Two hollow long and thin coaxial conducting cylinders have radii $\mathrm{a}$ and $\mathrm{b}$ for inner and outer conductors respectively as shown in figure. The electric field between the two conductors has a maximum value ${\mathrm{E}}_{\max }$. Breakdown occurs when the electric field reaches value ${\mathrm{E}}_{\max }$.

The breakdown voltage in terms of maximum electric field E_max is ${\mathrm{V}}_{\mathrm{b}}-{\mathrm{V}}_{\mathrm{a}}=-{\mathrm{E}}_{\max }\mathrm{\hspace{0.17em}}\mathrm{a}\mathrm{\hspace{0.17em}}\ln \frac{\mathrm{b}}{\mathrm{a}}$

The breakdown voltage in terms of maximum electric field E_max is ${\mathrm{V}}_{\mathrm{b}}-{\mathrm{V}}_{\mathrm{a}}={\mathrm{E}}_{\max }\hspace{0.17em}\hspace{0.17em}\ln \frac{\mathrm{b}}{\mathrm{a}}$

The maximum value of breakdown voltage occurs (keeping E_max and b fixed for varying a) when $\left(\frac{{\displaystyle \mathrm{b}}}{{\displaystyle \mathrm{a}}}\right)={\mathrm{e}}^2$

The maximum value of breakdown voltage occurs (keeping E_max and b fixed for varying a) when $\frac{\mathrm{b}}{\mathrm{a}}=\mathrm{e}$