The classical electron radius is a combination of fundamental physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation. A classical charged sphere producing an electric field with energy equal to the electron's rest mass energy would have a radius equal to the classical electron radius. It links the classical electrostatic self-interaction energy of a homogeneous charge distribution to the electron's rest mass energy. According to modern understanding, the electron has no internal structure, and hence no size attributable to it. Nevertheless, it is useful to define a length that characterizes electron interactions in atomic-scale problems. The CODATA value for the classical electron radius is

where ${\displaystyle e}$ is the elementary charge, ${\displaystyle m_{\text{e}}}$ is the electron mass, ${\displaystyle c}$ is the speed of light, and ${\displaystyle \varepsilon _{0}}$ is the permittivity of free space. This is about three times larger than the charge radius of the proton.

The classical electron radius is sometimes known as the Lorentz radius or the Thomson scattering length. It is one of a trio of related scales of length, the other two being the Bohr radius ${\displaystyle a_{0}}$ and the reduced Compton wavelength of the electron ⁠${\displaystyle \lambda \!\!\!{\bar {}}_{\text{e}}}$⁠. Any one of these three length scales can be written in terms of any other using the fine-structure constant ${\displaystyle \alpha }$:

The classical electron radius length scale can be motivated by considering the energy necessary to assemble an amount of charge ${\displaystyle q}$ into a sphere of a given radius ⁠${\displaystyle r}$⁠, with the charge uniformly distributed throughout the volume. The electrostatic potential at a distance ${\displaystyle r}$ from a charge ${\displaystyle q}$ is

To bring an additional amount of charge ${\displaystyle dq}$ from infinity adds energy ⁠${\displaystyle dU}$⁠ to the system:

If the sphere is assumed to have constant charge density, ⁠${\displaystyle \rho }$⁠, then

Integrating ⁠${\displaystyle dU}$⁠ for ${\displaystyle r}$ from zero to a final radius ${\displaystyle r'}$ yields the expression for the total energy ⁠${\displaystyle U}$⁠, necessary to assemble the total charge ${\displaystyle q'}$ uniformly into a sphere of radius ⁠${\displaystyle r'}$⁠:

This is called the electrostatic self-energy of the object. Interpreting the charge ${\displaystyle q'}$ as the electron charge, ⁠${\displaystyle -e}$⁠, and equating the total energy ${\displaystyle U}$ with the energy-equivalent of the electron's rest mass, ⁠${\displaystyle m_{\text{e}}c^{2}}$⁠, and solving for ⁠${\displaystyle r'}$⁠: