Passive proximity detection using metal plate-capacity
Have you looked at this: https://en.wikipedia.org/wiki/Capacitive_sensing there are some references at the bottom that should lead you to the correct formulas. 

Capacitive sensing works based on capacity CHANGE due to the presence of another conductor. For 50 - 100 cm I think you would need a large plate (also 50 cm in diameter at least ?), also the object needs to be large and (somewhat) conductive to get a decent amount of change in capacitance. The main problem I see is the distance. On a smartphone's touchscreen your finger is only a few mm away so the sensing conductor can be small. That is not so in your application.

I work at a company where we make a distance sensor based on radar, it has no problem detecting objects at 50 - 100 cm. It is still in development though and not cheap.