The Lorentz force can be approximated only to the electric force when the velocity (v) of the particle less than or equals the light speed. Why the magnetic force is considered to be very small?
The magnetic field is limited by the velocity of the charge, for example the current in a wire (v=some cm/s) or by the distance from the source. An electron radius (r=some 10^-15).
In reality only the EM-waves have a magnetic field equal to the electric field, otherwise the B-field is much smaller then the E-field.
The force acting on charged particle consists of two parts. The one created by electric field is independent form particle velocity, while the Lorentz force is proportional to particle velocity. Thus at v=0 the electric part of force acting on particle is infinitely stronger than magnetic part.