# Energy of Charged Cylinder

What is the energy (per unit length) that is stored in an infinitely long cylinder with charge density (constant) $rho$?

Consider the cylinder above (black) with a cylindrical Gaussian surface (red) that just surround it. Apply Gauss law (on the lateral surface of the cylinder – on the bases the flux of the field cancel)

$E(r)*S(r)=(ρ*V)/ϵ_0$     where $S(r)=2πrL$  is the lateral surface and $V=πr^2 L$
$E(r)*2πrL=(ρπr^2 L)/ϵ_0$    so that $E(r)=ρr/(2ϵ_0 )$

Compute the potential inside the cylinder (potential is φ to make distinction from volume V):
$φ(r)=-∫_0^r E*d r=-(ρr^2)/(2ϵ_0 )+φ(0)=-(ρr^2)/(2ϵ_0)$  if we take $φ(0)=0$

Energy for all cylinder is the energy required to gather all the charge inside it from infinity initial. In cylindrical coordinates the element of volume is $d V=r*d r*dθ*dz$

$U=(1/2)*∫ ρφ*d V=-(1/2)*[ρ^2/(4ϵ_0)] ∫_0^R r^3 d r ∮dθ∫_0^L d z=$

$= (-1/2)*[ρ^2/(4ϵ_0)]*(R^4/4)*2π L$

So that the energy per unit length is

$u=U/L=-(πρ^2 R^4)/(16ϵ_0 )$