= Concentric-sphere field grading
{title2=$V_2^*=\alpha V_1/(\alpha+\beta)$}
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 $V_2^*=\alpha V_1/(\alpha+\beta)$ and $E_{\max}^*=\alpha\beta V_1/(\alpha+\beta)$. Equal potential drops are generally not optimal.
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