Question #43410

A hypothetical 'atomic galvanometer' is constructed using a single Hydrogen atom ($Z=1$). The 'coil' of this galvanometer is the single electron orbiting in the $n^{\text{th}}$ Bohr orbit. The atom is placed in a uniform radial magnetic field $B$ such that the plane of the electron's orbit is always parallel to the magnetic field lines. A microscopic restoring mechanism provides a restoring torque $\tau = -k \theta$, where $\theta$ is the angular deflection of the orbital plane and $k$ is the torsional constant. If the torsional constant is given by $k = \frac{|E_n|}{\pi}$, where $E_n$ is the total mechanical energy of the electron in the $n^{\text{th}}$ orbit, what is the magnitude of the magnetic field $B$ required to produce a steady deflection of $\theta = \pi$ radians?

(Given: $m_e$ is the mass of the electron, $e$ is the elementary charge, $\hbar$ is the reduced Planck's constant, and $\epsilon_0$ is the vacuum permittivity. Assume the magnetic field does not alter the Bohr orbit.)