A method of avoiding all infrared divergencies in the radiative effects of a constant electromagnetic field is developed. It is used, without the necessity of resumming a nonintegrable series, to calculate the one photon mass operator up to the third order in the external field. The magnetic susceptibility and field dependence of the rest mass of an electron are thereby exhibited to lowest order. The result is then shown in a simple way to provide charged particles with strong short-range forces against the Coulomb field. Particles of unrealistically small mass and charge would carry a large repulsive core, if Coulomb attractive, or a large attractive shell, if Coulomb repulsive.