We　investigate　a　system　of　N　Brownian　particles　with　the　Coulomb　interaction　in　any　dimension　d　＞=　2,　and　we　assume　that　the　initial　data　are　independent　and　identically　distributed　with　a　common　density　rho(0)　satisfying　integral(Rd)　rho(0)　ln　rho(0)　dx　＜　infinity　and　rho(0)　is　an　element　of　L2d/d+2　(R-d)　boolean　AND　L-1(R-d,　(1　+　vertical　bar　x　vertical　bar(2))　dx).　We　prove　that　there　exists　a　unique　global　strong　solution　for　this　interacting　partsicle　system　and　there　is　no　collision　among　particles　almost　surely.　For　d　=　2,　we　rigorously　prove　the　propagation　of　chaos　for　this　particle　system　globally　in　time　without　any　cutoff　in　the　following　sense.　When　N　-＞　infinity,　the　empirical　measure　of　the　particle　system　converges　in　law　to　a　probability　measure　and　this　measure　possesses　a　density　which　is　the　unique　weak　solution　to　the　mean-field　Poisson-Nernst-Planck　equation　of　single　component.