Magnetic fields of current loops and charged particles around black holes in modified gravity

We investigate the magnetic field produced by a stationary current loop around a static black hole in scalar–tensor–vector gravity (STVG) as a modified gravity (MOG) and its influence on electrically charged test particles. By solving Maxwell’s equations for the retained magnetic dipole mode, we derive exact piecewise radial solutions and assess their accuracy as an approximation to the localized current-loop field through comparison with higher multipoles. The loop is located at the innermost stable circular orbit (ISCO) of a neutral test particle; this orbit shifts outward as the MOG parameter α increases. The magnetic dipole moment grows with α , while the interior field, normalized to its Schwarzschild value, decreases. The retained mode reduces to the Schwarzschild dipole solution for α → 0, while its exterior branch approaches the Newtonian dipole form at large distances. Using the derived vector potential, we study circular motion through the effective potential, specific angular momentum, specific energy, ISCO radius, and ISCO binding-energy proxy. Because of the piecewise field structure, the magnetic coupling parameter ω affects orbital properties in opposite ways inside and outside the loop. For the adopted magnetic orientation, positive ω lowers the interior ISCO energy, while the interior unit-energy boundary occurs at negative coupling. At α = 0.2 and ω = 0.01 , the interior binding-energy proxy is approximately 8.01%, compared with 5.72% in the uncoupled Schwarzschild limit. We also formulate the center-of-mass energy for four charge configurations, retaining the STVG vector interaction and the domain restrictions of the piecewise magnetic field. Its dependence on charge and on the MOG parameter is conditional on the specified incident trajectories.

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