We show that the electrostatic potential of an atomic nucleus “seen” by a fast charged projectile at short distances is quantum mechanically smeared due to nucleus motion around the atomic center of inertia. For example, the size of the “positive charge cloud” in the Hydrogen ground state is much larger than the proper proton size. For target atoms in excited initial states, the effect is even larger. The elastic scattering at large angles is generally weaker than the Rutherford scattering since the effective potential at short distances is softer than the Colombian one due to a natural “cutoff”. In addition, the large-angle scattering leads to target atom excitations due to pushing the nucleus (⇒ inelastic processes). The Rutherford cross section is in fact inclusive rather than elastic. These results are analogous to those from QED. Non-relativistic atomic calculations are presented. The difference and the value of these calculations arise from nonperturbatively (exact) nucleus “dressing” that immediately leads to correct physical results and to significant technical simplifications. In these respects a nucleus bound in an atom is a simple but rather realistic model of a “dressed” charge in the QFT. This idea is briefly demonstrated on a real electron model (electronium) which is free from infinities.