Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 11 / a / Solution

Take the dipoles to point along $+z$, and let their surface number density be $n_s$. An annulus of radius $\rho$ and width $d\rho$ contains $n_{s}2\pi \rho d\rho$ dipoles. The electron's electrostatic potential energy is thereforeFor $z\ne 0$, the integral without $2\pi$ equals $z/|z|$, givingThus the jump's magnitude isFor the chosen outward dipole orientation the electron-energy change from the negative to positive side is negative. The electrostatic voltage jump has the opposite sign. This distinguishes the requested magnitude from the sign of the work function change.