  The interaction of large scale magnetic fields with the event horizon of
rotating black holes (the Blandford-Znajek [1977] mechanism) forms the basis
for some models of the most relativistic jets. We explore a scenario in which
the central inward "plunging" region of the accretion flow enhances the
trapping of large scale poloidal field on the black hole. The study is carried
out using a fully relativistic treatment in Kerr spacetime, with the focus
being to determine the spin dependence of the Blandford-Znajek effect. We find
that large scale magnetic fields are enhanced on the black hole compared to the
inner accretion flow and that the ease with which this occurs for lower
prograde black hole spin, produces a spin dependence in the Blandford-Znajek
effect that has attractive applications to recent observations. Among these is
the correlation between inferred accretion rate and nuclear jet power observed
by Allen et al. (2006) in X-ray luminous elliptical galaxies. If the black hole
rotation in these elliptical galaxies is in the prograde sense compared with
that of the inner accretion disk, we show that both the absolute value and the
uniformity of the implied jet-production efficiency can be explained by the
flux-trapping model. The basic scenario that emerges from this study is that a
range of intermediate values of black hole spins could be powering these AGN.
We also suggest that the jets in the most energetic radio-galaxies may be
powered by accretion onto {\it retrograde} rapidly-rotating black holes.
