This paper brings further insight into the recently published N-body description of Debye shielding and Landau damping (Escande et al 2014 Plasma Phys. Control. Fusion57 025017). Its fundamental equation for the electrostatic potential is derived in a simpler and more rigorous way. Various physical consequences of the new approach are discussed, and this approach is compared with the seminal one by Pines and Bohm (1952 Phys. Rev. 85 338–53).