Charged sphere inside a neutral coaxial capacitor (${\Phi }_b=\frac{2R{V}_0}{l}\log (a/b)$)

An isolated conducting sphere of radius $R$ initially at potential $V_0$ has charge $Q=4\pi {ϵ}_{0}R{V}_0$. Inserting it without contact into the hollow inner conductor of a long coaxial cylindrical capacitor conserves that charge. The neutral inner conductor acquires $-Q$ on its cavity wall and $+Q$ outside. With the outer conductor grounded, the annular electric field is $Q/(2\pi {ϵ}_{0}lr)$ and the inner potential is the displayed result. The original electric potential is not conserved. End effects are neglected; exact cavity fields are geometry-dependent.