  I study the generation and evolution of magnetic fields in the plasma
surrounding a rotating black hole. Attention is focused on effects of the
gravitomagnetic potential. The gravitomagnetic force appears as battery term in
the generalized Ohm's law. The generated magnetic field should be stronger than
fields generated by the classical Biermann battery. The coupling of the
gravitomagnetic potential with electric fields appears as gravitomagnetic
current in Maxwell's equations. In the magnetohydrodynamic induction equation,
this current re-appears as source term for the poloidal magnetic field, which
can produce closed magnetic structures around an accreting black hole. In
principle, even self-excited axisymmetric dynamo action is possible, which
means that Cowling's anti dynamo theorem does not hold in the Kerr metric.
Finally, the structure of a black hole driven current is studied.
