  The evolution of magnetic fields frozen to a perfectly conducting plasma
fluid around a Kerr black hole is investigated. We focus on the plunging region
between the black hole horizon and the marginally stable circular orbit in the
equatorial plane. Adopting the kinematic approximation where the dynamical
effects of magnetic fields are ignored, we exactly solve Maxwell's equations
with the assumptions that the geodesic motion of the fluid is stationary and
axisymmetric, the magnetic field has only radial and azimuthal components and
depends only on time and radial coordinates. We show that the stationary state
of the magnetic field in the plunging region is uniquely determined by the
boundary conditions at the marginally stable circular orbit. If the magnetic
field at the marginally stable circular orbit is in a stationary state, the
magnetic field in the plunging region will quickly settle into a stationary
state if it is not so initially, in a time determined by the dynamical time
scale. The radial component of the magnetic field at the marginally stable
circular orbit is more important than the toroidal component in determining the
structure and evolution of the magnetic field in the plunging region. Even if
at the marginally stable circular orbit the toroidal component is zero, in the
plunging region a toroidal component is quickly generated from the radial
component by the shear motion of the fluid. Finally, we show that the dynamical
effects of magnetic fields are unimportant in the plunging region if they are
negligible on the marginally stable circular orbit. This supports the
``no-torque inner boundary condition'' of thin disks, contrary to the claim in
the recent literature.
