Question #43409

Consider a hypothetical hydrogen-like ion where the binding energy of the electron in its ground state ($n=1$) is exceptionally large, given by $E_b = 0.25 m_e c^2$, where $m_e$ is the rest mass of a free electron and $c$ is the speed of light. The electron (charge $-e$) revolves in a circular Bohr orbit. According to relativistic principles, the binding energy manifests as a mass defect, reducing the effective inertial mass of the bound electron. The ion is placed in a uniform external magnetic field of magnitude $B$. If the magnetic field vector makes an angle of $30^{\circ}$ with the orbital angular momentum vector of the electron, what is the magnitude of the magnetic torque experienced by the electron's orbit? (Assume the Bohr quantization condition $L = n\hbar$ holds, and the gyromagnetic ratio is determined by the electron's effective mass).