In this talk, we are interested in the connection between the two important physical systems where particles interact through the Coulomb force and are influenced by their self consistent electrostatic field. The Coulomb potential yields the famous Rutherford scattering cross section which is too singular in the angular variable for the Boltzmann equation to be meaningful. With an angular cutoff and a proper scaling as in [R. Alexandre and C. Villani, Ann. Inst. H. Poincar'e Anal. Non Lin'eaire, 21 (2004), pp. 61–95], we prove global well-posedness of VPB in algebraically weighted Sobolev spaces. To this end, we develop a new two-stage energy method and merely rely on the L^2_3/2 -norm (from the dissipation of the linearized Boltzmann operator) to control the nonlinear terms containing the electrostatic field. When the cutoff parameter tends to zero, convergence of VPB to VPL is established with an explicit rate.