Electrostatic energy

Electrostatic energy can be written as one half the charge-potential integral or epsilon-zero over two times the electric-field energy integral.

A first spherical shell with charge $q$ followed by shells of charges $2(-1{)}^{n+1}q$ leaves enclosed charge $(-1{)}^{n+1}q$ after each shell. Gauss's law therefore gives the same squared electric field $q^2/(16{\pi }^2{ϵ}_0^2{r}^4)$ in every gap outside the first radius. Integrating the electrostatic energy density over all space gives $W=q^2/(8\pi {ϵ}_0{r}_1)$, independent of the other shell radii.

A thin spherical shell of radius $a$ carrying total charge $Q$ has surface potentialand electrostatic self-energy

For distinct point charges $q_i$ at positions $r_i$, the electrostatic potential energy isWhen some charges are fixed, the terms depending on a movable charge form its potential energy in their electric potential.

For coaxial cylinders of radii $a<b$ and length $L\gg b$, carrying charges per unit length $\pm \lambda$,The field energy is $U={\lambda }^{2}L\log (b/a)/(4\pi {ϵ}_0)=QV/2$.