  The standard calculations of the synchrotron emission from charged particles
in magnetic fields does not apply when the energy losses of the particles are
so severe that their energy is appreciably degraded during one Larmor rotation.
In these conditions, the intensity and spectrum of the emitted radiation depend
on the observation time $T_{obs}$: the standard result is recovered only in the
limit $T_{obs}\ll T_{loss}$, where $T_{loss}$ is the time for synchrotron
losses. In this case the effects of the radiation backreaction cannot be
detected by the observer. We calculate the emitted power of the radiation in
the most general case, naturally including both the cases in which the
backreaction is relevant and the standard case, where the usual result is
recovered. Finally we propose several scenarios of astrophysical interest in
which the effects of the backreaction cannot and should not be ignored.
