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Astrophysical black holes embedded in organized magnetic fields: Case of a nonvanishing electric charge

Large scale magnetic fields pervade the cosmic environment where the astrophysical black holes are often embedded and influenced by the mutual interaction. In this contribution we outline the appropriate mathematical framework to describe magnetized black holes within General Relativity and we show several examples how these can be employed in the astrophysical context. In particular, we examine the magnetized black hole metric in terms of an exact solution of electro-vacuum Einstein-Maxwell equations under the influence of a non-vanishing electric charge. New effects emerge: the expulsion of the magnetic flux out of the black-hole horizon depends on the intensity of the imposed magnetic field.