  In the derivation of synchrotron radiation formulas it has been assumed that
the pitch angle of a charge remains constant during the radiation process.
However from the radiation reaction formula, while the component of the
velocity vector perpendicular to the magnetic field reduces in magnitude due to
radiative losses, the parallel component does not undergo any change during
radiation. Therefore there is a change in the ratio of the two components,
implying a change in the pitch angle. We derive the exact formula for the
change in energy of radiating electrons by taking into account the change of
the pitch angle due to radiative losses. From this we derive the characteristic
decay time of synchrotron electrons over which they turn from highly
relativistic into mildly relativistic ones.
