〈 Berger | Dillon 〉

The best problems are based on important results in real physics research, and this one is no exception.

okay, here's the solution: https://www.aapt.org/physicsteam/2020/upload/2019_USAPHO_revised-solutions.pdf … (scroll down to page 14) The key thing to realize is that the voltage across the membrane depends _non-monotonically_ on the charge.

The voltage is very small when the charge is close to zero, but also when the charge is large enough that the two sides of the membrane are brought very close together. In between those two extremes is a maximum voltage that the membrane can support.

When the threshold voltage is turned up larger than this maximum, the membrane abruptly collapses to near-zero thickness. I learned about this concept from this research paper: https://journals.aps.org/pre/abstract/10.1103/PhysRevE.80.011112 …