Concentric-sphere field grading ($V_2^{\ast }=\alpha V_1/(\alpha +\beta )$)

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.