Charged particles and QPO constraints in tidal charge black hole magnetosphere sourced by a current loop
We investigate the dynamics of charged particles in the dipolar electromagnetic field generated by a stationary current loop around a static black hole in the Randall--Sundrum braneworld model. The current loop is located at the innermost stable circular orbit (ISCO) of the tidal charged black hole, and the azimuthal electromagnetic four-potential is obtained as an exact analytical solution of the Maxwell equations in the curved background. Starting from the Hamilton--Jacobi equation, we derive the effective potential for charged particle motion in both the interior and exterior regions of the current loop, and analyze the dependence of the ISCO radius, angular momentum, energy, and orbital frequency on the tidal charge parameter $b$ and the magnetic coupling parameter $ω$. We further compute three fundamental frequencies and show that the dipolar loop field revives a nonzero nodal precession frequency even though the background is static and non-rotating. Using these frequencies we construct five twin-peak QPO models (RP, WD, and ER2--ER4) and perform a Markov chain Monte Carlo (MCMC) analysis to constrain the braneworld parameters $(b,ω)$ from the observed twin-peak QPO frequencies of six black hole sources spanning three mass regimes: GRO~J1655--40, XTE~J1550--564, GRS~1915$+$105, H~1743$+$322, M82~X-1, and Sgr~A$^{*}$. Adopting the epicyclic-resonance ER4 model as the representative framework, we find moderate, non-negligible tidal charges $b\simeq0.3$--$1.0$, while the magnetic coupling remains consistent with zero, $|ω|\lesssim0.05$. Our results therefore place upper bounds on the tidal charge and on the magnetic coupling strength from astrophysical QPO data.