= Coaxial capacitor with grounded inner and outer cylinders
{title2=$C'=2\pi\epsilon_0(1/\log\lambda+1/\log(2/\lambda))$}
A driven cylindrical conductor between two grounded coaxial conductors forms two <coaxial cylindrical capacitors> in parallel. For radii $a,\lambda a,2a$, the <capacitance per unit length> is the sum $C'=2\pi\epsilon_0[1/\log\lambda+1/\log(2/\lambda)]$. Writing $u=\log\lambda$ and $L=\log2$, the function $1/u+1/(L-u)$ is strictly <convex> on $(0,L)$ and symmetric about $L/2$. Hence $C'$ has its unique minimum at $\lambda=\sqrt2$, with value $8\pi\epsilon_0/\log2$. As $\lambda=1+\delta$ approaches the inner conductor, $C'\sim2\pi\epsilon_0/\delta$, agreeing with the local <parallel-plate capacitor> approximation.
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