We study, both theoretically and numerically, qualitative and quantitative properties of the solutions a Vlasov-Poisson system modeling the interaction between a plasma and a cylindrical Langmuir probe. In particular, we exhibit a class of radial solutions for which the electrostatic potential is increasing concave with a strong variation in the vicinity of the probe which scales as the inverse of the Debye length. These solutions are proven to exist provided the incoming distributions of particles from the plasma verify the so called generalized Bohm condition of plasma physics. It extends the study [1]. Small perturbations of the radial semi-Maxwellian incoming distributions are then investigated numerically. We notably observe potential barriers that lead to the existence of unpopulated trapped orbits and to the presence of particles that by bypass the probe. Curves of the collected current versus its applied voltage are also presented. This work is the continuity of the previous work [2] where the existence of solutions has been proven.