The energy if an electron can be described by the following equation,
E = -b / n^2
Where E is energy,
b is a constant equal to (2*pi^2 * m_e^4)/h^2, where m is the mass of the electron, and h is Plank's Constant, so evaluated, b = 2.18E-18 J,
and n is the principle quantum number describig which electron shell the electron resides in.
So as you can see, as the electron goes into higher and higher shells (higher n values), the energy goes to smaller and smaller negative values for energy.
The energy an electron has around an atoms is it potential energy from the nucleus. The farther away from the nucleus, the more potential energy it has, up until it is completely removed from the atom, at which case it has zero potential energy.
We know the nucleus has a positive charge, which is the heaviest, most dense part of an atom. We also know that that amount that it is positive only comes in "packets" of a certain charge to cancel out the charge on an electron, so you can't have 3.4 + charges since they come in whole numbers in the form of a proton, just like you can't have 3.4 negative charges since you can only have whole numbers of electrons.
Then we can also predict how many protons a certain element should have (Hrogen 1, Sodium 11, ...), but if you "weight" the atom, the weight usually comes out to be more than the predicted weight, sometimes double or more. So we know that there must be something else in the nucleus that is contributing to the overall weight, but no the charge, thus neutrons and not just one blob of positive charge.
