Electrostatic potential of charged macroparticles in plasma under conditions of thermal equilibrium

An exact analytical solution of the Poisson-Boltzmann equation (PBE) is found in the case of spherical, axial, or plane geometry, which describes the potential distribution around a charged macroparticle (wire or plane) in plasma under conditions of thermal equilibrium with an arbitrary relationship between the charge densities of macroparticles and plasma. The solution is obtained as a logarithm of the power series in several different forms. A recurrent relation is found for coefficients of the power series. The exact solution of the PBE for the case of plane geometry is already known; its identity to that found in this study is shown in a particular case. In the case of not too high charges of macroparticles, the exact solution is approximated by the solution of linearized PBE; for a solitary particle, this solution corresponds to the DLVO and Debye-Huckel approximations. For high charges, the exact solution deviates from an approximate one (very appreciably too) only in the vicinity of the macroparticle. An approximate solution is used to renormalize the macroparticle charge and to construct a simple model of electrostatic interaction of like-charged particles, which demonstrates the possibility of their attraction. The equilibrium distance of attracting particles (the position of minimum of interaction energy) agrees well with the position of maximum of the experimentally obtained pair correlation function for dust particles in thermal plasma.