The motion of charged particles can be interfaced with electric circuitry via the current induced in nearby pick-up electrodes. Here we show how the rotational and translational dynamics of levitated objects with arbitrary charge distributions can be coupled to a circuit and how the latter acts back on the particle motion. The ensuing cooling rates in series and parallel RLC circuits are determined, demonstrating that quadrupole ion traps are well suited for implementing all-electric cooling. We derive the effective macromotion potential for general trap geometries and illustrate how consecutive rotational and translational resistive cooling of a microscale particle can be achieved in linear Paul traps.