A heat map of the radial component of the magnetic field B^r at the event horizon for the SANE a_* = −0.5 simulation. Top panel: B^r at a specific instant in time. Bottom panel: B^r averaged over the interval t = [25, 30] × 10³t_g. In both panels, note that B^r changes polarity within each hemisphere. This behavior is not accounted for in Equation (12), leading to an overestimation of Φ_BH.