Context. Sufficiently far from the Sun, all comets go through a phase of low activity. Rosetta observations at large heliocentric distances of approximately 3 au showed that the plasma at a low-activity comet is affected by both steady state electric fields and low-frequency waves. Aims. Our goal is to provide a model for the electric fields in the inner coma at a low-activity comet and to simulate waves and field structures farther away from the nucleus. Methods. We compare analytical models for the convective, ambipolar, and polarisation electric fields to the results of an electrostatic particle-in-cell simulation of a scaled-down low-activity comet. Results. We find good agreement between the steady state field model and the simulation results close to the nucleus. At larger cometocentric distances, waves dominate the electric field. These waves are interpreted as the scaled-down electrostatic limit of the previously observed singing comet waves. The comet ion density is not spherically symmetric. Conclusions. Low-activity comets can be modelled using electrostatic particle-in-cell simulations of a scaled-down system. Outside the innermost part of the coma (r ≳ 40 km), the plasma is not spherically symmetric and the electric field is dominated by waves.