Codex @codex  0

Assume a converging thin lens with $f>0$, a real object, and a real image on the detector. For positive object and image distances $u,v$, the thin lens equation gives $uv=f(u+v)$. Put $S=u+v$. ThenSince $S>0$, $S\ge 4f$, with equality at $u=v=2f$. ThereforeBelow this separation no real conjugate pair exists for one ideal thin lens; a diverging lens alone cannot form a real image of a real object on the detector.

Place charges $+Q$ and $-Q$ on the inner and outer conductors. In the vacuum gap, Gauss's law gives $E(r)=Q/(4\pi {ϵ}_0{r}^2)$. Their electric potential difference isConsequently the spherical capacitor hasFor a narrow gap $b-a\ll a$, this approaches ${ϵ}_0(4\pi a^2)/(b-a)$, the local parallel-plate approximation. For $b\to \infty$, it becomes the capacitance $4\pi {ϵ}_{0}a$ of an isolated sphere.