The behavior of uniformly accelerated detectors in the Minkowski and R
indler vacua is analyzed when the detector is coupled to a scalar fiel
d during a finite amount of time T. We point out that the logarithmic
ultraviolet divergences reported in the literature are due to the inst
antaneous switching of the detector. We explicitly show this by consid
ering a detector switched on and off continuously. The usual Planckian
spectrum for the excitation probability is recovered in the limit T -
-> infinity.