My textbook asks me to derive an equation for the potential energy ($U$) of sphere ($r_0$) filled with an electric charge of uniform density ($\rho$), expressed in terms of the total charge $Q$. The equation can be derived by treating the potential energy ($U$) as the total work it would require to build the sphere and computing an integral (which I won't show unless somebody wants me to, that is not why I am asking this question).

Here is the kicker: In the next part of the question, it asks us to set our result

$$U = \frac{3Q^2}{5r_0}$$

equal to $mc^2$, and attempt to calculate the proper radius of an electron - it does not work, it is three orders of magnitude too large.

Now, it asks us to explain why this theory does not adequately analyze the proper radius of an electron, or to find out what the flaw in this theory is. I really do not know, I wouldn't have used this method in the first place. Any thoughts here?