Gauss' Law will tell you that the potential, V:
• outside the Earth's surface, V goes as 1/r, whiere r = distance from Earth's center,
• inside the Earth, V is proportional to the mass enclosed by a concentric sphere, divided by r, so that it will continue to diminish all the way to the center, bottoming out like a paraboloidal "bowl" at the center.
Because of gauge symmetry, the 0 of potential can be chosen arbitrarily.
Choosing V=0 at Earth's surface, will make V
If the Earth were constant density throughout, then V inside would be proportional to M(r)/r ~ r³/r = r².
In fact, Earth's density increases with depth; i.e., decreases with increasing r, so that V decreases a bit faster with depth than r².
