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We evaluate the stopping and image forces on a charged particle movingparallel to a doped sheet of graphene by using the dielectric responseformalism for graphene's $\pi$-electron bands in the random phase approximation(RPA). The forces are presented as functions of the particle speed and theparticle distance for a broad range of charge-carrier densities in graphene. Adetailed comparison with the results from a kinetic equation model reveal theimportance of inter-band single-particle excitations in the RPA model for highparticle speeds. We also consider the effects of a finite gap between grapheneand a supporting substrate, as well as the effects of a finite damping ratethat is included through the use of Mermin's procedure. The damping rate isestimated from a tentative comparison of the Mermin loss function with a HREELSexperiment. In the limit of low particle speeds, several analytical results areobtained for the friction coefficient that show an intricate relationshipbetween the charge-carrier density, the damping rate, and the particledistance, which may be relevant to surface processes and electrochemistryinvolving graphene.

Dynamic polarization of graphene by moving external charges: random phase approximation