Spherical shell of charge / electron

[Zaphodx57x]

I am having a problem understanding this problem which references this exercise .

I tried it a couple different ways. I used
$$\begin{multline*} \Delta U(Potential Energy) = Uf - Ui = Uf - U(r = infinity) = Uf - 0 \\ dU = Uf = -W = \int F * ds = \int E * Q * ds \\ Uf = -\int E * Q * dr = -\int \frac{kqQ}{r^2} = \frac{kqQ}{r} \end{multline*}$$
I also tried using the bare potential energy equation and using two arbitrary surface areas (dA) on opposite sides of the sphere with a distance between charges of 2r , then integrating each over just half of the sphere, which provided the same result.
The problem I have is that this result gives me a pretty decent result for the mass of an electron if I assume that q = Q = charge of an electron. But this assumption seems to say that the electron is pushing against itself by its full charge.
That is a force $\frac{QQ}{r^2}$ is present instead of a $\frac {(Q)(q}{r^2}$ where $q = \frac {Q}{n}$ or some other fractional or modified charge.
This almost makes sense to me in that if a sphere were to have a charge spread evenly over its surface and its radius is infinity, it still acts as if it has all its charge at its center and therefore would resist a decreasing radius of like charge.

If anyone can find a way to explain why this problem makes sense I would really appreciate it. I just don't understand how the Potential energy can contain the electron's charge twice, when there is only one charge to begin with.

 

[photon_mass]

I just don't understand how the Potential energy can contain the electron's charge twice, when there is only one charge to begin with.

there is a potential from one electron to BOTH other electrons

 

[Zaphodx57x]

If you keep increasing the amount of particles that the electron is composed of, the approximation for mass gets worse. Going to two particles and then eventually one gives you a better approximation, but the concept behind how it works physically eludes me. Its as if the electron focuses all its charge at its center and simultaneously has the same charge spread out over the shell of charge.

Perhaps the cute mathematical answer is just an ends to a means, and the accumulation of potential energy from the radius decreasing from infinity to r just happens to simplify to a very concise answer. Anybody have any ideas?