  We present a detailed analysis of an astrophysical mechanism that generates
cosmological magnetic fields during the Epoch of Reionization. It is based on
the photoionization of the Intergalactic Medium by the first sources formed in
the Universe. First the induction equation is derived, then the characteristic
length and time scales of the mechanism are identified, and finally numerical
applications are carried out for first stars, primordial galaxies and distant
powerful quasars. In these simple examples, the strength of the generated
magnetic fields varies between the order of $10^{-23}$ G on hundreds of
kiloparsecs to $10^{-19}$ G on hundreds of parsecs in the neutral Intergalactic
Medium between the Str\"omgren spheres of the sources. Thus this mechanism
contributes to the premagnetization of the whole Universe before large scale
structures are in place. It operates with any ionizing source, at any time
during the Epoch of Reionization. Finally, the generated fields possess a
characteristic spatial configuration which may help discriminate these seeds
from those produced by different mechanisms.
