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The existence of the radius of marginal stability means that accretion flows around black holes invariably undergo a transition from a magnetohydrodynamic (MHD) turbulent disklike flow to an inward-plunging flow. We argue that the plunging inflow can greatly enhance the trapping of large-scale magnetic fields on the black hole, and therefore may increase the importance of the Blandford-Znajek (BZ) effect relative to previous estimates that ignore the plunge region. We support this hypothesis by constructing and analyzing a toy model of the dragging and trapping of a large-scale field by a black hole disk, revealing a strong dependence of this effect on the effective magnetic Prandtl number of the MHD turbulent disk. Furthermore, we show that the enhancement of the BZ effect depends on the geometric thickness of the accretion disk. This may be, at least in part, the physical underpinnings of the empirical relation …