 Solution

For the displayed gauge field,which already vanishes at infinity. A static future-directed particle has $U=K/\sqrt{f}$, soIf $qQ<0$, the first term tends to zero at $r_{+}$ while the second remains negative. Lowering the particle sufficiently close to the horizon therefore produces $E<0$. Dropping it into the hole reduces the black-hole mass by $|E|$, while the external agent receives the corresponding positive work: this is charged-particle black-hole energy extraction.