  A toy model for the Blandford-Znajek mechanism is investigated: a Kerr black
hole with a toroidal electric current residing in a thin disk around the black
hole. The toroidal electric current generates a poloidal magnetic field
threading the black hole and disk. Due to the interaction of the magnetic field
with remote charged particles, the rotation of the black hole and disk induces
an electromotive force, which can power an astrophysical load at remote
distance. The power of the black hole and disk is calculated. It is found that,
for a wide range of parameters specifying the rotation of the black hole and
the distribution of the electric current in the disk, the power of the disk
exceeds the power of the black hole. The torque provided by the black hole and
disk is also calculated. The torque of the disk is comparable to the torque of
the black hole. As the disk loses its angular momentum, the mass of the disk
gradually drifts towards the black hole and gets accreted. Ultimately the power
comes from the gravitational binding energy between the disk and the black
hole, as in the standard theory of accretion disk, instead of the rotational
energy of the black hole. This suggests that the Blandford-Znajek mechanism may
be less efficient in extracting energy from a rotating black hole with a thin
disk. The limitations of our simple model and possible improvements deserved
for future work are also discussed.
