= Breakdown-limited energy of a conducting sphere
{title2=$U_{\max}=2\pi\epsilon_0 a^3E_{\rm bd}^2$}
For an isolated smooth <conducting sphere> of radius $a$, <Gauss's law> gives surface <electric field> $E_s=V/a$. Combining its <capacitance> $4\pi\epsilon_0a$ with <capacitor energy> $CV^2/2$ gives the displayed bound when the surrounding medium has <dielectric strength> $E_{\rm bd}$ and approximately vacuum <permittivity>. Nearby conductors, surface roughness and discharge geometry can lower the practical limit.
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