  (abridged) The interpretation of Faraday rotation measure maps of AGNs within
galaxy clusters has revealed regions, $\sim 50-100$ kpc, that are populated
with large, $\sim 30 \mu$ G magnetic fields. The magnetic energy of these
coherent regions is $\sim 10^{59-60}$ ergs, and the total magnetic energy over
the whole cluster ($\sim 1$ Mpc across) is expected to be even larger. A
sequence of physical processes that are responsible for the production,
redistribution and dissipation of these magnetic fields is proposed. These
fields are associated with single AGNs within the cluster and therefore with
all galaxies during their AGN phase, simply because only the central
supermassive black holes ($\sim 10^8 M_{\odot}$) have an accessible energy,
$\sim 10^{61}$ ergs. We propose an $\alpha-\Omega$ dynamo process in an
accretion disk. The disk rotation naturally provides a large winding number,
$\sim 10^{11}$ turns, sufficient to make both large gain and large flux. The
helicity of the dynamo can be generated by the differential plume rotation
derived from star-disk collisions. This helicity generation process has been
demonstrated in the laboratory and the dynamo gain was simulated numerically. A
liquid sodium analog of the dynamo is being built. Speculations are that the
back reaction of the saturated dynamo will lead to the formation of a
force-free magnetic helix, which will carry the energy and flux of the dynamo
away from the accretion disk and redistribute the field within the clusters and
galaxy walls. The magnetic reconnection of a small fraction of this energy
logically is the source of the AGN luminosity, and the remainder of the field
energy should then dominate the free energy of the present-day universe.
