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 understanding 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 the extensive air shower (EAS). We also study the electric field of a special case of the transition radiation mechanism when the EAS particles cross the air-soil boundary. We introduce the sudden death pulse, the direct emission caused by the coherent deceleration of the shower front at the boundary, as a first approximation to the whole electric field for the air-soil transition, and study its properties. We show that at frequencies larger than 20 MHz and distances larger than 100 m, the standard far-field approximation for the horizontal polarizations of the field is always accurate at the 1

**Published by D. García-Fernández, B. Revenu, D. Charrier, R. Dallier, A. Escudié, L. Martin, in Physical Review D, Volume 97, May 2018, Issue 10, id.103010**
https://doi.org/10.1103/PhysRevD.97.103010