On a PCB, what does an Earth Ground prong (from a 3-prong plug) connect to?
The Earth/Safety Ground/Protoective Earth wire would normally be connected to exposed metal parts of an appliance. It is intended to hold the exposed metal parts at Ground potential to protect users from shock.
It must not be directly connected to the AC Neutral and is normally not connected to current-carrying parts of the circuit.
If your device has no exposed metal parts, there is no need for a three-wire power cord.