Editing Classical electron radius (section)

== Derivation ==
The classical electron radius length scale can be motivated by considering the energy necessary to assemble an amount of charge <math>q</math> into a sphere of a given radius <math>r</math>.<ref>
{{cite book
 | title=University Physics, 11th Ed.
 | last=Young | first=Hugh
 | publisher=Addison Wesley
 | year=2004
 | isbn=0-8053-8684-X
 | location=
 | pages=873
 | quote=
}}</ref>  The electrostatic potential at a distance <math>r</math> from a charge <math>q</math> is
: <math>V(r) = \frac{1}{4\pi\varepsilon_0}\frac{q}{r} .</math>

To bring an additional amount of charge <math>dq</math> from infinity necessitates putting energy into the system, {{tmath| dU }},  by an amount 
: <math>dU = V(r) dq .</math>   

If the sphere is ''assumed'' to have constant [[charge density]], <math>\rho</math>, then 
: <math>q = \rho \frac{4}{3} \pi r^3</math> and <math>dq = \rho 4 \pi r^2 dr .</math> 

Integrating for <math>r</math> from zero to the final radius <math>r</math> yields the expression for the total energy {{tmath| U }}, necessary to assemble the total charge <math>q</math> into a uniform sphere of radius {{tmath| r }}:
: <math>U = \frac{1}{4\pi\varepsilon_0} \frac{3}{5} \frac{q^2}{r} .</math>  

This is called the electrostatic [[self-energy]] of the object. The charge <math>q</math> is now interpreted as the electron charge, {{tmath| e }}, and the energy <math>U</math> is set equal to the energy-equivalent of the electron's rest mass, {{tmath| m_\text{e} c^2 }}, and the numerical factor 3/5 is ignored as being specific to the special case of a uniform charge density. The radius <math>r</math> is then ''defined'' to be the classical electron radius, {{tmath| r_\text{e} }}, and one arrives at the expression given above.

Note that this derivation does not say that <math>r_\text{e}</math> is the actual radius of an electron.   It only establishes a link between electrostatic self-energy and the energy due to the rest mass of the electron.