A capacitor consists of three long concentric cylinders of radii $a, \lambda a$ and $2 a$ respectively, where $1<\lambda<2$. The inner and outer cylinders are earthed (i.e. held at zero potential); the cylinder of radius $\lambda a$ is raised to a potential $V$. Find the electrostatic potential in the regions between the cylinders and deduce the capacitance, $C(\lambda)$ per unit length, of the system.

For $\lambda=1+\delta$ with $0<\delta \ll 1$ find $C(\lambda)$ correct to leading order in $\delta$ and comment on your result.

Find also the value of $\lambda$ at which $C(\lambda)$ has an extremum. Is the extremum a maximum or a minimum? Justify your answer.