For a fixed point charge, the electric field line is not change with time.
And we will measure steady E field around the charge.
When the charge starts to oscillate , the Electric field line will change with time and produce electro-magnetic field.
This applet simulate a charged particle under simple harmonic motion and the electric field line is propagated.
Virtual Physics laboratory Original URL
Fu-Kwun Hwang, Dept. Of Physics, National Taiwan Normal University.