# How strong can the magnetic field of a black hole be if it accretes matter?

I've read some papers where black holes have magnetic fields on the order of 10^6 Gauss, of course this compared to the magnetic field of neutrons stars or magnetars is quite small. So I'm curious whether there have been any observed black holes that have magnetic fields that surpass that of magnetars. I know that black holes can only have magnetic fields if they accrete matter, so my question is, is it possible for one to have such high accretion rate that its magnetic field is like a magnetar or even stronger?

Yes, absolutely. The magnetic field of a BH, as you describe, requires there to be active accretion. In particular, the accretion pressure determines how strong of a magnetic field can be confined. The magnetic pressure, which scales as $B^2$ (for magnetic field strength $B$), can at most be equal to the accretion pressure, scaling like $\rho v^2$ (for density of material $\rho$ and in-falling velocity $v$). This is called 'equipartition'.
Typically maximum accretion rates are the given by the Eddington Accretion Rate, so you can solve for the typical maximum magnetic field strengths which are theoretically something like $10^{17} G$ for a roughly solar-mass BH (just slightly larger than that of a NS because of the slightly larger escape velocity---the speed of light).