  We study the energetics of a black hole-accretion disk system with magnetic
connection: a Keplerian disk is connected to a Kerr black hole by a large-scale
magnetic field going through the transition region. We assume that the magnetic
field is locked to the inner boundary of the disk and corotates with the inner
boundary, the accretion rate is low but the accretion from the disk can still
provide enough amount of cold plasma particles in the transition region so that
the magnetohydrodynamics approximation is valid. Then, the magnetic field is
dynamically important in the transition region and affects the transportation
of energy and angular momentum. Close to the equatorial plane, the motion of
particles is governed by a one-dimensional radial momentum equation, which
contains a fast critical point as the only intrinsic singularity. By finding
solutions that smoothly pass the fast critical point, we find that a system
with a fast rotating black hole and that with a slow rotating black hole behave
very differently. For a black hole with $a > a_{\rm cr}\equiv 0.3594 M$, where
$M$ is the mass, $a$ the specific angular momentum of the black hole, the
spinning energy of the black hole is efficiently extracted by the magnetic
field and transported to the disk, increasing the radiation efficiency of the
disk by many orders of magnitude. For a black hole with $0\leq a < a_{\rm cr}$,
the inner region of the disk is disrupted by the magnetic field and the inner
boundary of the disk moves out to a radius where the Keplerian angular velocity
of the disk is equal to the spinning angular velocity of the black hole (which
is at infinity if the black hole is nonrotating). As a result, the disk may
have an extremely low radiation efficiency if $0\leq a/M \ll 1$.
