A capacitor consists of three perfectly conducting coaxial cylinders of radii $a, b$ and $c$ where $0<a<b<c$, and length $L$ where $L \gg c$ so that end effects may be ignored. The inner and outer cylinders are maintained at zero potential, while the middle cylinder is held at potential $V$. Assuming its cylindrical symmetry, compute the electrostatic potential within the capacitor, the charge per unit length on the middle cylinder, the capacitance and the electrostatic energy, both per unit length.

Next assume that the radii $a$ and $c$ are fixed, as is the potential $V$, while the radius $b$ is allowed to vary. Show that the energy achieves a minimum when $b$ is the geometric mean of $a$ and $c$.