The effect of external electric field pulses on the ion atmosphere around linear polyelectrolytes is characterized by Brownian dynamics simulations. The polymer is fixed at the center of a cube and is surrounded by mobile counterions and byions; electrostatic interactions beyond the cube are evaluated using periodic boundary conditions and the Ewald summation technique; hydrodynamic interactions are not included in the simulation. The polymers are constructed in analogy; to DNA double helices with a constant charge spacing of 0.17 nm. Electric fields applied parallel to the long axis of the polymer induce a dipole moment parallel to this axis with time constants around 10 ns (at ion concentrations around 1 mM). The dipole of a polymer with 40 charged residues approaches saturation at field strengths in the range around 50 kV/cm. Parallel to the dipole rise, there is a dissociation of counterions from the polymer reflected by an increase of the root mean square distance [S-c(2))(1/2)of the counterions from the center of the cube. The [S-c(2)](1/2) value increases linearly with the field strength E up to E approximate to 100 kV/cm; this effect and its dependence on E are in close analogy to the "dissociation field effect". The dipole moment at a given field strength and ion concentration does not increase with more than the square of the chain length under the conditions of our present simulations. A relatively small decrease of the dipole moment is observed when the ion concentration is increased.