  The radio technique for the detection of high energy cosmic rays consists in
measuring the electric field created by the particle showers created inside a
medium by the primary cosmic ray. The electric field is then used to infer the
properties of the primary particle. Nowadays, the radio technique is a
standard, well-established technique. While most current experiments measure
the field at frequencies above $20$ MHz, several experiments have reported a
large emission at low frequencies, below $10$ MHz. The EXTASIS experiment aims
at measuring again and understand this low-frequency electric field. Since at
low frequencies the standard far-field approximation for the calculation of the
electric field does not necessarily hold, in order to comprehend the
low-frequency emission we need to go beyond the far-field approximation. We
present in this work a formula for the electric field created by a particle
track inside a dielectric medium that is valid for all frequencies. We then
implement this formula in the SELFAS Monte Carlo code and calculate the
low-frequency electric field of EAS. We also introduce the sudden death pulse
(SDP), an emission caused by the coherent deceleration of the shower front, and
study its properties.
