Flaw in the radius of a spherical electron model (classical electron radius)

Question

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 came from the standard charge through a sphere formula (if you feel the need for a diagram or intermediate mathematical steps let me know)]:

($Work$) = $\int_0^r ((4\pi\rho)^2r^4\cfrac13)~dr$ which means that:

($Q$) = $4\pi\cfrac13r_0^3\rho$ , and from the relationship between potential energy and Work, we obtain:

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

The next question (based off of the equation we just derived) asks us to set our answer equal to $mc^2$, and attempt to calculate the proper radius of an electron (the value we receive is, I now know is the "classical radius" of the electron, or the "Lorentz radius" $\approx$($2.8 *10^{-15}\;\rm m$).

Finally we arrive at our problem :

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? Is looking at an electron as a sphere inncorrect? I actually attempted to research this matter on my own, and could not ascertain the reason that the classical electron's radius is a poor theory, only that it does not work well in quantum mechanics.

Answer 1

There are several problems with the classical electron radius of: $$ r_e = 3 \times 10^{-15} \text{ m} $$ first and foremost, and perhaps the only real reason that is important in physics, is that it is wrong. Wrong in the sense that we can do detailed experiments to try to measure this radius, and to date, we have failed to ever observe any structure to the electron inconsistent with it being a point like particle. One reference I could find is this 1988 paper: A Single Atomic Particle Forever Floating at Rest in Free Space: New Value for Electron Radius by Hans Dehmelt [doi] in which he summarizes the then to date experimental bound on the observed radius of the electron to be: $$ r_e < 10^{-22} \text{ m} $$

But, beside cold hard experimental fact, are there some theoretical reasons we might be suspicious of the value we get for the classical electron radius?

Well, to start with, the model we used to find the radius is just one where the electron is a small uniform ball of charge of the specified size. The first question this raises is? What keeps that ball from exploding? There would have to be some other force that manages to keep this 0.5 MeV bomb together. We don't have a good idea for what that force could be.

Secondly, we used classical E&M in order to estimate the radius of the electron. But, this presupposes that the classical theory of E&M works on length scales of $10^{-15} \text{ m}$. What reason do we have to trust that this is so? From a modern perspective, we know this is erroneous, as quantum mechanical effects in the form of quantum electrodynamics become important on those kinds of small length scales.