  The explicit solution for a Kerr-Newman black hole immersed in an external
magnetic field, sometimes called the Melvin-Kerr-Newman black hole, has been
derived by Ernst and Wild in 1976. In this paper, we clarify the first law and
Smarr formula for black holes in a magnetic field. We then define the unique
mass which is integrable and reduces to the Kerr-Newman mass in the absence of
magnetic field. This defines the thermodynamic potentials of the black hole.
Quite strikingly, the mass coincides with the standard Christodoulou-Ruffini
mass of a black hole as a function of the entropy, angular momentum and
electric charge.
