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 (the value we receive is the "classical radius" of the electron, or the "Lorentz radius: $10^{-15}\;\rm m$).

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?

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.

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 (the value we receive is the "classical radius" of the electron, or the "Lorentz radius. $${10^-15 m}$$

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?

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 (the value we receive is the "classical radius" of the electron, or the "Lorentz radius: $10^{-15}\;\rm m$).

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?

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?

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 (the value we receive is the "classical radius" of the electron, or the "Lorentz radius. $${10^-15 m}$$

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?