the maximum distance the penny could ever be.

Here is a specific question: how do I determine the force of attraction between the penny and the electromagnet given the distance and the voltage of the magnet?

This is still in the earliest stages, so any advice at all will be appreciated!