  The expulsion of axisymmetric magnetic field from the event horizons of
rapidly rotating black holes has been seen as an astrophysically important
effect that may significantly reduce or even nullify the efficiency of the
Blandford-Znajek mechanism of powering the relativistic jets in Active Galactic
Nuclei and Gamma Ray Bursts. However, this Meissner-like effect is seen in
vacuum solutions of black hole electrodynamics whereas the Blandford-Znajek
mechanism is concerned with plasma-filled magnetospheres. In this paper we
argue that conductivity dramatically changes the properties of axisymmetric
electromagnetic solutions -- even for a maximally rotating Kerr black hole the
magnetic field is pulled inside the event horizon. Moreover, the conditions
resulting in outgoing Poynting flux in the Blandford-Znajek mechanism exist not
on the event horizon but everywhere within the black hole ergosphere. Thus, the
Meissner effect is unlikely to be of interest in astrophysics of black holes,
at least not in the way this has been suggested so far. These conclusions are
supported by the results of time-dependent numerical simulations with three
different computer codes. The test problems involve black holes with the
rotation parameter ranging from $a=0.999$ to $a=1$. The pure electrodynamic
simulations deal with the structure of conductive magnetospheres of black holes
placed in a uniform-at-infinity magnetic field (Wald's problem) and the
magnetohydrodynamic simulations are used to study the magnetospheres arising in
the problem of disk accretion.
