My guess is potential difference $V_b=V_c=0$ and $V_a$ is same as battery. When the electron is in equilibrium in capacitor. I set positive terminal of battery as 0V.
The idea that as you move through the earth you get a symmetric cancelling of gravitational acceleration which approaches zero acceleration due to gravity at the center of the earth makes a lot of sense I think. I also think that is a pretty easy idea to understand physically.
On the other hand,...
Density is a function of position isn't it. If you model the Earth by a ball of constant density then ρ(r) is constant for r < R and zero for r > R, where R is the radius of the Earth.
Suppose you are inside the Earth at a distance r from the centre, then we can divide the Earth into two parts. The part of the Earth farther from the centre than you forms a shell with internal radius r and external radius R. Because this is a shell the gravity inside it is zero, so the end result is that you do not feel any gravity from the part of the Earth outsde you.
so the gravitational influence I experience is of the mass that is between me and the centre, which is less than the whole of the mass of earth. so the a decreases ? yes ?(if I got it right)