  Binary systems of compact objects with electromagnetic field are modeled by
helically symmetric Einstein-Maxwell spacetimes with charged and magnetized
perfect fluids. Previously derived thermodynamic laws for helically-symmetric
perfect-fluid spacetimes are extended to include the electromagnetic fields,
and electric currents and charges; the first law is written as a relation
between the change in the asymptotic Noether charge $\dl Q$ and the changes in
the area and electric charge of black holes, and in the vorticity, baryon rest
mass, entropy, charge and magnetic flux of the magnetized fluid. Using the
conservation laws of the circulation of magnetized flow found by Bekenstein and
Oron for the ideal magnetohydrodynamic (MHD) fluid, and also for the flow with
zero conducting current, we show that, for nearby equilibria that conserve the
quantities mentioned above, the relation $\dl Q=0$ is satisfied. We also
discuss a formulation for computing numerical solutions of magnetized binary
compact objects in equilibrium with emphasis on a first integral of the ideal
MHD-Euler equation.
