  The radio detection of cosmic rays consists in the estimation of the
properties of a primary cosmic ray by observing the electric field emitted by
the extensive air shower (EAS) created when the primary cosmic ray enters the
atmosphere. This technique is fully operative nowadays and presents a good
degree of maturity. In addition, several projects intend to employ this
technique for the detection of neutrinos. In order for the technique to be
useful, accurate methods for computing the electric field created by a particle
shower in the context of a particular experiment must exist. Although current
ground-based radio experiments lie on the air-soil interface and some planned
experiments on the South Pole envision antennas near the air-ice interface,
most of the analytical approaches and Monte Carlo codes used for calculating
the electric field either do not take into account the effect of the boundary
or calculate the radiation fields only (direct, reflected and transmitted
radiation fields). When the particle shower and the antenna are close to the
boundary, compared to the observation wavelength, the far-field approximation
breaks down, which is the case for the low-frequency EXTASIS experiment, for
instance. We present in this work a new formula for calculating the exact field
emitted by a particle track in two semi-infinite media separated by a planar
boundary. We also explore the validity of the far-field approximation and make
some predictions for EAS using a simple shower model.
