In the Wikipedia article Classical electron radius in calculation of radius of electron. Charge distribution of electron described as $$\rho(r)= \frac{q}{4{\pi}Rr^2}$$for $r\leq R$. This is variable density why it is not constant like volume charge density $\frac{3q}{4{\pi}R^3}$ I know this will change the value of radius but my question is how such charge distribution is concluded for electron.
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There are currently three derivations in that article: for a sphere of constant charge density, for a spherical shell of charge, and for the charge distribution you give. The difference is the numerical factor $f$ in the electrostatic energy $E=f\frac{1}{4\pi\epsilon_0}\frac{e^2}{ r_e}$ (which is 3/5, 1/2, and 1 for the three example charge distributions).
That particular charge distribution is a terrible choice since it suggests a rigorous process was used to come up with a "model classical electron" with $f=1$. The logic goes the other direction. The relationship $$ m_\text ec^2 = \frac{1}{4\pi\epsilon_0}\frac{e^2}{r_\text e} = \frac{\alpha\hbar c}{r_\text e} $$ is chosen purely by dimensional analysis; the only reason to consider a uniform sphere or a spherical shell of charge is to confirm the bias that the dimensional result is correct to within a factor of about two. Since there are not any features of the electron interaction that change at femtometer-ish length scales, it doesn't make sense to say that the electron has a "classical" radius at all, and the factor of two is moot.
I note that the article has been "flagged as having multiple issues" for seven years. Maybe one of us should clean it up.
