We　consider　smooth　periodic　solutions　for　the　Euler-Maxwell　equations,　which　are　a　symmetrizable　hyperbolic　system　of　balance　laws.　We　proved　that　as　the　relaxation　time　tends　to　zero,　the　Euler-Maxwell　system　converges　to　the　drift-diffusion　equations　at　least　locally　in　time.　The　global　existence　of　smooth　solutions　is　established　near　a　constant　state　with　an　asymptotic　stability　property.