Spherical capacitor ($C=4\pi {ϵ}_{0}ab/(b-a)$)

Two concentric conducting spherical shells of radii $a<b$ with charges $Q,-Q$ have an electric field only in the intervening vacuum: $E_r=Q/(4\pi {ϵ}_0{r}^2)$ by Gauss's law. Integrating gives a potential difference $Q(1/a-1/b)/(4\pi {ϵ}_0)$, and hence the displayed capacitance. The electrostatic energy is $Q^2/(2C)$, either by charging work or by integrating ${ϵ}_0|E{|}^2/2$ over the gap.

A biased intermediate concentric conductor redistributes the potential drop between two gaps. For radii $a<c<b$ and boundary voltages $V_1,V_2,0$, the peak gap fields are $\alpha |V_1-V_2|$ and $\beta |V_2|$, where $\alpha =c/[a(c-a)]$ and $\beta =b/[c(b-c)]$. For positive $V_1$, moving $V_2$ into $[0,V_1]$ cannot increase either field. Within that interval, one field falls and the other rises, so the maximum is minimized at equality. Consequently $V_2^{\ast }=\alpha V_1/(\alpha +\beta )$ and $E_{\max }^{\ast }=\alpha \beta V_1/(\alpha +\beta )$. Equal potential drops are generally not optimal.